TOPOLOGICAL METHODS FOR NONLINEAR DIFFER- ENTIAL …vdvorst/lectures-2013.pdf · A.5 Posets, lattices and Boolean algebras 211 A.6 Homology and cohomology 214 B Solutions 225 Bibliography

R . VA N D E R V O R S T

T O P O L O G I C A L M E T H O D SF O R N O N L I N E A R D I F F E R -E N T I A L E Q U AT I O N S

C O U R S E N O T E S 2 0 1 3

Copyright © 2013 R. Vandervorst

vu course notes 2013

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Licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in com-pliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0. Unless required by applicable law or agreed to in writing, software distributed under theLicense is distributed on an “as is” basis, without warranties or conditions of any kind, eitherexpress or implied. See the License for the specific language governing permissions and limitations underthe License.

First printing, November 2013

Contents

1 Smooth degree theory 13

1.1 Notation 13

1.2 The C1-mapping degree 16

1.3 The general homotopy principle 22

1.4 Integral representations 24

1.5 Proper mappings 31

1.6 Mappings between smooth manifolds and the mapping degree 37

1.7 Applications 38

1.8 Notes 43

2 The Brouwer degree 47

2.1 The Brouwer degree 47

2.2 Properties and axioms for the Brouwer degree 49

2.3 Boundary dependence of the degree 55

2.4 The Brouwer fixed point theorem 58

2.5 The homological defintion of the Brouwer degree 60

2.6 The Lefschetz Fixed Point Theorem 60

3 Elementary homotopy theory 61

3.1 Homotopy types and Hopf’s Theorem 62

3.2 The extension problem for mappings on a ball 66

3.3 The general extension problem 68

4

3.4 Framed cobordisms 71

3.5 Pontryagin manifolds 73

3.6 Framed cobordism classes and homotopy types 79

3.7 Algebraic structures and obstruction theory 83

4 The Leray-Schauder degree 85

4.1 Notation 86

4.2 Compact and finite rank maps 89

4.3 Definition of the Leray-Schauder degree 91

4.4 Properties of the Leray-Schauder degree 94

4.5 Compact homotopies 96

4.6 Stable cohomotopy 98

4.7 Semi-linear elliptic equations and a priori estimates 98

4.8 Positive solutions of elliptic PDE’s 101

5 Dynamical Systems 103

5.1 Preliminaries on Dynamical systems 103

5.2 Attractors and repellers 108

5.3 Attractor-repeller pairs 112

5.4 Some properties of attractors and repellers 114

6 Decompositions of Dynamical Systems 121

6.1 Morse decompositions 121

6.2 Attractor lattices and Morse decompositions 124

6.3 A Dynamical version of Birkhoff’s Representation Theorem 133

6.4 Morse tilings 135

7 The Conley Index 139

7.1 Blocks 139

7.2 Wazewski’s Principle 142

5

7.3 The Homological Conley Index for Morse sets 145

7.4 The Morse relations 148

7.5 Connecting orbits 151

7.6 Isolated invariant sets and the Conley index 153

7.7 Continuation of the Conley index 155

8 Discrete Parabolic Dynamics 157

8.1 Parabolic recurrence relations 157

8.2 Parabolic flows and braids 159

8.3 Isolating blocks and braid classes 162

8.4 Morse decompositions, Morse relations and connecting orbits 166

8.5 Stabilization and global invariants 166

8.6 Scalar parabolic PDE’s 166

9 Morse theory 167

9.1 Gradient-like flows on compact spaces 167

9.2 Palais-Smale functions and compactness 169

9.3 The Morse relations for critical points 171

9.4 The deformation lemma 173

9.5 Homotopy types and the Morse index 175

9.6 Other homology invariants and the Morse inequalities 176

10 Morse Theory for elliptic equations 177

10.1 Variational principles and critical points 177

10.2 Solutions via Morse Theory 180

10.3 Multiplicity results for critical points 183

10.4 Functions lacking compactness 184

11 Strongly indefinite systems 189

11.1 Strongly indefinite elliptic systems 189

11.2 The Weinstein Conjecture for compact contact type hypersurfaces 189

6

A Appendix 191

A.1 Differentiable mappings 191

A.2 Basic Nemytskii maps 193

A.3 Sobolev Spaces 198

A.4 Partitions of unity 207

A.5 Posets, lattices and Boolean algebras 211

A.6 Homology and cohomology 214

B Solutions 225

Bibliography 233

List of Figures

1.1 The graphs f and g. 13

1.2 A bounded open set, or domain W with boundary ∂W. 13

1.3 The unit ball in the norms | · |1 and | · |• respectively. 14

1.4 The pre-image of small neighborhood Be

(p) is the union of small neigh-borhoods N

e

(xj) ⇢ W diffeomorphic to Be

(p). 16

1.5 Two different orientations with respect to the points p1 and p2. 17

1.6 The opposite points connected by a curve have the same sign of theJacobian, and the points at t = 0, or t = 1 connected by a curvehave opposite sign of the Jacobian. The same holds for any regularsection. 19

1.7 The disc is folded to the right half plane and the boundary of the im-age is not given by f (∂D2). 20

1.8 Also in the case of general domains W ⇢ [0, 1]⇥Rn the oppositepoints connected by a curve have the same sign of the Jacobian. 23

1.9 The support of the weight function w in a connected component Dof Rn \ f (∂W). 26

1.10 The covering of supp(w) [ supp(w0) by Kn ⇢ D. 27

1.11 The image f ([�3, 3]) locally covers the regular value p = 1 andno covering at p = �20. 33

1.12 Two linking embeddings of S1. One circle intersects any ‘filing’ ofthe other circle, which yields a non-zero linking number. 41

2.1 For small perturbations g of f , the point p is not contained in g(∂W)

[left]. The same holds for homotopies ht. The second figure showsf (W) and ht(W) for t 2 [0, 1] [right]. 48

2.2 The intersection mappings l± : B1(0)! ∂B1(0). 59

3.1 The zeroes of g are contained in U ⇢ W and are connected by anon-intersecting path g [left]. The path g ⇢ U can be covered bya union of open balls V ⇢ U [right]. 63

3.2 Deformation of ∂W via the normalized gradient flow on H. 64

3.3 The map j = f |∂W maps to the unit circle in R2\{0}, and can be

viewed as constant vector field along the torus. Clearly the homo-topy type of j is non-trivial. 68

8

3.4 The semi-circle is a framed cobordism between N and N0 = ?. Theframing given by rF. 82

3.5 Connecting the m positively oriented zeroes across and the connectthe remaining m0 positively and negatively oriented zeroes of g. 82

4.1 Zeroes for p-values restricted to the subspace F. 92

5.1 (a) Phase portrait for Example 5.4.10. The set of bounded orbits isthe unit square S whose interior is filled with connecting orbits frome4 to e1. (b) Acyclic directed graph representation of the dynamicsin X. 118

5.2 Posets (Att,✓) and (Rep,✓) for Example 5.4.10. 118

6.1 Coarsening of the partial order by identifying two consecutive ele-ments by a double line (a), the new partial order (b), and the Morsesets in S for M2 (c). 122

6.2 The poset P = {1, 2, 3, 4} and the lattice of attracting interval O(P).The join-irreducible elements are solid dots. 124

7.1 The Morse set S is given by the attractors A0 ( A and correspondsto the Morse decomposition A0 < S < A⇤. This yields the Morsedecomposition M(X). In the lattice of attractors [right] A0M and A0M0

are the attractors given by A0M = n(A0, M) and A0M0 = n(A0, M0). 152

8.1 Linking sequences [left] and non-transverse intersections, or tangen-cies [right]. 159

8.2 The behavior of parabolic flows with respect to intersections and braidclasses. 161

8.3 A improper class [left] and a proper class [right]. 162

8.4 The braid of Example 1 [left] and the associated configuration spacewith parabolic flow [middle]. On the right is an expanded view ofW2 rel y where the fixed points of the flow correspond to the fourfixed strands in the skeleton v. The braid classes adjacent to thesefixed points are not proper. 164

8.5 The braid of Example 8.3.4 and the configuration space B. 165

8.6 The lifted skeleton of Example 1 with one free strand. 165

A.1 The functions f1 and f2. 208

A.2 A locally finite covering [left], and a refinement [right]. 209

A.3 Constructing a nested covering U from a covering with balls B [left],and a locally finite covering obtained from the previous covering [right].

210

A.4 The different stages of constructing the sets Wp colored with differ-ent shades of grey. 211

List of Tables

Preface

These notes where composed of lectures given at the VU Univer-sity in 2006, 2007, 2009 and 2013. The lectures notes use the de-sign of Edward Tufte’s books1 and the use of the tufte-book and 1 Edward R. Tufte. The Visual Display of

Quantitative Information. Graphics Press,Cheshire, Connecticut, 2001; Edward R.Tufte. Envisioning Information. GraphicsPress, Cheshire, Connecticut, 1990;Edward R. Tufte. Visual Explanations.Graphics Press, Cheshire, Connecticut,1997; and Edward R. Tufte. BeautifulEvidence. Graphics Press, LLC, firstedition, May 2006

tufte-handout document classes. The chapters on Conley theory arebased on lecture notes with W.D. Kalies and K. Mischaikow.

1Smooth degree theory

The mapping degree is a topological tool that can be used to findzeroes of functions from Rn to Rn. Consider the functions f (x, l) =

x4� 2x2 + 1� l, and g(x, l) = x3� x� l. In both cases, for l = 0, thefunctions have only non-degenerate zeroes. Assign either ±1 to eachroot depending on the sign of derivative of the function at a zero anddefine the degree to be the sum of the signs. For f this number isequal to zero and for g it is equal to 1. By varying the parameter l,the degree can be computed in most cases, i.e. when the zeroes areall non-degenerate. Notice that for f the answer is always 0 and for gthe answer is always 1, see Fig. 1.1.

Figure 1.1: The graphs f and g.

In the latter case there is at least one zero, while f does not needto have zeroes at all. In Section 1.2 this idea is formalized for C1-mappings f : Rn ! Rn.

1.1 Notation

1.1.1 Let W ⇢ Rn be a bounded,1 open subset of Rn, which be 1 Regard Rn with the standard Eu-clidean metric. See 1.1.5 for moredetails.

will referred to as a bounded domain. Its closure is denoted by Wand the boundary is defined as ∂W = W\W. The closure W is acompact set. Points x 2 W are represented in coordinates as follows;x = (x1, · · · , xn). Super-indices will be used to label points in Rn, seeFig. 1.2.

Figure 1.2: A bounded open set, ordomain W with boundary ∂W.

1.1.2 The class of functions f : W ⇢ Rn ! Rn that are continuouson W is denoted by C0(W; Rn), or C0(W) for short. Functions that arecontinuous on W are denoted by C0(W; Rn). If f : W ⇢ Rn ! Rn

is uniformly continuous, then f can be extended to a continuousfunction on W. Therefore C0(W) ⇢ C0(W), which is also referred to asthe subspace of uniformly continuous functions on W. A function fis said to be k-times continuously differentiable on W if f and all itsderivatives up to order k are continuous on W. This class is denoted

14 topological methods for nonlinear differential equations

by Ck(W; Rn). A function f is k-times continuously differentiable onW if f and all derivatives up to order k are uniformly continuous,and thus extend continuously to W. The class k-times continuouslydifferentiable on W is denoted by Ck(W; Rn).

In order to extend degree theory to unbounded domains an ap-propriate class of admissible mappings is needed. Let W ⇢ Rn be anunbounded domain. A continuous mapping f : W ⇢ Rn ! Rn issaid to be proper if f�1(K) = {x 2 W | f (x) 2 K} is compact for anycompact set K ⇢ Rn. Proper mappings are closed, i.e. a mapping f iscalled a closed mapping if it maps closed sets A ⇢ W to closed setsf (A) ⇢ Rn.1.1.3 Exercise. Show that a proper mapping is a closed mapping.

If W is a bounded domain, then f : W ⇢ Rn ! Rn is a propermapping since f�1(K) ⇢ W is a closed subset and thus compact.Proper mappings on non-compact domains are therefore a naturalextension of continuous mappings on compact domains.

1.1.4 The Jacobian of f 2 C1(W) at a point x 2 W is defined byJ f (x) = det

�

f 0(x)�

, where f 0(x) is the n⇥ n matrix of partial deriva-tives, i.e. if f = ( f1, · · · , fn), then

f 0(x) =

0

B

B

@

∂ f1∂x1

· · · ∂ f1∂xn

.... . .

...∂ fn∂x1

· · · ∂ fn∂xn

1

C

C

A

.

1.1.5 It is sometimes useful to measure distance in Rn using the so-called p-norms, which are defined as follows; for x = (x1, · · · , xn) 2Rn,

|x|p =⇣

Âi|xi|p

⌘1/p, 1 p < •, and |x|• = max

i{|xi|}.

The latter is also referred to as the supremum norm. Since Rn is finitedimensional all these norms are equivalent.

Figure 1.3: The unit ball in the norms| · |1 and | · |• respectively.

1.1.6 Exercise. Prove that the p-norms defined above are all equivalent normson Rn.

In the case that no subscript is given, | · | indicates the 2-norm, orEuclidean norm. The 2-norm can be associated to an inner product.For x, y 2 Rn, define hx, yi = Âi xiyi, and |x|2 = hx, xi. The normsgiven above can also be used to define the notion of distance. For anytwo points x, y 2 Rn define the distance to be dp(x, y) = |x � y|p.The distance is also referred to as a metric and Rn is a metric space.The distance between a set W and a point x is defined by dp(x, W) =

smooth degree theory 15

infy2W dp(x, y), and more generally, the distance between two sets W,and W0 is then given by dp(W0, W) = infx2W0 dp(x, W). The distanceis symmetric in W and W0. If no subscript is indicated, d(x, y) is thedistance associated to the standard Euclidean norm. An open ball inRn of radius r and center x is denoted by Br(x) = {y 2 Rn | |x� y| <r}.

1.1.7 The linear spaces of Ck-functions can be regarded as a normedspace. For k = 0 the norm is given by

k f kC0 = maxx2W

| f (x)|•.

and for functions f 2 C1 the norm k f kC1 = k f kC0 +max1in k∂xi f kC0 ,where ∂xi f denotes the partial derivative with respect to the ith coor-dinate. The norms for k � 2 are defined similarly by considering thehigher derivatives in the supremum norm. On these normed linearspaces the norm can be used to define a distance, or metric as ex-plained above for Rn. Since W is compact the spaces Ck(W), equippedwith the norms described above, are complete and are therefore Ba-nach spaces. For function f 2 Ck(W) the support is defined as theclosed set

supp( f ) = {x 2 W | f (x) 6= 0}.

Functions whose support is contained in W are denoted by Ck0(W) =

{ f 2 Ck(W) | supp( f ) ⇢ W} and form a linear subspace of Ck(W).As matter of fact Ck

0(W) is a closed subspace and therefore again aBanach space with respect to the norm of Ck(W).

1.1.8 A value p = f (x) is called a regular value of f if J f (x) 6= 0 forall x 2 f�1(p) = {y 2 W | f (y) = p}, and p is called a critical valueif J f (x) = 0 for some x 2 f�1(p). The points x 2 f�1(p) for whichJ f (x) 6= 0 are called regular points, and those for which J f (x) = 0 arecalled critical points. The set of all critical points of f , i.e. all pointsx 2 W for which J f (x) = 0, is denoted by Crit f (W), or Crit f for short.

1.1.9 Remark. The notions of regular and singular values can alsobe defined for functions f : Rn ! Rm, n, m � 1. In that casef 0(x) replaces the role of the Jacobian, i.e. p is regular if f 0(x) isof maximal rank for all x 2 f�1(p) and singular if f 0(x) is not ofmaximal rank for some x 2 f�1(p). A regular point is therefore apoint for which f 0(x) is of maximal rank and a singular point is apoint for which f 0(x) is not of maximal rank. In the special case offunctions f : Rn ! R, the critical points are those points for whichf 0(x) = 0.


1.2 The C1-mapping degree

The definition of the C1-mapping degree is carried out in two steps.The first step is to define the degree in the generic case — regularvalues —, and secondly the extension to singular values, using thehomotopy invariance of the degree. In Section 1.4 a direct definitionof the C1-mapping degree is given via an integral representationthat does not require a distinction between regular and singularvalues. Because both approaches are common these two equivalentdefinitions are explained here.

Regular values

1.2.1 Let f : W ⇢ Rn ! Rn be a differentiable mapping, i.e. f 2C1(W), and let p 2 Rn be a regular value, i.e. f�1(p) \ Crit f = ?.Since W is compact, and J f (x) is non-zero for all x 2 f�1(p), theInverse Function Theorem implies that f�1(p) is a finite set.

1.2.2 Exercise. Let p 62 f (∂W). Show, using the Inverse Function Theorem, thatf�1(p) ⇢ W consists of finitely many isolated points whenever p is a regularvalue.

1.2.3 Definition. For a regular value p 62 f (∂W), define the C1-mapping degree by

deg( f , W, p) := Âx2 f�1(p)

sign⇣

J f (x)⌘

,

which takes values in Z.

1.2.4 Exercise. Explain that when p 2 f (∂W) the degree is not stable undersmall perturbations.

1.2.5 Exercise. (Local continuity/stability of the degree in p) Show, that if pis regular with p 62 f (∂W), there exists an e > 0 such that all p0 2 B

e

(p) areregular values for f . Use this to prove that deg( f , W, p0) = deg( f , W, p) for allp0 2 B

e

(p) with e > 0 is small enough so that p0 62 f (∂W) for all p0 2 Be

(p).

1.2.6 Exercise. (Local continuity of the degree in f ) Let p be a regular valuefor f with p 62 f (∂W). Show that there exists an e > 0 such that all for g 2C1(W), with k f � gkC1 < e, p is a regular value for g. Use this to prove thatdeg( f , W, p) = deg(g, W, p) for all k f � gkC1 < e with 0 < e 1

2 d(p, f (∂W))

small enough so that p is a regular value for all such g.

Figure 1.4: The pre-image of smallneighborhood B

e

(p) is the union ofsmall neighborhoods N

e

(xj) ⇢ Wdiffeomorphic to B

e

(p).

Definition 1.2.3 of degree was used in the prelude to this chapterand gives a convenient way of computing the mapping degree inthe case of regular values p. The condition p 62 f (∂W) is an isola-tion condition and makes W a set that strictly contains solutions of


f (x) = p on W, i.e. W isolates the solution set f�1(p). This isolationrequirement in the definition of degree equips the mapping degreewith various robustness properties, see Exercises 1.2.4 - 1.2.6.

1.2.7 The definition yields a number of crucial properties. For theidentity map f = id the degree is easily computed, i.e. if p 2 W, then

deg(id, W, p) = 1, (1.1)

and for p 62 W, deg(id, W, p) = 0. Another important propertythat follows from the definition is that the equations f (x) = p andf (x)� p = 0 have the same solution set and J f = J f�p. Therefore

deg( f , W, p) = deg( f � p, W, 0). (1.2)

If W1, W2 ⇢ W are two disjoint, open subsets, such that p 62f�

W\(W1 [W2)�

, then2

2 Louis Nirenberg. Topics in nonlinearfunctional analysis, volume 6 of CourantLecture Notes in Mathematics. NewYork University Courant Institute ofMathematical Sciences, New York, 2001.Chapter 6 by E. Zehnder, Notes by R.A. Artino, Revised reprint of the 1974

original

deg( f , W, p) = deg( f , W1, p) + deg( f , W2, p). (1.3)

Indeed, since p 62 f�

W\(W1 [ W2)�

, f�1(p) ⇢ W1 [ W2. FromDefinition 1.2.3 and the fact that W1 \W2 = ?, Equation (1.3) follows.

1.2.8 Example. Consider the mapping f : D2 ⇢ R2 ! R2 defined byf (x1, x2) = (2x2

1 � 1, 2x1x2). This mapping gives a 2-fold coveringof the disc. The boundary ∂D2 = S1 winds around the origin twiceunder the image of the map f . For the value (0, 0), the pre-imageconsists of the points x1 = (� 1

2p

2, 0) and x2 = ( 12p

2, 0), and

f 0(x1) =

�2p

2 00 �

p2

!

, f 0(x2) =

2p

2 00

p2

!

.

Therefore (0, 0) is a regular value for f , and since J f (x1) = J f (x2) =

+1, the degree is given by deg( f , D2, 0) = 2.

1.2.9 Example. Consider the mapping f (x1, x2) = (2x1x2, x1) onW = D2, and the image points p1 = (0,�1/2), and p2 = (0, 1/2).Then, as in Example 1.2.8, deg( f , D2, p1) = �deg( f , D2, p2) = 1. Thepositive degree corresponds to a counter clockwise rotation aroundp1, and the negative degree corresponds to a clockwise rotationaround p2, see Figure 1.5.

Figure 1.5: Two different orientationswith respect to the points p1 and p2.

1.2.10 For regular values p the degree is a count of the elements inf�1(p) with orientation, i.e. a point xj 2 f�1(p) is counted witheither +1 or �1 whenever f is locally orientation preserving ororientation reversing respectively. The degree counts how manytimes the image f (W) covers p counted with multiplicity. This is apurely local but stable property for regular values, see also Section


1.5. Of course, whether p is a regular value of a given function f ornot is not always straightforward to decide. Sard’s Theorem (seeAppendix A.1) claims that a value p is regular with ‘probability’equal to 1. This fact can be used to extend the definition of degree toarbitrary values p (see Chapter 2).

1.2.11 Remark. A rougher version of degree is the so-called mod-2degree and is defined as follows; deg2( f , W, p) = #

⇣

f�1(p)⌘

mod 2.This version of the mapping degree contains less information thanthe degree defined in Definition (1.2.3), but will be of importance formappings between non-orientable spaces.3 3 John W. Milnor. Topology from the

differentiable viewpoint. PrincetonLandmarks in Mathematics. PrincetonUniversity Press, Princeton, NJ, 1997.Based on notes by David W. Weaver,Revised reprint of the 1965 original

Homotopy invariance

1.2.12 A crucial property of the C1-mapping degree is the homotopyinvariance with respect to f . Large perturbations f which do notdestroy the isolation along a homotopy leave the degree unchanged.

1.2.13 Proposition. Let t 7! ft, t 2 [0, 1] be a continuous path inC1(W), with p 62 ft(∂W) for all t 2 [0, 1] and let p be a regular valuefor both f0 and f1. Then deg( f0, W, p) = deg( f1, W, p).

Proof: Let F(t, ·) = ft and consider the equation F(t, x) = p. Byassumption p is a regular value for both f0 and f1. From TheoremA.1.3 it follows that F can be approximated arbitrarily close in C0 bya function eF 2 C•([0, 1]⇥W) such that eft = eF(t, ·) is arbitrary closeto ft in C1, uniformly in t 2 [0, 1]. By Sard’s Theorem (see TheoremA.1.5) we can choose a value p0 arbitrary close to p which is regularfor both F and eF. By the local stability of the degree (Exercise 1.2.5)there exists an e > 0 such that deg( f0, W, p0) = deg( f0, W, p) anddeg( f1, W, p0) = deg( f1, W, p) for all p0 2 B

e

(p). Using the localcontinuity of the degree, see Exercise 1.2.6, there exists a d > 0 suchthat deg( ef0, W, p0) = deg( f0, W, p) and deg( ef1, W, p0) = deg( f1, W, p)for all p0 2 B

e

(p), and all maxt2[0,1] k eft � ftkC1 < d. For a regularvalue p0 the solution set eF�1(p0) of the equation

eF(t, x) = p0,

is a smooth 1-dimensional manifold with boundary given by ∂

eF�1(p0) =⇥

ef�10 (p0) ⇥ {0}

⇤

[⇥

ef�11 (p0) ⇥ {1}

⇤

(see Appendix A.1). Since p 62ft(∂W) it holds that p0 62 eft(∂W) and consequently eF�1(p0) ⇢[0, 1]⇥W. Therefore, the 1-dimensional components diffeomorphicto [0, 1] are curves connecting elements in ∂

eF�1(p0) and componentsdiffeomorphic to S1 are contained in (0, 1)⇥W, since p0 is a regularvalue for both ef0 and ef1. It is worth mentioning that by the Transver-sality Theorem (see Appendix A.1) p0 is a regular value for eft for


almost every t 2 [0, 1]. The manifold eF�1(p0) can be given a canonicalorientation as follows. Since p0 is regular the matrix

eF0(t, x) =

0

B

B

@

∂teF1 ∂x1eF1 · · · ∂xn

eF1...

.... . .

...∂teFn ∂x1

eFn · · · ∂xneFn

1

C

C

A

,

has maximal rank. The rows define the column vectors x

j, j =

1, · · · , n and the tangent space TeF�1(p0) is spanned by X = X(t, x)The vector X satisfies F0(t, x)X = 0 and X ? x

j. Consider the (n + 1)-form dx = dt ^ dx1 ^ · · · ^ dxn. Then the 1-form a = dx(x1, · · · , x

n)

defines an orientation on eF�1(p0). The vector X(t, x) can be identifiedwith a = X0dt + X1dx1 + · · · Xndxn, and a(X) = |X|2. The componentX0 in the t-direction is given by

X0(t, x) = a(e0) = Jeft(x).

Figure 1.6: The opposite points con-nected by a curve have the same signof the Jacobian, and the points at t = 0,or t = 1 connected by a curve haveopposite sign of the Jacobian. The sameholds for any regular section.

If two points x, x0 2 ef�10 (p0) are connected by a curve in eF�1(p0),

then Jef0(x) and J

ef0(x0) have opposite signs by the induced orientation

a, and do not contribute to the sum Âx2 ef�10

sign�

Jef0(x)

�

. For two

points xj 2 ef�10 (p0) and xj0 2 ef�1

1 (p0) connected by a curve ineF�1(p0), it holds that J

ef0(x) and J

ef1(x0) have the same sign by the

induced orientation a. Since all points in ∂

eF�1(p0) are connected, thecontributing terms in ef�1

0 and ef�11 are in one-to-one correspondence

and the Jacobians have the same signs. It immediately follows nowthat

Âx2 ef�1

0

sign⇣

Jef0(x)

⌘

= Âx02 ef�1

1

sign⇣

Jef1(x0)

⌘

,

which proves the homotopy property.

1.2.14 Remark. If the assumption p 62 ft(∂W), for all t 2 [0, 1], isremoved the above proof may fail at a number of points. Most impor-tant to mention in this context is that without the isolation propertythe points in ∂

eF�1(p0) are not necessarily connected by a curve andthe contributing terms in ef�1

0 and ef�11 are not necessarily in one-to-

one correspondence, see also Exercise 1.2.4.

1.2.15 Exercise. Prove the homotopy property of the mapping degree via thecontinuity of the degree established in Exercise 1.2.6.

1.2.16 Let D ⇢ Rn\ f (∂W) be any connected component,4 then the 4 Open subsets of Rn are connected ifand only if they are path-connected.degree deg( f , W, p) is independent of p 2 D. This easily follows from

the homotopy principle.


1.2.17 Proposition. For any curve t 7! pt 2 D, t 2 [0, 1], with p0 andp1 regular values, it holds that deg( f , W, p0) = deg( f , W, p1).

Proof: From Equation (1.2) it follows that deg( f , W, p0) = deg( f �p0, W, 0), and deg( f , W, p1) = deg( f � p0, W, 0). It holds that pt 2 D ifand only if pt 62 f (∂W). The homotopy ft = f � pt therefore satisfiesthe requirements of Proposition 1.2.13, and

deg( f , W, p0) = deg( f � p0, W, 0) = deg( f � p1, W, 0) = deg( f , W, p1),

which proves the statement.

1.2.18 Example. Consider the mapping f (x, y) = (x2, y) on the thestandard 2-dics D

2 in the plane. The image of D2 under f is the

‘folded pancake’ f (D2) = {p = (p1, p2) 2 R2 | p1 + p2

2 = 1, p1 � 0}.The image of the boundary S1 = ∂D2 is homeomorphic to a semi-circle and R2\ f (D2) is connected. Note that f (∂D2) 6= ∂ f (D2

)! Bythe homotopy invariance the degree can be evaluated by choosingany p 2 R2\ f (∂D2). Since D

2 is compact, so is the image. Wecan therefore choose a value p1 2 R2\ f (∂D2) which does not liein f (D2

). This implies that deg( f , D2, p) = 0. If we choose p2 =

(1/4, 0), then f�1(p2) = {(±1/2, 0)}, which gives a positive and anegative determinant. The sum is zero which confirms the previouscalculation.

Figure 1.7: The disc is folded to theright half plane and the boundary ofthe image is not given by f (∂D2).

If we choose a path t 7! pt connecting the regular values p1

and p2 and which lies in R2\ f (∂D2), then pt crosses the boundary∂ f (D2

) in the vertical. However, pt 62 f (∂D2) for all t 2 [0, 1] andf�1(pt) 2 D2 for all t 2 [0, 1]. The values in f (D2

) on the verticalare necessarily singular. This again shows that the boundary of theimage should not be considered as a restriction on p. In the nextsubsection we show that the degree is defined for all p in R2\ f (∂D2).

The previous example yields the following property of the map-ping degree.

1.2.19 Proposition. Suppose that Rn\ f (∂W) is connected, then for anyregular value p 2 Rn\ f (∂W) it holds that deg( f , W, p) = 0.

Proof: See Example 1.2.18.

The degree for arbitrary values

1.2.20 The homotopy invariance established in the previous subsec-tion can be used to extend the definition of the C1-mapping degree toarbitrary values p 2 D, for any connected component D of Rn\ f (∂W).


1.2.21 Definition. Let p 2 D, with D a connected component ofRn\ f (∂W). Then

deg( f , W, p) := deg( f , W, p0),

for any regular value p0 2 D. The justifies the notation deg( f , W, p) =deg( f , W, D).

By Sard’s Theorem (Appendix A.1, Theorem A.1.5) the regularvalues in D lie dense in D. By Proposition 1.2.17 the choice of regularvalue p0 does not matter and therefore the extension of the degreeas given by Definition 1.2.21 is well-defined. The properties of thegeneric degree listed in Equations (1.1) - (1.3) and Proposition 1.2.13

also hold for the general C1-mapping degree and are the fundamen-tal axioms that define a degree theory, see Section 2.2.

1.2.22 Theorem. The degree function deg( f , W, p) in Definition 1.2.21

satisfies the following axioms:5 5 N. G. Lloyd. Degree theory. CambridgeUniversity Press, Cambridge, 1978.Cambridge Tracts in Mathematics, No.73

(A1) if p 2 W, then deg(id, W, p) = 1;

(A2) for W1, W2 ⇢ W, disjoint open subsets of W and p 62 f�

W\(W1 [W2)

�

, it holds that deg( f , W, p) = deg( f , W1, p) + deg( f , W2, p);

(A3) for any continuous path t 7! ft, ft 2 C1(W), with p 62 ft(∂W), itholds that deg( ft, W, p) is independent of t 2 [0, 1];

(A4) deg( f , W, p) = deg( f � p, W, 0).

The application ( f , W, p) 7! deg( f , W, p) is called a C1-degree theory.

Proof: Axiom (A1) follows immediately from Equation (1.1). Asfor Axiom (A2), by assumption, f�1(p) ⇢ W1 [ W2 and thereforef�1(p0) ⇢ W1 [W2 for any regular value p0 sufficiently close to p.Consequently,

deg( f , W, p) = deg( f , W, p0) = deg( f , W1, p0) + deg( f , W2, p0)

= deg( f , W1, p) + deg( f , W2, p).

Choose a value p0 which is regular for both f0 and f1. If p0 is cho-sen sufficiently close to p, then p0 62 ft(∂W), and thus by Proposition1.2.13 and Definition 1.2.21

deg( f0, W, p) = deg( f0, W, p0) = deg( f1, W, p0) = deg( f1, W, p).

By considering the homotopy t 7! ft0t it follows that deg( f0, W, p) =deg( ft0 , W, p), for any t0 2 [0, 1], which proves Axiom (A3).

Finally, let p0 be a regular value sufficiently close to p, then byEquation (1.3), deg( f , W, p) = deg( f , W, p0) = deg( f � p0, W, 0).


Consider the homotopy ft = (1� t)( f � p) + t( f � p0) = f � (1�t)p� tp0. Since p0 is close to p, the line-segment {(1� t)p + tp0}t2[0,1]does not intersect f (∂W), and herefore 0 62 ft(∂W). From Axiom (A3)it then follows that

deg( f , W, p) = deg( f , W, p0) = deg( f � p0, W, 0) = deg( f � p, W, 0),

which proves Axiom (A4), and thereby completing the proof.

1.3 The general homotopy principle

The homotopy invariance established in the previous section allowsfor deformations in both f and p. Using the axioms of a degree the-ory one can prove that the domain W can also be varied. In Section2.2 the homotopy principle will be derived from the axioms. In thissection a direct proof using the definition will be given.

Variations in domains

1.3.1 Let W ⇢ Rn ⇥ [0, 1] be bounded and relatively open subset ofRn ⇥ [0, 1]. Define the t-slices by

Wt = {x | (x, t) 2 W}, t 2 [0, 1]

and their boundaries by (∂W)t = {x | (x, t) 2 ∂W} =�

W�

t \Wt.Note that Wt ⇢

�

W�

t which is essential for the definition of (∂W)t.

1.3.2 Definition. Two triples ( f , W f , p) and (g, Wg, q) are said to behomotopic, or cobordant, if there exists a bounded and relativelyopen subset W ⇢ Rn ⇥ [0, 1] and a continuous function F : W ! Rn,with ft = F(·, t) 2 C1(Wt), such that

(i) f0 = f , and f1 = g;

(ii) W0 = W f , and W1 = Wg;

(iii) there exists a continuous path t 7! pt, such that pt 62 ft((∂W)t)

for all t 2 [0, 1], and p0 = p and p1 = q.

Notation ( f , W f , p) ⇠ (g, Wg, q). Triples ( f , W f , Df ) and (g, Wg, Dg),for which the above requirements are met, are also called homotopic.


1.3.3 Theorem. Let ( f , W f , p) and (g, Wg, q) be homotopic triples, then

deg( f , W f , p) = deg(g, Wg, q).

In particular, deg( ft, Wt, pt) is constant in t 2 [0, 1], where ft is ahomotopy as defined in Definition 1.3.2.

Proof: From the definition of the degree we can choose valuesp0 and q0 which are regular values of f and g respectively. Thevalues p0 and q0 can be chosen arbitrary close to p and q. Thendeg( f , W f , p) = deg( f , W f , p0) and deg(g, Wg, q) = deg(g, Wg, q0).Let p0t be a continuous path in W connecting p0 and q0 and which liesin an e-neighborhood of the path pt. Therefore, if e is chosen smallenough, p0t 62 ft((∂W)t), for all t 2 [0, 1]. By Axiom (A4) it holds that

deg( ft, Wt, p0t) = deg( ft � p0t, Wt, 0),

for all t 2 [0, 1]. Now consider the equation

G(t, x) = F(t, x)� p0t.

Figure 1.8: Also in the case of generaldomains W ⇢ [0, 1]⇥Rn the oppositepoints connected by a curve have thesame sign of the Jacobian.

The proof now follows along the same lines as the proof of Propo-sition 1.2.13. The only difference is the domain W. By assumption,the solution set G�1(0) is contained in W, i.e. G�1(0) \ (∂W)t = ? forall t 2 [0, 1]. Choose a C•-perturbation eF, which yields eG = eF � p0t.By Sard’s Theorem choose a regular value 00, arbitrary close to 0, andthe solution set eG�1(00) can be described in exactly the same way asin the proof of Proposition 1.2.13. Figure 1.8 below shows the slightlydifferent situation with Proposition 1.2.13. The fact that deg( ft, Wt, pt)

is constant in t in the same way as Axiom (A3).

1.3.4 Remark. For t 2 [0, 1], let Dt be the connected component ofRn\ ft((∂W)t) containing pt. Then the result of Theorem 1.3.3 can bereformulated as

deg( ft, Wt, Dt) = const., (1.4)

which establishes continuity of the degree in f , W and p.

The index of isolated zeroes

1.3.5 It the case that a mapping has only isolated zeroes, and thusfinitely many, Property (A3) gives the degree as a sum of the local de-grees. More precisely, let xi 2 W be the zeroes of f and let Wi ⇢ W besufficiently small neighborhoods of xi, such that xi the only solutionof f (x) = p in Wi for all i. Then deg( f , W, p) = Âi deg( f , Wi, p) andwe define

i( f , xi, p) := deg( f , Wi, p),


which is called the index of an isolated zero of f . The index forisolated zero does not depend on the domain Wi. Indeed, if Wi andeWi are both neighborhoods of xi for which xi is the only zero off (x) = p, then we can define a cobordism between ( f , Wi, p) and( f , eWi, p) as follows. Let W =

S

t2[0,1] Wt with

Wt =

8

>

>

<

>

>

:

Wi for t < 12 ,

Wi \ eWi for t = 12 ,

eWi for t > 12 ,

and ft = F(·, t) = f , pt = p. By Theorem 1.3.3 deg( f , Wi, p) =

deg( f , eWi, p). The expression for the degree becomes

deg( f , W, p) = Âx2 f�1(p)

i( f , x, p). (1.5)

It is not hard to find mappings with isolated zeroes of arbitraryinteger index.

1.3.6 Exercise. Show that if x 2 f�1(p) is a non-degenerate zero of f , theni( f , x, p) = (�1)b, where b = #{negative real eigenvalues} (counted withmultiplicity).

1.3.7 Let V be a real linear vectorspace of dimension n. A continuousmapping f : W ⇢ V ! V is differentiable on W if ef : D ⇢ Rn ! Rn

is differentiable on D, where ef = q � f � q�1, q : V ! Rn is a linearisomorphism (linear chart), and D = q(W). A value p 2 V is calledregular for f if and only if ep = q(p) 2 Rn is regular for ef . We nowdefine the C1-mapping degree by

deg( f , W, p; V) := deg( ef , D, ep), (1.6)

for p 62 f (∂W). The definition is independent of the chosen isomor-phism q since the determinants in the expression of the C1-mappingdegree do not depend on the particular isomorphism, and the zeroesof ef and bf 6 are in 1-1 correspondence. 6 The transformed mapping under a

different isomorphism.1.3.8 Exercise. Show that the degree deg( f , W, p; V) is well-defined for allp 62 f (∂W).

1.4 Integral representations

The expression for the C1-mapping degree for regular values pointsto an obvious integral definition of the degree which allows for aformulation of of the C1-degree without distinguishing betweenregular and singular values. The integral formulation is is also usefulsometimes for establishing various properties.


Regular integrals

1.4.1 Let w : Rn ! R be a continuous function with supp(w) =

Be

(p). Choose e > 0 small enough such that supp(w) ⇢ Rn\ f (∂W)

and is a coordinate neighborhood of p with respect to the changeof coordinates p = f (x), see Figure 1.4. The weight function can benormalized by

Z

Rnw(x)dx = 1.

A function w that satisfies the above conditions is called a weightfunction, or test function. In the calculations that follow it is con-venient to use the notation of differential forms on Rn. Write dx =

dx1 ^ · · · ^ dxn as the standard n-form on Rn and consider the differ-ential n-forms

w = w(y)dy, and f ⇤w = w( f (x))J f (x)dx.

The latter is called the pullback under f , where y = f (x). The n-formdx provides Rn with a standard orientation. With this notation a lotof the calculations simplify considerably. The space of compactlysupported continuous n-forms on Rn is denoted by Gn

c (Rn).

1.4.2 Lemma. Let p 62 f (∂W) be a regular value and w a weightfunction as defined above. Then the integral I represents the C1-mapping degree;

I =Z

Wf ⇤w = deg( f , W, p). (1.7)

Proof: As pointed out before f�1(p) is a finite set strictly containedin W. Since J f is non-zero at points in f�1(p) = {x1, ..., xk}, theInverse Function Theorem gives that f maps neighborhoods N

e

(xj)

of points in xj 2 f�1(p) diffeomorphically onto Be

(p), see Figure1.4. Thus f is a local change of coordinates near every point in xj 2f�1(p). Indeed, the fact that J f is non-zero at points in xj 2 f�1(p),implies that J f is also non-zero at the points in N

e

(xj), provided thate is small enough. The integral I splits in k local integrals

Z

Wf ⇤w = Â

j

Z

Ne

(xj)f ⇤w = Â

jsign

⇣

J f (xj)⌘

Z

Be

(p)w

= Âj

sign⇣

J f (xj)⌘

= deg( f , W, p),

which proves that both I is independent of w and represents thedegree defined in Definition 1.2.3. The above calculation uses that lo-cally f is a coordinate transformation y = f (x) and

R

w( f (x))J f (x)dx =

sign⇣

J f (xi)⌘

R

w(y)dy.


1.4.3 Exercise. Verify the above change of coordinates formula given byy = f (x).

1.4.4 Remark. If in the above lemma we choose weight functions w

with the property thatR

W w 6= 0, then

deg( f , W, p) ·Z

Rnw =

Z

Wf ⇤w.

See also Remark 1.4.17.

A general representation

1.4.5 The integral characterization of the degree in the generic casemotivates a representation of the C1-degree in general, i.e. regardlesswhether p is regular or not. In order for the integral representationin Equation (1.7) to serve as a definition of degree for general p, theindependence on w needs to be established. As before let w be acontinuous weight function on Rn with the properties

Figure 1.9: The support of the weightfunction w in a connected component Dof Rn \ f (∂W).

supp(w) ⇢ D ⇢ Rn\ f (∂W), andZ

Rnw = 1,

where D is the connected component of Rn\ f (∂W) containing p. Thefirst property allows for a larger class of weight functions in the sensethat supp(w) is not necessarily a local coordinate neighborhood of p.The space of continuous n-forms w = w(x)dx, with supp(w) ⇢ D,are denoted by Gn

c (D). For w 2 Gnc (D) and

R

Rn w = 1 we define theintegral over W by

I( f , W, D) :=Z

Wf ⇤w. (1.8)

The notation is justified by the following lemmas which show thatthe integral does not depend on w, but does depend on the com-ponent D in which its support lies. Moreover, we establish that I isinteger valued. For regular p 2 D, and supp(w) = B

e

(p), a localcoordinate neighborhood, the integral representation in Equation (1.7)is retrieved.

1.4.6 Lemma. Let w, w0 2 Gnc (D) be two compactly supported n-forms

on D, withR

Rn w =R

Rn w0 = 1 and supp(w), supp(w0) ⇢ Kn ⇢ D.7 7 As for a function on Rn the support ofa k-form l is given as the set of pointssupp(l) = {x 2 Rn | l 6= 0} ⇢ Kn ⇢ D,where Kn is an n-dimensional cube.

ThenZ

Wf ⇤w =

Z

Wf ⇤w0.

Proof: Divergence form. Define the function µ = w

0 � w. Clearly,supp(µ) ⇢ supp(w) [ supp(w0) ⇢ Kn ⇢ D and since

R

Rn w =R

Rn w0 = 1 it holds thatR

D µ(x)dx = 0. For compactly supportedn-forms with integral equal to zero the following version of thePoincaré Lemma applies.


1.4.7 Lemma. Let µ be a compactly supported n-form on Rn withR

Rn µ = 0 and supp(µ) ⇢ Kn. Then there exists a compactly sup-ported (n� 1)-form q on Rn, with supp(q) ⇢ Kn such that µ = dq,where

q =n

Âi=1

(�1)i�1ci(x)dx1 ^ · · · ^ dxi�1 ^ dxi+1 ^ · · · ^ dxn,

with ci 2 C10(R

n), and supp(ci) ⇢ Kn.

Proof: Establishing µ = dq is equivalent to finding a vector field c

such that µ = divc. Indeed, in terms of differential forms, if we setµj = µ

j(x)dx, then

q =n

Âi=1

(�1)i�1ci(x)dx1 ^ · · · ^ dxi�1 ^ dxi+1 ^ · · · ^ dxn.

Figure 1.10: The covering of supp(w) [supp(w0) by Kn ⇢ D.

For n = 1 we take c(x) =R x�• µ(s)ds. Suppose the above

statement is true in dimension n � 1. Write x = (y, xn), wherey = (x1, ..., xn�1), and define a(y) =

R

R µ(y, xn)dxn. By the inductionhypothesis a is of divergence form, i.e. a = divx, for some vectorfield x, with supp(xi) ⇢ Kn�1. Let t 2 C•(R) with supp(t) ⇢ K anddefine

cn(y, xn) =Z xn

�•

⇣

µ(y, z)� t(z)a(y)⌘

dz.

By construction supp(cn) ⇢ Kn, and ∂cn∂xn

= µ(x)� t(xn)a(y). Now let

c(x) =�

x(y)t(xn), cn(y, xn)�

,

then

divc(x) = t(xn)divx(y) +∂cn∂xn

(x)

= t(xn)a(y) + µ(x)� t(xn)a(y)

= µ(x),

and supp(ci) ⇢ Kn.

Now apply Lemma 1.4.7 to the form µ = w0 �w with support inKn ⇢ D. Therefore, µ = w0 �w = dq for some compactly supported(n� 1)-form q. Moreover, the support of the form q is contained inKn ⇢ D.

Independence. The cube Kn ⇢ D has a piecewise smooth boundaryand therefore by Stokes’ Theorem

Z

Wf ⇤w0 �

Z

Wf ⇤w =

Z

Wf ⇤(w0 �w) =

Z

Wf ⇤µ

=Z

Wf ⇤dq =

Z

f�1(Kn)f ⇤dq

=Z

f�1(Kn)d( f ⇤q) =

Z

∂ f�1(Kn)f ⇤q = 0,

since supp( f ⇤q) ⇢ f�1(Kn) ⇢ W. This proves the lemma.


1.4.9 Exercise. Check, using differential forms calculus, that f ⇤dq = d( f ⇤q)(Hint: show this first for C2-functions).

1.4.10 Remark. The first step of the proof of Lemma 1.4.6 holds forany two n-forms w and w0 with compact support. This is essentiallythe Poincaré Lemma as given by Lemma 1.4.7. The second step isStokes’ Theorem and uses in an essential way that both w and w0

have their supports in D. If the latter does not hold then the integralsR

W f ⇤w andR

W f ⇤w0 are not necessarily equal. As a consequence itwill become clear from the forthcoming discussion that p 62 f (∂W)

is an essential condition in order for the integral definition of thedegree to hold.

1.4.11 Lemma. Let w 2 Gnc (D) with

R

Rn w = 1 and supp(w) ⇢ Kn ⇢D. Then

Z

Wf ⇤w = deg( f , W, p) 2 Z,

for any regular value p 2 supp(w).

Proof: By Sard’s Theorem choose a regular value p 2 supp(w)

and choose a coordinate neighborhood Be

(p) ⇢ Kn. As beforechoose w0 with supp(w0) = B

e

(p). From Lemma 1.4.2 it follows thatR

W f ⇤w0 = deg( f , W, p) and from Lemma 1.4.6Z

Wf ⇤w =

Z

Wf ⇤w0 = deg( f , W, p),

which proves the lemma.

The next lemma shows that for any form w 2 Gnc (D), with support

in some cube Kn ⇢ D, the integrals are the same.

1.4.12 Lemma. Let w, w0 2 Gnc (D) be two compactly supported n-

forms on D, withR

Rn w =R

Rn w0 = 1, and supp(w) ⇢ Kn ⇢ D andsupp(w0) ⇢ K0n ⇢ D. Then

Z

Wf ⇤w =

Z

Wf ⇤w0,

and thereforeR

W f ⇤w does not depend on p 2 D, but only on theconnected component D.

Proof: Choose two balls Be

(p) ⇢ supp(w) and Be

0(p0) ⇢ supp(w0)and a curve g 2 D connecting p and p0. Cover g by finitely manysmall balls B

ej(pj), j = 1, · · · , k, such that for any two conseccutiveballs it holds that

Bej(pj) [ B

ej+1(pj+1) ⇢ Knj ⇢ D,

for some cube Knj . Let wj be forms with supp(wj) = B

ej(pj). Then byLemma 1.4.6

Z

Wf ⇤wj =

Z

Wf ⇤wj+1,


and thereforeZ

Wf ⇤w =

Z

Wf ⇤w1 = · · · =

Z

Wf ⇤wk =

Z

Wf ⇤w0,


Lemma 1.4.12 justifies the notation I( f , W, D) and by Lemma 1.4.11

the integral is integer valued. In particular, the above considerationsprove that:

1.4.13 Proposition. For any regular values p, p0 2 D ⇢ Rn\ f (∂W) itholds that

deg( f , W, p) = deg( f , W, p0),

and I( f , W, D) = deg( f , W, p) for any regular value p 2 D.

It is is clear from the previous considerations that the degree isindependent of p 2 D and coincides with the definition of degreein the regular case; Definition 1.2.3. The advantage of the integralrepresentation is that a lot of properties of the degree can be obtainedvia fairly simple proofs. The final step is to show that one can useany compactly supported form w 2 Gn

c (D) to represent the mappingdegree.

1.4.14 Proposition. Let w 2 Gnc (D) with

R

Rn w = 1. ThenR

W f ⇤w =

I( f , W, D) is independent of w.

Proof: Since supp(w) ⇢ D is compact there exists a finite coveringof open balls Uj = B

ej(pj) with the additional property that Uj ⇢Kn

j ⇢ D for all j. Let {h

j} be a partition of unity subordinate to{Uj} and define the n-forms wj = h

jw (see Appendix). It holds thatÂj wj = w, supp(wj) ⇢ Uj. If

R

D wj 6= 0, then by Proposition 1.4.14

and Remark 1.4.4

I( f , W, D) ·Z

Dwj = deg( f , W, pj) ·

Z

Dwj =

Z

Wf ⇤wj. (1.9)

IfR

D wj = 0, then by Lemma 1.4.7, wj = dq

j andZ

Wf ⇤wj =

Z

Wf ⇤dq

j =Z

Wd f ⇤q j = 0.

Therefore, Equation (1.9) holds for all j. Sum over j in Equation (1.9),which proves the lemma.

This leads to the following alternative definition of the mappingdegree for arbitrary values p 2 D.

1.4.15 Definition. Let p 2 D ⇢ Rn\ f (∂W) and w 2 Gnc (D), with

R

Rn w = 1. Define

deg( f , W, p) := I( f , W, D) =Z

Wf ⇤w,

as the C1-mapping degree.


1.4.16 Exercise. Prove that deg( f , W, p) =R

W f ⇤w/R

D w for any p 2 D ⇢Rn\ f (∂W) and any w 2 Gn

c (D), withR

D w 6= 0, i.e. w not exact.

1.4.17 Remark. The definition of the C1-mapping degree can formu-lated in terms of compactly supported cohomology; H⇤c . Considerf : W ! Rn. By considering a connected component D ⇢ Rn\ f (∂W)

the map f yields a homomorphism f ⇤ in compactly supported coho-mology via pull-back;

f ⇤ : Hkc�

D) �! Hkc (W), [w] 7! [ f ⇤w],

where [w] is a non-trivial cohomology class in Hkc (D). Choose k = n,

then following diagram is a commutative diagram

Hnc (D)

f ⇤��! Hnc (W)

⇠=?

?

y

?

?

y

R

W

Rdeg( f ,W,D)��! R

where the mapR

W : Hnc (W) ! R is onto and the isomorphism

R

D : Hnc (D) ! R is given by [w] !

R

D w. In the case that W isconnected, then

R

W is an isomorphism between Hnc (W) and R. The

commutativity of the diagram gives the relation

deg( f , W, D)Z

Dw =

Z

Wf ⇤w,

which is exactly the definition of the C1-mapping degree in Defini-tion 1.4.15.

Homotopy invariance

1.4.18 The degree deg( f , W, p) is independent of p 2 D, with D ⇢Rn\ f (W), a connected component. Therefore, for any curve t 7! pt

in D, deg( f , W, pt) is a constant function of t; the degree is invariantunder homotopies in p.

The integral representation of the degree can be used to establishhomotopy invariance of the degree with respect to f . In particular,since in the definition of degree the domain W isolates the solutionset f�1(p), the degree is stable under small perturbations of the mapf , see Exercise 1.2.5. The general homotopy invariance of the degreewill be proved in several steps. The key ingredient is the continuity ofthe integral representation with respect to f .


1.4.19 Lemma. The function f 7!R

W f ⇤w =R

W w( f (x))J f (x)dx iscontinuous with respect to the C1-topology.

Proof: By the continuity of w(x), k f � gkC1 < d, implies that|w( f (x))�w(g(x))| < e uniformly for x 2 W. Similarly, since J f (x) isa polynomial term in ∂ fi

∂xj, k f � gkC1 < d implies that |J f (x)� Jg(x)| <

e, uniformly in x 2 W. These estimates combined yield the continuityof the integral

R

W f ⇤w with respect to f .

1.4.20 Lemma. Let t 7! ft and t 7! wt, t 2 [0, 1] be a continuous pathsin and assume that supp(wt) \ ft(∂W) = ? for all t 2 [0, 1], thenR

W f ⇤t wt = const.

Proof: By assumption, for each t 2 [0, 1] the integral represents adegree, i.e.

R

W f ⇤t wt = deg( ft, W, pt) for some pt 2 supp(w). There-fore the integral is integer valued. On the other hand by Lemma1.4.19 the integral is a continuous function of t and therefore constant.

1.4.21 We can use these lemmas to prove the general homotopyprinciple as given in Theorem 1.3.3.

1.4.22 Proposition. Let t 7! ft and t 7! pt, t 2 [0, 1] be a contin-uous paths and assume that pt 62 ft(∂W) for all t 2 [0, 1]. Then,deg( ft, W, pt) is a continuous function of t and is therefore constantalong ( ft, W, pt).

Proof: Choose an e > 0 small enough such that Be

(pt) ⇢ Rn\ ft(∂W).Define a form w = w(x)dx such that supp(w) = B

e

(0) and setwt = w(x � pt)dx. Consequently t 7! wt is a continuous path withsupp(wt) \ ft(∂W) = ? for all t 2 [0, 1] and

R

W f ⇤t wt = deg( ft, W, pt).By Lemma 1.4.20 the integral

R

W f ⇤t wt is constant, which proves thelemma.

1.5 Proper mappings

So far the mapping degree has been defined for mappings onbounded domains W. For unbounded domains W the generic con-struction of the degree does not make sense in general due to thepossible non-compactness of the set f�1(p). However, if a mappingis proper the C1-degree can be defined in the usual manner. Letf : W ⇢ Rn ! Rn be a proper mappings and W an unboundeddomain. If p 2 Rn is a regular value then the degree deg( f , W, p) isgiven by Definition 1.2.3. The degree can be extended to arbitraryvalues p following the procedures in Section 1.2.


Local and global degree

1.5.1 The integral representation in Section 1.4 can be used the definethe C1-mapping degree for proper mappings for arbitrary values pdirectly. Proper mappings are the natural morphisms that inducethe homomorphisms on compactly supported cohomology, seee.g. 8. Using the construction in Section 1.4 yields to the following 8 Raoul Bott and Loring W. Tu. Dif-

ferential forms in algebraic topology,volume 82 of Graduate Texts in Math-ematics. Springer-Verlag, New York,1982

definition. Consider triples ( f , W, p), where W ⇢ Rn is open, f : W ⇢Rn ! Rn proper, and p 62 Rn\ f (∂W), and let w 2 Gn

c (D), withR

D w = 1, where D ⇢ Rn\ f (∂W) a connected component containingp. Then

deg( f , W, p) :=Z

Wf ⇤w.

As before the degree is independent of p 2 D and is thereforesometimes written as deg( f , W, D). If p is a regular value then thedegree is given by Definition 1.2.3. In particular, for proper mappingsfrom Rn to Rn the degree does not depend on p and is defined asdeg( f ). The identity map on Rn is an example of a proper map, anddeg(Id) = 1. The theory discussed in the remainder this chapter willmainly concern bounded sets W. However, if f : W ⇢ Rn ! Rn

is a proper mapping the degree deg( f , W, p) can be computed bychoosing a bounded domain W0 ⇢ W containing f�1(p). In thatcase deg( f , W, p) = deg( f , W0, p), which is useful for translatingvarious properties of the degree for proper mappings. The degree( f , W, p) 7! deg( f , W, p) is a local degree, or degree over p (cf. 9, IV, 9 Albrecht Dold. Lectures on algebraic

topology. Classics in Mathematics.Springer-Verlag, Berlin, 1995. Reprint ofthe 1972 edition

§5). As pointed out before, for a regular value p the local degreecounts the number times the set f�1(p) covers p (counted withmultiplicity) under the mapping f , see Figure 1.11. The degreedepends on p (per connected component of Rn\ f (∂W)).

1.5.2 Example. Consider the function f : [�3, 3] ⇢ R ! R given byf (x) = x3 � 3x. The value p = 1 is regular and deg( f , [�3, 3], 1) = 1.The Figure 1.11 below shows how the image f ([�3, 3]) covers p = 1.If we consider p = �20 (see Figure 1.11) then p is not covered by theimage under f and the degree is zero.

1.5.4 In the case W = Rn and f : Rn ! Rn is proper, then themapping degree deg( f , Rn, p) is independent of p and is denotedby deg( f ). In this case we refer to the global mapping degree (cf. 10, 10 Albrecht Dold. Lectures on algebraic


IV, §4). The global mapping degree counts how many times f (Rn)

covers Rn counted with multiplicity. The notion of local and globaldegree will be discussed in more depth in Section 2.5.


Figure 1.11: The image f ([�3, 3]) locallycovers the regular value p = 1 and nocovering at p = �20.

1.5.5 Example. Consider the polynomial function f (x) = x4 � 2x2 + 1defined on R is a proper mapping. Choose a weight function

w(x) =

8

<

:

(1� |x|) when x 2 [�1, 1],

0 otherwise.

Via the integral definition the degree is given by

deg( f ) =Z

Rf ⇤w =

Z

p2

0

⇥

1� |x4 � 2x2 + 1|⇤⇥

4x3 � 4x⇤

dx = 0,

which also follows from counting f�1(p) for any regular value p.

1.5.6 Example. The polynomial function f (x) = x3 � x/2 defined onR is also a proper mapping. Let 2w(2x) be a weight function, w asabove, then

deg( f ) =Z

Rf ⇤w =

Z 1

�1

⇥

1� 2|x3 � x|⇤⇥

3x2 � 1/2⇤

dx = 1,

which proves that f (x) = p has at least one zero for any p 2 R.Functions w with finite mass and which decrease monotonically tozero as |x| ! •, can be used to approximate weight functions. Forexample take w = e�x2 and consider the mapping f (x) = x3. ThenR

R w(x) =R

R e�x2=p

p, and

deg( f ) =1pp

Z

Rf ⇤w =

3pp

Z

Rx2e�x6

dx = 1,

which is of course the same answer as before.

The examples suggest that for proper mappings from Rn to Rn thedegree is related to surjectivity.

1.5.7 Proposition. Let f : Rn ! Rn be a proper C1-mapping.

(i) If deg( f ) 6= 0, then f is surjective.

(ii) If f is not surjective, then deg( f ) = 0.


When f is surjective deg( f ) counts (with multiplicity) how manytimes the image under f covers Rn.

Proof: If deg( f ) 6= 0, then the equation f (x) = p has a solution forany p 2 Rn and thus f is surjective, proving (i).

On the other, if f is not surjective then there exists a p and e > 0such that f (Rn) \ B

e

(p) = ?, since proper mappings are closedmappings. Now choose w, with supp(w) ⇢ B

e

(p). Then deg( f ) =R

R f ⇤w = 0, which proves (ii) and therefore the lemma.

1.5.8 Exercise. Give an example of an improper function f : R ! R such thatf�1(p) 6= ? for all p 2 R.

1.5.9 Exercise. Show that the formula deg( f ) = 1p

n/2

R

Rn e�| f (x)�p|2 J f (x)dx,p 2 Rn, is an alternative expression for the mapping degree for propermappings f on Rn.

1.5.10 Remark. In terms of compactly supported De Rham coho-mology (Remark 1.4.17) the local degree is again expressed via[ f ⇤w] 2 Hn

c (W) with [w] 2 Hnc (D), p 2 D a connected component

of Rn\ f (∂W). Properness is needed to ensure that f ⇤w deterem-ines a cohomology class in Hn

c (W). In the case of the global degree[ f ⇤w] = deg( f )[w]. The compactly supported De Rham cohomologyis homotopy invariant with respect to proper homotopies (cf. 11, §44, 11 James R. Munkres. Elements of algebraic

topology. Addison-Wesley PublishingCompany, Menlo Park, CA, 1984

12, I, §2) and deg( f ) is a homotopy invariant.12 Raoul Bott and Loring W. Tu. Dif-ferential forms in algebraic topology,volume 82 of Graduate Texts in Math-ematics. Springer-Verlag, New York,1982

Proper mappings on open subsets

1.5.11 We can restate the degree theory developed in this chapterfor smooth mappings between open subsets of Rn. Let N, M ⇢ Rn

be open subsets. Note that we do not assume boundedness, norconnectedness of N and M. Let f : N ! M be a mapping of class C1;f 2 C1(N; M). We start with remarking that when f�1(p) is compactthen deg( f , N, p) is defined.

1.5.12 Exercise. Give an example of a function f : R ! R for which f�1(0) iscompact and for which f�1(p) is unbounded for any p 6= 0 close to 0.

1.5.13 Let Be

�

f�1(p)�

⇢ M, then the compactness of ∂Be

yieldsthat p 62 f (∂B

e

). Define the local degree over p by deg( f , N, p) :=deg( f , B

e

, p). This definition holds for any compact neighborhood(open interior is needed) K

e

⇢ N that contains f�1(p) in its interiorand is independent of K

e

by Theorem 1.3.3 (compare the argumentsfor the index of an isolated zero in Section 1.3). This shows thatdeg( f , N, p) is well-defined. We now give a general definition of localdegree over compact sets K ⇢ M (cf. 13, IV, §5 and VIII, §4, where the 13 Albrecht Dold. Lectures on algebraic



local degree is defined for any continuous mapping, see also Section2.5).

1.5.14 Definition. Let K ⇢ M ( 6= ?) be compact, connected andf�1(K) is compact. Then the local degree over K is defined bydeg( f , N, K) := deg( f , N, p) for any p 2 K.

1.5.15 Lemma. The local degree deg( f , N, K) is well-defined.

Proof: Let K0 ⇢ N be a compact neighborhood that contains f�1(K)in its interior and consider the (restriction) mapping f : K0 ⇢ N ! M.By the compactness of f (∂K0) and K it follows that d( f (∂K0), K) �d > 0 and thus K \ f (∂K0) = ?. Since K is connected it lies in aconnected component D of M\ f (∂K0). From Proposition 1.2.19 andDefinition 1.2.21 it follows that deg( f , K0, D) only depends on D.Since for p 2 K ⇢ D it holds that deg( f , N, p) = deg( f , K0, p) weshowed that deg( f , N, p) is the same for every p 2 K.

1.5.16 Exercise. Show that d( f (∂K0), K) � d > 0 in the proof of Lemma 1.5.15.

1.5.17 For any two connected sets K0 ⇢ K ⇢ M it holds thatdeg( f , N, K0) = deg( f , N, K). This definition is reminiscent of thedegree as presented in Definition 1.2.21. The above considerationsreveal that the local degree deg( f , N, K) can also be characterized interms of the integral representation. Let D ⇢ M\ f (∂K0) be the con-nected component containing K and w 2 Gn

c (D) such thatR

M w = 1,then deg( f , N, K) =

R

K0 f ⇤w.A mapping f : N ! M is said to be proper over M0 ⇢ M if

f�1(K) is compact for all compact subsets K ⇢ M0. The degreedeg( f , N, K) is well-defined for all compact sets K ⇢ M0. If M0 isopen and connected then the local degree deg( f , N, p) = deg( f , N, p0)for any p, p0 2 M0 (connect p and p0 by a path g which is a compactset). In this case we have the degree deg( f , N, M0). From the latterthe degree for bounded domains follows as a special case.

1.5.18 Example. Let W be a bounded domain and let f : W ⇢ Rn ! Rn

be a C1-mapping, then f : W! Rn is a smooth mapping with N = Wand M = Rn. Let D ⇢ Rn\ f (∂W) be a connected component, thenfor any compact subset K ⇢ D it holds that f�1(K) is closed andcontained in W which implies that f�1(K) is compact. Thereforef 2 C1(W; Rn) is proper over D and the local degree deg( f , W, D) iswell-defined.


1.5.19 Definition. Let f : N ! M be a proper C1-mapping and letM be connected. Then the global degree is defined by deg( f ) =

deg( f , N, K) for any compact subset K with K ⇢ M (not necessarilyconnected).

1.5.20 The global degree is well-defined since deg( f , N, K) is inde-pendent of K (connected) and by the above arguments deg( f , N, K1 [K2) = deg( f , N, K1) = deg( f , N, K2) (connect by a path). The de-gree deg( f ) counts how many times f (N) covers M counted withmultiplicity, i.e. each sheet is either positively or negatively oriented.

1.5.21 Example. Let W, W0 ⇢ Rn be bounded domains and f : W ⇢Rn ! W0 ⇢ Rn be a C1-mapping. Suppose now that f : W! W0 withN = W and M = W0, and f (∂W) ⇢ ∂W0. For any compact set K ⇢ W0

it holds that f�1(K) ⇢ W and is compact. The mapping f 2 C1(W, W0)is proper and the degree deg( f ) = deg( f , W, K) is a global degree. Ifwe only assume that f (∂W) ⇢ ∂W0, then f : W\ f�1(∂W0) ! W0. Alsoin this case f is proper and deg( f ) is the global mapping degree.

1.5.22 The homotopy of the global degree is exactly as explainedbefore is we consider proper homotopies. For the local degree oneneeds to consider homotopies ft for which f�1

t (K) is compact alongthe homotopy. Other properties stay more or less the same and wewill come back to this in a more general setting in Section 2.5. Formappings that a proper over an open set M0 ⇢ M the degree canalso be expressed via the integral representation. Let w 2 Gn

c (M0)with

R

M0 w = 1, then deg( f , N, M0) =R

N f ⇤w. For the global degree(M0 = M) this gives deg( f ) =

R

N f ⇤w. In Example 1.5.21 the degreeis of course given by deg( f ) =

R

W f ⇤w where w 2 Gnc (W0) and

R

W0 w = 1.

1.5.23 Remark. In the case that W and W0 are bounded domains withsmooth boundary and f (∂W) ⇢ ∂W0 we have the following com-muting diagram. By the latter condition f is a mappings of pairs, i.e.f : (W, ∂W)! (W0, ∂W0) and

Hnc (W)

⇠= �� Hn(W, ∂W)f ⇤ �� Hn(W0, ∂W0)

⇠= �� Hnc (W0)

⇠=?

?

y

?

?

y

⇠=

Hn�1(∂W)y

⇤ �� Hn�1(∂W0)

where y = f |∂W. From this diagram it follows that [ f ⇤w] = deg( f )[w]

and [y⇤q] = deg(y)[q] and thus deg( f ) = deg(y). In Section 2.3 wewill come back to the boundary dependence of the degree. In Section2.5 we will give a more detailed account of the algebraic topology.


1.6 Mappings between smooth manifolds and the mapping degree

So far the mapping degree is discussed for continuous mappingsbetween compact subsets of Euclidean spaces. Most of the ideas andproofs translate to the setting of continuous mappings between com-pact subsets of orientable differentiable manifolds. In the previoussection we already discussed the degree for maps from Sn�1 ⇢ Rn

into Rn. Before discussing the degree for maps between manifoldssome preliminary notation and definitions needs to be introduced.

The C1-mapping degree for mappings between manifolds

1.6.1 Let f : W ⇢ N ! M be continuous and C1 on W and N andm smooth orientable manifolds with dim N = dim M = n. Thedefinition of the degree works in exactly the same way as for subsetsof Rn. One starts with the regular case. A value p 2 M is regular iff 0(x) has maximal rank for all x 2 f�1(p), i.e. f 0(x) is invertible. IfW is compact then f�1(p) is a discrete set by the Inverse FunctionTheorem and

deg( f , W, p) = Âx2 f�1(p)

sign⇣

J f (x)⌘

,

where J f (x) = det( f 0(x)) as before. If f is proper, or if N is compactthen deg( f ) := deg( f , N, p) is well-defined for any regular valuep 2 M. The construction of the mapping degree via integration canbe repeated verbatim for compactly supported n-forms w on M andtheir pull-back f ⇤w under f . We conclude, for any p 2 D ⇢ M\ f (∂W)

and w 2 Gnc (D) with

R

M w = 1, that deg( f , W, p) =R

W f ⇤w. Thelatter definition of degree is homotopy invariant with respect to pand f and retrieves the sign-definition given above. This extends theC1-degree to arbitrary values p 2 M. In the case that M is compactand f is proper, or when also N is compact, then any n-form onM is compactly supported and deg( f ) := deg( f , N, p) =

R

N f ⇤w(here

R

M w = 1). This generalizes the degree for mappings from∂W! Sn�1 as described in Section 2.3.

1.6.2 Proposition. Let f : N ! M with N and M compact, orientablemanifolds of dimension dim N = dim M = n. Then

(i) If deg( f ) 6= 0, then f is surjective.

(ii) If f is not surjective, then deg( f ) = 0.

The degree gives the number of times the image f (N) covers Mcounted with orientation.

Proof: See proof of Proposition 1.5.7.


Local degree and proper mappings

1.6.3 The local mapping degree for mappings between open subsetsof Rn as explained in Section 1.5 extends without modification tothe case of continuous mappings between orientable n-dimensionalmanifolds. We start with the case of manifolds (without boundary).The case of manifolds with boundary will be discussed thereafter.

Let f : N ! M be a C1-mapping, i.e. f 2 C1(N; M) and K ⇢ Ma non-empty compact and connected subset such that f�1(K) ⇢N is compact. Then the local degree deg( f , N, K) is defined as inDefintion 1.5.14. The local degree can be expressed via the integralrepresentation as described in in Sections 1.5 and 1.6. For a propermapping f 2 C1(N; M), M connected, the global degree deg( f ) =

deg( f , N, K) for any compact subset K ⇢ M. In the case N andM are compact ∂-manifolds and f (∂N) ⇢ ∂M, then the mappingf : N\ f�1(∂M) ! M14 is proper and the global degree is given 14 The interior N\∂N is denoted by N

and N\ f�1(∂M) ⇢ N since ∂N ⇢f�1(∂M).

by deg( f ) := deg�

f , N\ f�1(∂M), M�

. In Section 2.5 we will comeback to these notion in context of continuous mappings and directdefinition using homology.

1.7 Applications

In this section we will discuss a number of applications of the map-ping degree.

The degree for holomorphic functions

1.7.1 The Brouwer degree can also be used in complex functiontheory. A complex function f : C ! C can be regarded as a mappingfrom R2 to R2 via the following correspondence. Set z = x1 + ix2 andf (z) = u(x1, x2) + iv(x1, x2), then f : R2 ! R2 is defined by

(x1, x2) 7! f (x1, x2) =�

u(x1, x2), v(x1, x2)�

.

A complex mapping f is holomorphic on a bounded open set W ⇢ C,if the Cauchy-Riemann equations are satisfied, i.e. ∂ f = 0, which isequivalent to

∂u∂x1

=∂v∂x2

,∂v∂x1

= � ∂u∂x2

,

for all z = x1 + ix2 2 W. The Brouwer degree for the triple ( f , W, z),with z 2 C\ f (∂W), is defined as the degree of the mapping f = (u, v)on R2.

From complex function theory it follows that zeroes of holomor-phic functions are isolated, or the function is identically equal to


zero. This leads to a following result about the mapping degree forholomorphic functions.

1.7.2 Proposition. Let f : W ⇢ C ! C be a holomorphic function,not identically equal to zero, and f (z0) = 0, for some z0 2 W. Thenthere exists an e > 0, and a ball B

e

(z0) ⇢ W such that f (z) 6= 0, for allz 2 B

e

(z0)\{z0}, and

deg( f , Be

(z0), 0) = m � 1,

where m is the order of z0, i.e. f (z) = (z� z0)mg(z), z 2 Be

(z0), and|g(z)| � a > 0, for all z 2 B

e

(z0).

Proof: Since f is not identically equal to zero, z0 is an isolated zero off , and there there exists a ball B

e

(z0) ⇢ W on which f is non-zero,except at z0. Also, by analyticity, it follows that z0 is a finite orderzero; f (z) = (z� z0)mg(z), m � 1, and |g(z)| � a > 0 in B

e

(z0). Theseconsideration make that the degree deg( f , B

e

(z0), 0) is well-defined,since | f (z)| = e

ma > 0, for z 2 ∂Be

(z0). In the case m = 1 thedegree can be easily computed from the definition. In general, for ahomolomorphic function, J f (z) = 1

2kr f k2. Since 0 is a regular value,J f (z0) can be computed as follows: f (z) = (z � z0)g(z), and thusf 0(z) = g(z) + (z� z0)g0(z). Therefore

J f (z0) = |g(z0)|2 = a2 > 0,

and deg( f , Be

(z0), 0) = 1.Consider the holomorphic function p(z) = (z � z0)mg(z0), and

the homotopy fl

(z) = l f (z) + (1� l)p(z), l 2 [0, 1], which is ahomotopy of holomorphic functions. Choose e > 0 small enoughsuch that |g(z) � g(z0)| < 1

2 |g(z0)|, for all z 2 Be

(z0). In order touse the homotopy property of the degree it needs to be verified that0 62 ∂ f (B

e

(z0)), for all l 2 [0, 1]. Let |z� z0| = e, then

| fl

(z)| =�

�

l(z� z0)mg(z) + (1� l)(z� z0)mg(z0)�

�

= e

m��

lg(z) + (1� l)g(z0)�

�

= e

m��g(z0) + l(g(z)� g(z0))

�

�

� e

m��g(z0)

�

�� l

�

�g(z)� g(z0)�

�

� 12

e

m|g(z0)|.

If we choose d = 14 e

m|g(z0)|, then fl

(z) = d has no solutions on∂B

e

(z0), for all l 2 [0, 1]. Consequently,

deg( f , Be

(z0), d) = deg(p, Be

(z0), d).

It remains to evaluate deg(p, Be

(z0), d). The associated equation is

p(z) = (z� z0)mg(z0) = d =14

e

m|g(z0)|.


This implies that zeroes lie on |z� z0| = e4�1m . For the arguments it

holds that

m arg(z� z0) + arg(g(z0)) = 2pn, n 2 Z.

It follows immediately that the above equation has exactly m non-degenerate solutions, and therefore, deg(p, B

e

(z0), d) = m.At the boundary ∂B

e

(z0), |(z � z0)|m|g(z)| = e

ma. Consider thepath x

l

= 12 le

ma, then f (z) 6= x

l

on ∂Be

(z0), for all l 2 [0, 1].Consequently, d( f , B

e

(z0), x

l

) is constant all l 2 [0, 1], and

deg( f , Be

(z0), 0) = deg( f , Be

(z0), d) = deg(p, Bd

(z0), d) = m,

which completes the proof.

1.7.3 A direct consequence of the above proposition is a result onmapping degree holomorphic functions in general.

Theorem 1.7.4 Let f : W ⇢ C ! C be a holomorphic function. Assumethat 0 62 f (∂W). Then

d( f , W, 0) � 0.

Proof: By analyticity f has only isolated zeroes zi 2 W. Let Be

(zi) ⇢W be sufficiently small neighborhoods containing exactly one zeroeach. The excision and summation properties of the degree then give

d( f , W, 0) = d( f ,[iBe

(zi), 0) = Âi

d( f , Be

(zi), 0) = Âi

mi,

where the numbers mi � 1 are the orders of the zeroes zi. SinceÂi mi � 0 this yields the desired result.

Another consequence of Porposition 1.7.2 is the FundamentalTheorem of Algebra.

Theorem 1.7.5 Any polynomial p(z) = zn + an�1zn�1 + · · ·+ a0, withreal coefficients ai, has exactly n complex roots counted with multiplicity.

Proof: Write p(z) = zn + r(z), then |p(z) + r(z)| ��

�|p(z)|� |r(z)|�

�.On the circle |z| = R > 0, for R sufficiently large, we have |r(z)| CRn�1, and thus

|p(z) + r(z)| ��

�|p(z)|� |r(z)�

� � Rn � CRn�1 > 0,

which proves that all zeroes are contained in the ball BR(0), anddeg( f , BR(0), 0) is well-defined. The same holds for the homotopyp

l

(z) = zn + lr(z), l 2 [0, 1]. This gives

deg(p, BR(0), 0) = deg(zn, BR(0), 0) = n > 0,

implying that p(z) = 0 has at least one solution z1 in BR(0). Nowrepeat the argument for the polynomial p1(z) = p(z)

z�z1 . This againproduces a zero z2. This process terminates after n steps, proving thedesired result.


Linking numbers

1.7.6 The usual example of linking are two tangled closed loopsin R3, but also the winding of a closed loop around a point in theplane is an example of linking in R2. Similarly, a compact orientablesurface in R3 separating the inside from the outside is an exampleof lining in R3. The concept of linking can be formulated in terms ofdegree degree for objects of higher dimension as well.

Let K, L ⇢ Rn be smooth embedded manifolds of dimension k and` respectively. Assume that both K and L are compact and orientable.Moreover K \ L = ? and

k + ` = n� 1.

Define the mapping

Y : K⇥ L ⇢ R2n ! Sn�1 ⇢ Rn, (x, y) 7! y� x|y� x| ,

which is a continuous mapping between orientable manifolds. Theorientation on K⇥ L is the product orientation and the orientation onSn�1 the orientation induced by Rn.

1.7.7 Definition. For two disjoint, smoothly embedded compactand orientable submanifolds K and L in Rn, the linking number isdefined by

link(K, L) := deg(Y),

provided that k + ` = n� 1.

For the traditional linking of embedded circles in R3 we cancompute some simple examples, see Fig. ??.

Figure 1.12: Two linking embeddings ofS1. One circle intersects any ‘filing’ ofthe other circle, which yields a non-zerolinking number.

1.7.8 Example. Consider embedded circles K and L in R3. In order tocompute the linking number we need to compute the degree of themap Y : K⇥ L ⇠= T2 ! S2. We start with a volume form on S2. Definew = i

n

dx, where dx = dx1 ^ dx2 ^ dx3 and n = x1∂

∂x1+ x2

∂

∂x2+ x3

∂

∂x3

the unit normal vector field to S2 ⇢ R3, then

w = x1dx2 ^ dx3 � x2dx1 ^ dx3 + x3dx1 ^ dx2.

The integralR

S2 w = 4p gives the area (volume) of S2. The map Yis a composition of the F(x, y) = y� x : K ⇥ L ! R3\{0} and theretraction r(x) = x

|x| : R3\{0}! S2.Now

r

⇤w(x, h) = w(r⇤(x), r⇤(h))

=x1|x|3 dx2 ^ dx3(x, h)� x2

|x|3 dx1 ^ dx3(x, h) +x3|x|3 dx1 ^ dx2(x, h)

=det(x, x, h)

|x|3 ,


where we used the fact that for x, h 2 TxS2 it holds that r⇤(x) = 1|x| x

and r⇤(h) = 1|x|h. Under the map F we obtain

F⇤w(x, h) = w(�x, h) = �w(x, h)

= �det(y� x, x, h) = det(x� y, x, h).

For the map Y this implies that

Y⇤w(x, h) =det(x� y, x, h)

|x� y|3 .

Parametrize K and L and denote the parametrizations by k and l

respectively. Then,Z

K⇥LY⇤w =

Z 2p

0

Z 2p

0

det(k(t)� l(s), k

0(t), l

0(s))|k(t)� l(s)|3 dtds.

The linking number is given by

link(K, L) =1

4p

Z 2p

0

Z 2p

0

det(k(t)� l(s), k

0(t), l

0(s))|k(t)� l(s)|3 dtds. (1.10)

This integral may be hard to compute. Consider an example of twocircles in the x1, x2-plane, then

t 7! (cos(t), sin(t), 0), s 7! (2 cos(s), 2 sin(s), 0),

and det(k(t)� l(s), k

0(t), l

0(s)) = 0, which shows that link(K, L) = 0.

Before doing some more elaborate examples let us derive someproperties of the linking number.

1.7.9 Theorem. The linking number satisfies the following properties:

(i) link(L, K) = (�1)(k+1)(`+1) link(K, L);

(ii) if K and L are separated by a hyperplane in Rn, then link(K, L) =0;

(iii) let Kt and Lt be 1-parameter families of embedded circles suchthat Kt \ Lt = ? for all t 2 [0, 1], then link(K0, L0) = link(K1, L1).

As a matter of fact the linking number is an isotopy invariant.

Proof: For the pair L, K we have the map eY(y, x) = x�y|x�y| . Define

the maps r(x, y) = (y, x) and a(x, y) = (�x,�y). Then deg(r) =

(�1)k` and deg(a) = (�1)k+`+1. For the map eY it holds that eY =

r�1 � Y � a and and by the composition property of the degree wederive the desired statement in (i).

As (ii) if a hyperplane exists then Y is not surjective onto Sn�1 andtherefore deg(Y) = 0, which proves (ii).

Property (iii) is a direct consequence of he homotopy principle forthe degree.


1.7.10 Example. Consider the circles K = {x 2 R3 | x21 + x2

2 = 1, x3 =

0} and L = {x 2 R3 | (x2 � 1)2 + x23 = 1, x1 = 0}. On K consider the

orientation form qK = �x2dx1 + x1dx2 and on L the orientation formqL = �x3dx2 + (x2 � 1)dx3. Choose the following parametrizations

t 7! (� sin(t), cos(t), 0), s 7! (0, 1 + cos(s), sin(s)),

again denoted by k and l respectively. Upon substitution in Equation(1.10) yields the following expression

link(K, L) =1

4p

Z 2p

0

Z 2p

0

cos(s)� cos(t) cos(s)� cos(t)(3 + 2 cos(s)� 2 cos(t) cos(s)� 2 cos(t))3/2 dtds.

Under the mapping Y the inverse image of a value p 2 S2 is charac-terized by the following relation

Y�1(p) = {(x, y) 2 K⇥ L | y� x = µp, µ > 0}.

Such a value is regular if Y⇤w is nondegenerate at points in Y�1(p).By our previous calculations this means when det(x � y, x, h) 6=0, where x 2 TxK and h 2 TyL. If we choose p to be a regularvalue, then the degree can be computed by adding the signs of thedeterminants at points in Y�1(p). Let us carry out this calculationfor the above situation. Choose p = (0, 1, 0), then Y�1(p) consists ofthe point pairs (0, 1, 0) 2 K, (0, 2, 0) 2 L, (0,�1, 0) 2 K, (0, 2, 0) 2 Land (0,�1, 0) 2 K, (0, 0, 0) 2 L. The determinants are �1, +2 and �1respectively, and therefore link(K, L) = �1.

1.7.11 Remark. In order to compute link(K, L) in Example 1.7.10 onecan also try to evaluate the integral with brute force. We the help ofMaple we obtain that

link(K, L) =1

4p

Z 2p

0

Z 2p

0

cos(s)� cos(t) cos(s)� cos(t)(3 + 2 cos(s)� 2 cos(t) cos(s)� 2 cos(t))3/2 dsdt

=1

4p

Z 2p

0

2 · EllipticK⇣

2p

�1 + cos(t)⌘

5� 4 cos(t)

�6 · EllipticE

⇣

2p

cos(t)� 1⌘

5� 4 cos(t)dt = �1,

which follows by numerically integrating the elliptic integrals.

1.8 Notes

The C1-mapping degree as considered in this first chapter was in-troduced by Nagumo in 1951

15. In his paper Nagumo diverts from 15 Mitio Nagumo. A theory of degreeof mapping based on infinitesimalanalysis. Amer. J. Math., 73:485–496, 1951


the definition of the mapping the for continuous mappings by ap-proximation via simplicial mappings, by approximation via smoothmappings and defines the mapping degree for smooth mappings aswas given in Definition 1.2.3. The mapping degree for continuousfunctions, or Brouwer degree was developed by Brouwer 16 and is 16 L. E. J. Brouwer. Über Abbildung

von Mannigfaltigkeiten. Math. Ann.,71(1):97–115, 1911

treated in Chapter 2, where we follow the approach of Nagumo.In a series of papers Nagumo also treats the degree in a more gen-eral settings such as the Leray-Schauder degree 17, see Chapter 4. 17 Mitio Nagumo. Degree of mapping in

convex linear topological spaces. Amer. J.Math., 73:497–511, 1951; Mitio Nagumo.A theory of degree of mapping basedon infinitesimal analysis. Amer. J. Math.,73:485–496, 1951; and Mitio Nagumo.A note on the theory of degree ofmapping in Euclidean spaces. OsakaMath. J., 4:1–9, 1952

The generic definition of the C1-mapping degree is useful often forcomputing the degree in specific situations. This definition of themapping degree ties in with the homotopy argument in Proposition1.2.13 and can found for example in 18 and 19. The homotopy princi-

18 John W. Milnor. Topology from thedifferentiable viewpoint. PrincetonLandmarks in Mathematics. PrincetonUniversity Press, Princeton, NJ, 1997.Based on notes by David W. Weaver,Revised reprint of the 1965 original19 Mitio Nagumo. A note on the theoryof degree of mapping in Euclideanspaces. Osaka Math. J., 4:1–9, 1952

ple can be applied in many situations, see Chapter 2. The propertiesproved in Theorem 1.2.22 are axioms for a degree theory and canbe used in a much broader context, see Chapter 2. In 20 Amman &Weiss show that the properties, or axioms uniquely determine thedegree. Heinz 21 gave an integral formulation of the smooth mappingdegree. In Section 1.4 we essentially followed the treatments in 22 and23. The integral representation in Section 1.4 provides a definitionof the degree for smooth functions without having to worry aboutregular versus non-regular values. This definition is based on thedefinition using compactly supported De Rham cohomology. Thedefinition of degree extends to mappings between smooth manifoldsand to mappings on unbounded domains — proper mappings, seeChapter 2. In Chapter 2 a direct definition of the degree for contin-uous functions is linked to a homological definition of the degree.Further elementary accounts of the degree for smooth functions inthe context of bounded domains can be found in many books onnonlinear analysis. We mention in particular the books by Berger24, Nirenberg 25 and Schwartz 26. See also Guillemin & Pollack 27,Malchiodi & Ambrosetti 28, Brown 29, Lloyd 30, Bott & Tu 31.

Exercises

1 By identifying C and R2 the application z 7! zn can be identifiedwith a smooth mapping f on R2. Show that i( f , 0) = n. Find aclass of mappings on R2 for 0 is an isolated zero and i( f , 0) = �n.

2 Let f 2 C1(W), with W ⇢ Rn a bounded domain and f is one-to-one. Prove that deg( f , W, p) = ±1.

3 Let f : B1(0) ! Rn and f (x) 6= µx for µ � 0 and for all x 2 ∂B1(0).Show that f (x) = 0 has a non-trivial solution in B1(0).

4 Let f (x) = anxn + an�1xn�1 + · · ·+ a1x + a0 be a polynomial withan 6= 0.


(i) Show that for fixed coefficients a0, · · · , an there exists an r > 0such that f�1(0) 2 (�r, r).

(ii) Prove for n odd that deg( f , (�r, r), 0) = 1.

(iii) Prove that n even that deg( f , (�r, r), 0) = 0.

(Hint: use the integral representation of the degree with w(x) =

1� x2 on (�1, 1) and zero outside).

5 Prove Proposition 1.2.19.

6* (Borsuk’s Theorem) Let W ⇢ Rn be a bounded domain satisfyingthe property that x 2 W implies that �x 2 W. Let j : ∂W ⇢ Rn !Rn\{0} such that j(�x) = �j(x). Prove that for any continuousextension f : W ⇢ Rn ! Rn of j it holds that deg( f , W, 0) is anodd integer.

2The Brouwer degree

The C1-mapping degree defined in Chapter 1 uses the fact that fis differentiable. The homotopy invariance of the C1-degree can beused to extend the degree to the class of continuous functions onRn, which is essentially the approach due to Nagumo 1. At the core 1 Mitio Nagumo. A theory of degree

of mapping based on infinitesimalanalysis. Amer. J. Math., 73:485–496, 1951

of the definition of the C0-mapping degree, or Brouwer degree isthe fact that C1-functions can be approximated by C0-functions. Wewill discuss the Brouwer for bounded and unbounded domains,as well as for functions between manifolds. Another aspect is theaxiomatic approach towards degree theory. This will discussedfor the Brouwer degree. At the end of this chapter we will alsodiscuss the homological definition of the degree which allows a directdefinition of the Brouwer degree.

2.1 The Brouwer degree

Using approximations of f via smooth mappings and homotopyinvariance leads to the definition of the C0-degree, or Brouwer degree.

2.1.1 Definition. Let f 2 C0(W) and let p 62 f (∂W). Then, for anysequence f k 2 C1(W) converging to f in C0, define

deg( f , W, p) := limk!•

deg( f k, W, p),

as the Brouwer degree of the pair ( f , W, p).

2.1.2 The properties of the C1-mapping degree imply that this def-inition makes sense, i.e. the limit exists and is independent of thechosen sequence f k. Approximating sequences exist by virtue ofTheorem A.1.3. Since p 2 Rn\ f (∂W) it holds that d = d(p, f (∂W)) > 0(compactness of f (∂W)).2 Let g, g 2 C1(W) be approximations of 2 The continuous image of a compact

set is compact, which implies thatd = d(p, f (∂W)) > 0.

f such that kg � f kC0 , kg � f kC0 < d/2. Consider the homotopy


ht(x) = (1� t)g(x) + tg(x), t 2 [0, 1]. The choices of g and g give

kht � f kC0 (1� t)kg� f kC0 + tkg� f kC0

< (1� t) d/2 + t d/2 = d/2,

and for x 2 ∂W it holds that

|ht(x)� p| � | f (x)� p|� |ht(x)� f (x)| � d/2.

Figure 2.1: For small perturbations g off , the point p is not contained in g(∂W)[left]. The same holds for homotopiesht. The second figure shows f (W) andht(W) for t 2 [0, 1] [right].

Therefore, p 62 ht(∂W) for all t 2 [0, 1] and the degree deg(ht, W, p)is constant in t by the homotopy invariance of the degree (e.g. Propo-sition 1.4.22). We conclude that deg(g, W, p) = deg(g, W, p). For anyapproximating sequence f k it holds that k f k � f kC0 < d/2, for klarge enough. Therefore, we may assume in the above definition, thatp 62 f k(∂W). These observations prove that the limit in Definition2.1.1 exists and is independent of the chosen sequence f k.

2.1.3 Remark. In approximating C0-fucntions via C1-functions it isnot necessary to assume that p is a regular value for the sequencef k. Approximations can always be chosen such that this is the case,which can be useful sometimes.

2.1.4 Exercise. Let p 2 Rn\ f (∂W). Show that one can always approximate fwith C1-maps f k with the additional property that p is regular value for allf k.

2.1.5 Proposition. The Brouwer degree deg( f , W, p) is continuous inf 2 C0(W).

Proof: Let g 2 C0(W) be any continuous mapping such that kg�f kC0 < d/4. Then deg(g, W, p) well-defined, since, for x 2 ∂W, itholds that |g(x)� p| � | f (x)� p|� |g(x)� f (x)| � 3d/4 and thusd(p, g(∂W)) > 3d/4 which implies that p 62 g(∂W).

Let f k 2 C1(W) and gk(W) 2 C1(W) be sequences that converge tof and g respectively. Choose k large enough such that k f k � f kC0 <

d/4, and kgk � gkC0 < d/4. Since kgk � gkC0 < d/4 < 3d/8 wehave that deg(g, W, p) = deg(gk, W, p), and similarly deg( f , W, p) =

deg( f k, W, p), since k f k � f kC0 < d/4 < d/2.On the other hand

kgk � f kC0 kg� f kC0 + kgk � gkC0 < d/2,

which implies that deg( f , W, p) = deg(gk, W, p), and thereforedeg( f , W, p) = deg(g, W, p), establishing the continuity of deg withrespect to f .

Using the continuity of the degree in f the invariance undercontinuous homotopies can be derived.

the brouwer degree 49

2.1.6 Proposition. For any continuous path t 7! ft in C0(W), withf0 = f and p 62 ft(∂W), t 2 [0, 1], it holds that deg( ft, W, p) =

deg( f , W, p) for all t 2 [0, 1].

Proof: By definition t 7! ft is continuous in C0(W) and thereforeby Proposition 2.1.5, deg( ft, W, p) depends continuously on t 2 [0, 1].Since the degree is integer valued it has to be constant along thehomotopy ft.

2.1.7 Proposition. The Brouwer degree satisfies the translation prop-erty, i.e. for any q 2 Rn it holds that deg( f � q, W, p� q) = f ( f , W, p).

Proof: Choose a sufficiently small perturbation g 2 C1(W) of f ,then Axiom (A4) implies that

deg(g� q, W, p� q) = deg(g� q� (p� q), W, 0)

= deg(g� p, W, 0) = deg(g, W, p).

By definition deg( f � q, W, p � q) = deg(g � q, W, p � q) anddeg( f , W, p) = deg(g, W, p), which proves the lemma.

2.1.8 Remark. If t 7! pt is a continuous path such that pt 62 ft(∂W),then the translation property of the degree, Proposition 2.1.7, shows,since ft � pt is a homotopy, that

deg( ft, W, pt) = deg( ft � pt, W, 0) = deg( f � p, W, 0) = deg( f , W, p).

Therefore, the Brouwer degree is an invariant for cobordant triples( f , W, p) ⇠ (g, W, q), or ( f , W, D) ⇠ (g, W, D0). In the Section 2.2 themore general version will be given allowing variations in W.

2.2 Properties and axioms for the Brouwer degree

2.2.1 In this section a number of useful properties of the mappingdegree will be given. In principle these properties can be proved us-ing the definitions of the C1-mapping degree and the Brouwer degree.Another approach is to single out the most fundamental proper-ties and show that these determine the Brouwer degree uniquely,and that all properties can be derived from the axioms. Considertriples ( f , W, p), where W ⇢ Rn are open sets, f 2 C(W), andRn 3 p 62 f (∂W). Such triple are called admissible, and the mapping( f , W, p)! deg( f , W, p), which satisfies the following three axioms;

(A1) if p 2 W, then deg(Id, W, p) = 1;

(A2) for W1, W2 ⇢ W, disjoint open subsets of W, and p 62 f�

W\(W1 [W2)

�

, it holds that deg( f , W, p) = deg( f , W1, p) + deg( f , W2, p);


(A3) for any continuous paths t 7! ft, ft 2 C0(W) and t 7! pt, withpt 62 ft(∂W), i.e. ( ft, W, pt) is admissible for all t, it holds thatdeg( ft, W, pt) is independent of t 2 [0, 1];

is called a degree theory.

2.2.2 Theorem. The Brouwer degree deg( f , W, p) for admissibletriples ( f , W, p) satisfies the Axioms (A1)-(A3), i.e. the Brouwerdegree is a degree theory.

Proof: In order to verify Axiom (A1) consider the equation x = p.Clearly, there exists a unique solution and JId(x) = Id, which proves(A1). Axiom (A3) follows from Proposition 2.1.6 and Remark 2.1.8.

Since f is continuous and W\(W1 [W2) is compact, we have that

| f (x)� p| � d > 0, 8x 2 W\(W1 [W2).

As a consequence the open ball Bd

(p) ⇢ f (W\(W1 [ W2)). If g 2C1(W), with k f � gkC0 < d/2, then |g(x) � p| � d/2 and thusB

d/2(p) ⇢ W\(W1 [W2). We now have deg( f , W, p) = deg(g, W, p).Choose a regular value p0 2 B

d/2(p) such that deg(g, W, p) =

deg(g, W, p0). By 1.2.7, deg(g, W, p0) = deg(g, W1, p0) + deg(g, W2, p0),and deg(g, Wi, p0) = deg(g, Wi, p), i = 1, 2. By the choices of d and g,deg( f , Wi, p) = deg(g, Wi, p), which proves the theorem.

2.2.3 Remark. For the Brouwer degree for maps from Rn to Rn, Axiom(A3) is equivalent to the following two alternative axioms:

(A3

0) for any continuous path t 7! ft, ft 2 C0(W) and p 62 ft(∂W), itholds that deg( ft, W, p) is independent of t 2 [0, 1];

(A4) deg( f , W, p) = deg( f � p, W, 0).If the degree is considered for mappings between manifolds Axiom(A4) need not be well-defined.

2.2.4 Exercise. Show that the (A3

0) and (A4) combined are equivalent to (A3).

The above theorem shows that there exists a degree theory satis-fying Axioms (A1)-(A3); the Brouwer degree. The remainder of thissection is a list of properties that are derived from Axioms (A1)-(A3)and a proof that the Brouwer degree is the only degree satisfying(A1)-(A3).

2.2.5 (Validity of the degree) If p 62 f (W), then deg( f , W, p) = 0.Conversely, if deg( f , W, p) 6= 0, then there exists a x 2 W, such thatf (x) = p.

Proof: By chosing W1 = W and W2 = ? it follows from Axiom (A2)that deg( f ,?, p) = 0. Now take W1 = W2 = ? in Axiom (A2), thendeg( f , W, p) = 2 · deg( f ,?, p) = 0.


Suppose that there exists no x 2 W, such that f (x) = p, i.e.f�1(p) = ?. Since p 62 f (∂W), it follows that p 62 f (W), and thusdeg( f , W, p) = 0, a contradiction.

2.2.6 (Continuity of the degree) The degree deg( f , W, p) is con-tinuous in f , i.e. there exists a d = d(p, f ) > 0, such that forall g satisfying k f � gkC0 < d, it holds that p 62 g(∂W), anddeg(g, W, p) = deg( f , W, p).

Proof: See Proposition 2.1.5.

2.2.7 (Dependence on path components) The degree only dependson the path components D ⇢ Rn\ f (∂W), i.e. for any two pointsp, q 2 D ⇢ Rn\ f (∂W) it holds that deg( f , W, p) = deg( f , W, q).For any path component D ⇢ Rn\ f (∂W) this justifies the notationdeg( f , W, D).

Proof: Let p and q be connected by a path t 7! pt in D, then byAxiom (A3), with ft = f , the degree deg( f , W, pt) is constant int 2 [0, 1].

2.2.8 (Translation invariance) The degree is invariant under trans-lation, i.e. for any q 2 Rn it holds that deg( f � q, W, p � q) =

deg( f , W, p).

Proof: The degree d( f � tq, W, p� tq) is well-defined for all t 2 [0, 1].Indeed, since p� tq 62 f (∂W)� tq, it follows from Axiom (A3) thatdeg( f � tq, W, p� tq) = deg( f , W, p), for all t 2 [0, 1].

2.2.9 (Excision) Let L ⇢ W be a closed subset in W, and p 62 f (L).Then, deg( f , W, p) = deg( f , W\L, p).

Proof: In Axiom (A2) set W1 = W\L and W2 = ?, then deg( f , W, p) =deg( f , W\L, p) + deg( f ,?, p) = deg( f , W\L, p).

2.2.10 (Additivity) Suppose that Wi ⇢ W, i = 1, · · · , k, are disjointopen subsets of W, and p 62 f

�

W\([iWi)�

, then deg( f , W, p) =

Âi deg( f , Wi, p).

Proof: The property holds trivially for k = 1. Now assume it holdsfor k� 1, then by Axiom (A2)

deg( f , W, p) = deg⇣

f ,k�1[

i=1Wi, p

⌘

+ deg( f , Wk, p) =k

Âi=1

deg( f , Wi, p),

by the induction hypothesis.


2.2.11 Exercise. Show that the above statement holds true for countablecollections of disjoint open subsets Wi of W.

Let W ⇢ Rn ⇥ [0, 1] be a bounded and relatively open subset ofRn ⇥ [0, 1] (Section 1.3), and let F : W! Rn a continuous function onW, with ft = F(·, t), such that

(i) f0 = f , and f1 = g;

(ii) W0 = W f , and W1 = Wg;

(iii) there exists a continuous path t 7! pt, p0 = p and p1 = q, suchthat ( ft, Wt, pt) is admissible for all t 2 [0, 1];

then ( f , W f , p) ⇠ (g, Wg, q) are homotopic, or corbordant (notation:( f , W f , p) ⇠ (g, Wg, q)), and ( ft, Wt, pt) is an admissible homotopy.Compare this with Definition 1.3.2 for the smooth mapping degree.

2.2.12 (Homotopy invariance) For an admissible homotopy ( ft, Wt, pt),the degree deg( ft, Wt, pt) is constant in t 2 [0, 1].

Proof: Following the proof Theorem 1.3.3 assume with loss ofgenerality that pt = p for all t. Choose a small ball B

e

(p), thenfollowing the reasoning in the proof of Theorem 1.3.3, f�1

t (Be

(p)) ⇢Cti , t 2 (ti � di, ti + di), for finitely many sets Cti ⇢ W. By excision,Property 2.2.9, it follows that deg( ft, Wt, p) = deg( ft, Cti , p), and sincethe sets Cti ⇥ (ti � di, ti + di) form an open covering of F�1�B

e

(p)⇥[0, 1]

�

, the degree deg( ft, Wt, p) is constant for all t 2 [0, 1].By (A4) deg( ft, Wt, pt) = deg( ft � pt, Wt, 0). Therefore, without loss

of generality, assume that pt = p is constant.Choose e > 0 small enough such that B

e

(p) ⇢ Rn\ [t2[0,1]f ((∂W)t). As before, write deg( ft, Wt, p) =

R

Wtf ⇤t w =

R

(W)tf ⇤t w,

where supp(w) = Be

(p). By assumption, the set F�1�Be

(p)⇥ [0, 1]�

2W is compact. At every t 2 [0, 1], the sets f�1

t (Be

(p)) can be coveredby open cylinders Ct ⇥ (t� d, t + d) ⇢ W. At each t, by compactnessand continuity of F, d can be small enough such that f�1

t0 (Be

(p)) ⇢Ct0 ⇥ (t0 � d, t0 + d) for all t0 2 (t � d, t + d). For all t 2 [0, 1] thesesets form an open covering of F�1�B

e

(p)⇥ [0, 1]�

, which has a finitesubcovering, Cti ⇥ (ti � di, ti + di), i = 1, · · · , k. Therefore, for givent0 2 (ti � di, ti + di),

R

Wtf ⇤t0w =

R

f�1t0 (B

e

(p)) f ⇤t0w =R

Ctif ⇤t0w, which is

continuous in t0 by Lemma 1.4.19 and thus constant in t0. Since thesets Cti ⇥ (ti � di, ti + di), i = 1, · · · , k, form an open covering, thedegree is the same for all t 2 [0, 1].

2.2.13 (Orientation) Let A 2 GL(Rn) and W any open neighborhoodof 0 2 Rn, then deg(A, W, 0) = sign det(A).

Proof: The group GL(Rn) consists of two path components GL+

and GL�. If A 2 GL+ choose a path t 7! At, connecting Id and


A. Clearly, (At, W, 0) is admissible for all t, and therefore by Ax-ioms (A1) and (A3) it follows that deg(A, W, 0) = deg(At, W, 0) =

deg(Id, W, 0) = 1, which proves the statement for A 2 GL+.For A 2 GL� choose a path t 7! At, connecting R = diag(�1, 1, · · · , 1)

and A. As before, (At, W, 0) is admissible for all t, and thus by Axiom(A3) it follows that deg(A, W, 0) = deg(At, W, 0) = deg(R, W, 0). Itremains therefore to determine deg(R, W, 0). Consider the homotopy

ft(x) =⇣

|x1|�12+ t, x2, · · · , xn

⌘

: (�1, 1)⇥W0 ! Rn,

where W0 ⇢ Rn�1 is an open neighborhood of 0 2 Rn�1. It is clearthat ( ft, (�1, 1) ⇥W0, 0) is admissible for all t 2 [0, 1], and thus byAxiom (A3)

deg( ft, (�1, 1)⇥W0, 0) = deg( f1, (�1, 1)⇥W0, 0) = 0,

since the equation f1(x) = 0 has no solutions (Property 2.2.5). Att = 0, the value 0 is a regular value and the equation f0(x) = 0 hasexactly two non-degenerate solutions x� =

�

� 12 , 0, · · · , 0

�

and x+ =� 1

2 , 0, · · · , 0�

. Choosing two sufficiently small open neighborhoodsW� and W+ of x� and x+ respectively, Axiom (A2) yields that

deg( f0, (�1, 1)⇥W0, 0) = deg( f0, W�, 0) + deg( f0, W+, 0) = 0.

For f0 it holds that f0(x) =�

�x1 � 12 , x2, · · · , xn

�

on W� and f0(x) =�

x1 � 12 , x2, · · · , xn

�

on W+. Set p =� 1

2 , 0, · · · , 0�

, then by Property2.2.8

deg( f0, W+, 0) = deg(Id� p, W+, 0) = deg(Id, W+, p) = 1.

The latter follows using Property 2.2.12, with W = {(x1� t/2, x2, · · · , xn) | x 2W+}, and pt =

� 1�t2 , 0, · · · , 0

�

, i.e. deg(Id, W+, p) = deg(Id, W1, 0) = 1by Axiom (A1). Similarly,

deg( f0, W�, 0) = deg(R� p, W�, 0) = deg(R, W�, p) = �deg( f0, W+, 0) = �1.

Using the homotopy property (Property 2.2.8) as above it follows thatdeg(R, W, 0) = �1.

2.2.14 Theorem. If ( f , W, p) is an admissible triple, with f 2 C1(W)

and p regular, then

deg( f , W, p) = Âx2 f�1(p)

sign⇣

J f (x)⌘

.

For an admissible triple in general there exists an admissible triple(g, W, q) with g 2 C1(W) and q regular, which homotopy to ( f , W, p).Moreover, deg( f , W, p) = deg(g, W, q).


Proof: For a regular value p is the inverse image f�1(p) = {xj}is a finite set in W (see Lemma 1.4.2). For e > 0 sufficiently small,f�1(B

e

(p)) consists of disjoint homeomorphic balls Ne

(xj) ⇢ W. Bythe additivity (Property 2.2.10)

deg( f , W, p) = Âj

deg( f , Ne

(xj), p).

If e > 0 is chosen small enough then

deg( f , Ne

(xj), p) = deg( f 0(xj), Be

0(0), 0) = sign⇣

J f (xj)⌘

,

which proves the first statement. The latter identity can be provedas follows. By assumption f 0(xj) is invertible for all j. Define thehomotopy ft = (1� t) f + tp + t f 0(xj)(x � xj), then for x 2 N

e

(xj)

it holds that ft � p = f 0(xj)(x � xj) + (1� t)R(x, xj), and kRk =

o(kx� xjk), for kx� xjk sufficiently small. This gives the estimate

k ft(x)� pk � k f 0(xj)(x� xj)k � (1� t)kR(x, xj)k� Ckx� xjk � o(kx� xjk),

and thus k ft � pk > 0, for all t 2 [0, 1], provided kx � xjk = e

0 issmall enough. Using Axiom (A3) it follows that deg( f , N

e

(xj), p) =deg( f 0(xj)(x � xj), B

e

0(xj), p). Now use the Properties 2.2.12 and2.2.13, as in the previous proof, to show that deg( f 0(xj)(x� xj), B

e

0(xj), p) =deg( f 0(xj), B

e

0(0), 0) = sign⇣

J f (xj)⌘

.In Section 2.1 it was proved that for each admissible triple ( f , W, p)

there exists an admissible triple (g, W, q), with g 2 C1(W) and q regu-lar, so that ( f , W, p) ⇠ (g, W, q). Then, by Axiom (A3), deg( f , W, p) =deg(g, W, q).

Theorem 2.2.14 shows that the Brouwer degree is the uniquedegree theory satisfying Axioms (A1)-(A3).

2.2.15 (Composition) Let f 2 C0(W), g 2 C0(L), with f (W) ⇢ L andW and L both bounded and open. Let Di be the path components ofL\ f (∂W). Assume that p 62 g(∂L) [ g( f (∂W)), then

deg(g � f , W, p) = Âi

deg(g, Di, p) · deg( f , W, Di),

which is finite sum.

Identify Rn with Rn1 ⇥Rn2 , and let W1 ⇢ Rn1 , and W2 ⇢ Rn2 , beopen and bounded subsets.

2.2.16 (Cartesian product) Let ( f , W1, p) and (g, W2, q), with f 2C0(W1), and g 2 C0(W2), be admissible triples. Then ( f ⇥ g, W1 ⇥W2, p⇥ q) is admissible, and deg( f ⇥ g, W1⇥W2, p⇥ q) = deg( f , W1, p) ·deg(g, W2, q).


Proof: By Theorem 2.2.14 it suffices to prove this statement forC1-functions f and g, and regular values p and q respectively. Theproduct f ⇥ g is also C1, p⇥ q a regular value, and ( f ⇥ g)�1(p⇥ q) =f�1(p)⇥ g�1(q) = {x

i, z

j}i,j. For the degree this yields

deg(( f ⇥ g, W1 ⇥W2, p⇥ q) = Âi,j

sign⇣

J f⇥g(xi, z

j)⌘

=

"

Âi

sign⇣

J f (xi)⌘

#

·"

Âj

sign⇣

Jg(zj)⌘

#

,

since for f ⇥ g it holds that J f⇥g(xi, z

j) = J f (xi) · Jg(z j), which

completes the proof of Property 2.2.16.

2.2.17 In 1.3.7 we explained how the mapping degree is definedfor mappings on a vectorspace V. The Brouwer degree extends tomappings on V by using approximation of C1-functions.

2.2.18 Exercise. Carry out the construction of the Brouwer mapping degree forcontinuous mappings on a real linear vectorspace V.

2.3 Boundary dependence of the degree

The homotopy invariance of the Brouwer degree can be used toprove that the degree on depends only on the restriction of f to theboundary.

2.3.1 Proposition. Let j : ∂W ! Rn \ {p} be a continuous mapping.Then, for any two continuous extensions3 f , g 2 C0(W), such that 3 Continuous extensions exist by virtue

of Tietze’s Extension Theorem, seeAppendix A.1, Theorem A.1.6f |

∂W = g|∂W = j,

it holds that deg( f , W, p) = deg(g, W, p) and therefore the Brouwerdegree only depends on the restriction f |

∂W to ∂W.

Proof: Consider the homotopy ht = (1� t) f + tg, t 2 [0, 1]. Then,since f = g = j on ∂W, it holds that ht = j on ∂W for all t 2 [0, 1]and therefore p 62 ht(∂W) = j(∂W) for all t 2 [0, 1]. Consequently,(ht, W, p) is an admissible homotopy and by the homotopy invarianceof the degree deg(ht, W, p) is independent of t 2 [0, 1].

2.3.2 Proposition 2.3.1 makes it possible to define a degree theoryfor continuous mappings on compact sets that occur as boundariesof bounded open sets in Rn. Continuous mappings from ∂W to Rn

cannot be surjective.4 Therefore assume, without loss of generality, 4 The continuous image of a compact setis compact!that maps j act from ∂W to Rn \ {p} for some p 62 j(∂W).


2.3.3 Definition. Let j : ∂W ! Rn \ {p} be a continuous mapping.The mapping degree is defined by

W∂W(j, p) := deg( f , W, p),

for any continuous extension5 f : W ⇢ Rn ! Rn, with f |∂W = j. The 5 A continuous extension of j to W

always exists by virtue of Tietze’sExtension Theorem A.1.6.

mapping degree for mappings j : ∂W! Rn \ {p} can be regarded asa generalized notion of the winding number.

Generalized winding numbers

2.3.4 From the translation property of the degree (see Property 2.2.8)it follows that deg( f , W, p) = deg( f � p, W, 0), which implies thatW

∂W(j, p) = W∂W(j� p, 0). Define the normalized mapping

y :=j� p|j� p| : ∂W! Sn�1, (2.1)

where Sn�1 ⇢ Rn denotes the standard unit sphere in Rn. Considerthe homotopy zt = (1� t)[j � p] + ty, then by Tietze’s ExtensionTheorem there exists a continuous homotopy ht on W ⇥ [0, 1] withht|

∂W = zt. From the homotopy property of the degree it then followsthat W

∂W(j, p) = deg(h0, W, 0) = deg(h1, W, 0) =: deg(y, ∂W, Sn�1),and thus

deg(y) := deg�

y, ∂W, Sn�1� = W∂W(j� p, 0),

which defines the degree for y. Homotopy defines an equivalencerelation on mappings j, called the homotopy type, and the degreeonly depends on the homotopy type of the map.

In order to derive an integral formula for the degree deg(y) as-sume now that f is a C1-mapping on W and ∂W is a piecewise C1-boundary.

2.3.5 Theorem. Let µ 2 Gn�1(Sn�1), withR

Sn�1 µ = 1, and let y be asdefined above. Then

deg(y) =Z

∂Wy

⇤µ,

where y

⇤µ 2 Gn�1(∂W).

Proof: Let f be as before, and since deg( f , W, p) does not dependon p, choose p to be a regular value, so that f�1(p) = {xj} is afinite set. Let B

e

(p) be a sufficiently small ball in Rn\ f (∂W) suchthat N ⇢ W, where N = f�1(B

e

(p)). Since p is regular, N = [jNj

(finite), where the sets Nj are all mutually disjoint and diffeomorphicto B

e

(p). Consider the disjoint open sets L = W\N ⇢ W and N ⇢ W.Then p 62 f

�

W\(L [ N)�

and by Property 2.2.10 deg( f , W, p) =


deg( f , L, p) + deg( f , N, p). Since p 62 f�

L�

it follows from Property2.2.5 that deg( f , L, p) = 0 and therefore deg( f , W, p) = deg( f , N, p).According to the Definition in 1.2.3 and Property 2.2.10 the degreeon N is given by deg( f , N, p) = Âj deg( f , Nj, p)and deg( f , Nj, p) =

sign⇣

J f (xj)⌘

.

The mapping y has a C1-extension to L denoted by Y and givenby the formula in (2.1), i.e. Y = ( f � p)/| f � p|. The restrictions to∂L is again denoted by y and the restriction to ∂Nj by y

j. Choosean (n � 1)-form µ 2 Gn�1(Sn�1), with

R

Sn�1 µ = 1, which can beregarded as the restriction of (n� 1)-form µ on an open neighborhoodof Sn�1 in Rn. The orientation on L induces the Stokes orientation6

6 The Stokes orientation is the inducedorientation on the boundary usingthe outward pointing normal. Theorientation of ∂N induced by L isopposite the orientation induced by N.This explains the notation ∂W� ∂N.

on ∂L = ∂W� ∂N. By Stokes’ Theorem

Z

∂Ly

⇤µ =Z

∂Wy

⇤µ�Z

∂Ny

⇤µ =Z

Ld�

Y⇤µ�

=Z

LY⇤dµ = 0.

The latter follows from the fact that Y⇤dµ = 0. Indeed, (Y⇤dµ)x(x1, · · · , x

n) =

dµ f (x)(Y⇤x1, · · · , Y⇤xn) = 0, since the set of tangent vectors�

(Y⇤x1, · · · , Y⇤xn

are linearly dependent. Consequently,

Z

∂Wy

⇤µ =Z

∂Ny

⇤µ = Âj

Z

∂Nj(yj)⇤µ.

Since f |Nj is a C1-change of variables that is either orientation pre-serving or reversing, the same holds for the renormalized restrictionsy

j via the Stokes orientation of ∂Nj. This yields the following identity

Z

∂Nj(yj)⇤µ = ±

Z

Sn�1µ = ±1 = sign

⇣

J f (xj)⌘

= deg( f , Nj, p).

Combining these identities finally gives

Z

∂Wy

⇤µ = Âj

Z

∂Nj(yj)⇤µ = Â

jdeg( f , Nj, p) = deg( f , N, p)

= deg( f , W, p) = deg(y),

which proves the theorem.

2.3.6 Remark. The integral representation can also be used to computethe degree of j as defined in Definition 2.3.3. Let µ 2 Gn�1(Rn \ {p})with

R

∂W µ = 1. Then W∂W(j, p) =

R

∂W j

⇤µ.

Winding numbers in the plane


2.3.7 Let W = B1(0) ⇢ R2, f : B1(0) ⇢ R2 ! R2 a continuous map-ping then if 0 6= f (∂B1), then the degree deg( f , B1, 0) is well-defined.Denote the restriction of f to ∂B1 = S1 by j, and the renormalizationby y = j

|j| : S1 ! S1, then deg(y) = deg( f , B1, 0). The degree of y

can be expressed as follows deg(y) =R

S1 y

⇤µ, where µ is a 1-form onS1. The 1-forms on S1 can be expressed as µ = (c + h(q))dq, whereh is 2p-periodic. Via polar coordinates x1 = r cos(q), x2 = r sin(q), µ

extends to R2\{(0, 0)} and is given by

µ = c�x2dx1 + x1dx2

x21 + x2

2+ dh(x1, x2).

By taking c = 1/2p it follows thatR

S1 µ = 1, and set q = 12p

�x2dx1+x1dx2x2

1+x22

,

as the standard ‘volume’ form on S1. A direct calculation shows thatR

S1 y

⇤q =R

S1 j

⇤q and therefore

W(j, 0) :=1

2p

Z

S1j

⇤q = deg(j) = deg(y), (2.2)

which is called the winding number j about 0. Conversely, startingwith a mapping j : S1 ! S1, Tietze’s extension theorem yields thatfor any extension f to B1(0) the degree deg( f , B1(0) is given by thewinding number defined in (2.2).

2.4 The Brouwer fixed point theorem

2.4.1 A classical application of the Brouwer degree is the Brouwerfixed point theorem for continuous maps of the n-disc. A fixed pointfor a mapping f : Rn ! Rn is a point x 2 Rn which satisfies theequation

f (x) = x.

As a matter of fact the Brouwer fixed point theorem can be stated forsets homeomorphic to the n-disc, or closed unit ball B1(0).

2.4.2 Theorem. Let W ⇢ Rn be an open subset such that W is home-omorphic to B1(0), and let f : W ! Rn be any continuous map. Iff (W) ⇢ W, then f has a fixed point in W.

Proof: Let j : W! B1(0) be a homeomorphism. Then the mappingg := j � f � j

�1 : B1(0)! B1(0) is continuous. The maps f and g areconjugate and thus f has a fixed point if and only g has a fixed point.


2.4.3 Exercise. Show the above claim for conjugate mappings.

Figure 2.2: The intersection mappingsl± : B1(0)! ∂B1(0).

The Brouwer fixed point theorem can be proved by showing thetheorem holds for g. Suppose that g has no fixed points in B1(0),then g(x) 6= x, for all x 2 B1(0). Consider the line y = x + l(g(x)� x),which intersects ∂B1(0) in exactly two points, see Fig. 2.2. If x 2∂B1(0), then l = 0 and if g(x) 2 ∂B1(0), then l = 1. Since g(x) 6=x for all x it follows that there are two solutions l�(x) 0 andl+(x) � 1 to the quadratic equation |x + l(g(x)� x)|2 = 1. Choosethe intersection corresponding to l� 0. The function x 7! l�(x)is continuous in x 2 B1(0). Indeed, by an application of the ImplicitFunction Theorem to the quadratic equation F(l; a, b, c) = al

2 + bl +

c = 0, continuity of l(a, b, c) holds provided

2al + b = 2[h(g(x)� x)l + x, g(x)� xi = 2hy, g(x)� xi 6= 0.

This holds since g(x) 6= x, and therefore the vector y cannot beorthogonal to g(x)� x. Upon substitution this yields the continuity ofthe mapping h : B1(0)! ∂B1(0), defined by h(x) = x + l�(x)(g(x)�x). Moreover, h(x) = x, for x 2 ∂B1(0), i.e. h|

∂B1 = Id. FromProposition 2.3.1 if follows that

deg(h, B1(0), 0) = deg(Id) = 1,

which implies that the equation h(x) = 0 has a solution. This isclearly a contradiction, since h(B1(0)) = ∂B1(0).

The proof of the Brouwer fixed point theorem is based on theobservation that continuous mappings from B1(0) to ∂B1(0), whichare the identity on ∂B1(0) do not exit. This uses the boundary depen-dence property of the Brouwer degree discussed in Section 2.3, andholds in a much more general setting of bounded and open subsetW ⇢ Rn.

2.4.4 Theorem. There are no continuous maps f : W ! ∂W, withf |

∂W = Id.

Proof: By Proposition 2.3.1 deg( f , W, p) = deg(Id) = 1, for anypoint p 2 W, which implies that the equation f (x) = p has a solution,a contradiction.

2.4.5 Another theorem worth mentioning in this context is the HairyBall Theorem, which, in dimension two, asserts that a 2-sphere ‘cov-ered with hair’ cannot be combed in a continuous manner. Here thetheorem is formulated for the embedded sphere Sn�1 = ∂B1(0). Con-sider a function X : Sn�1 ! Rn, with the property that hX(x), xi = 0,for all x 2 Sn�1. Such a function is called a tangent vector field onSn�1.


2.4.6 Theorem (Hairy Ball Theorem). The (n� 1)-sphere Sn�1 allows anon-vanishing tangent vector field X(x) 6= 0 if and only if n� 1 is odd.

Proof: If n� 1 is odd a non-vanishing vector field is easily given:

X(x) = (�x2, x1,�x4, x3, · · · ,�xn, xn�1),

which is clearly tangent to Sn�1 and non-vanishing.As for the converse argue as follows. Suppose there exists a non-

vanishing tangent vector field X(x) on Sn�1, then normalizationdefines a unit tangent vector field Y = X/kXk. Consider

ht = cos(pt)x + sin(pt)Y(x).

It is clear, since hx, Yi = 0, that khtk = 1 and ht : Sn�1 ! Sn�1

for all t 2 [0, 1]. Moreover, h0 = Id and h1 = �Id and are thushomotopic mappings. From 2.2.13 and Proposition 2.3.1 it followsthat deg(h1) = deg(�Id) = (�1)n. By the homotopy invariance ofthe degree, 1 = deg(Id) = deg(h0) = deg(h1) = (�1)n and thus n� 1is odd.

2.5 The homological defintion of the Brouwer degree

Give an account of a direct definition of the mapping degree forcontinuous maps using homology theory. Describe the treatment ofthe degree as was done in Dold. p. 266 and p. 66.

2.6 The Lefschetz Fixed Point Theorem

Discuss the relation to the Lefschetz Fixed Point Theorem.

3Elementary homotopy theory

For continuous mappings f from a compact domain W ⇢ Rn to Rn,the question of solvability of the f (x) = p is determined only bythe mapping degree, when formulated in the following setting. Itwas proved in Section 2.3 that the degree deg( f , W, p) is determinedonly by the degree of the boundary map j = f |

∂W : ∂W ! Rn\{p}.Non-triviality of deg(j) implies that any continuous extension f ofj to W has a solution to the equation f (x) = p. In this chapter it isproved that the converse also holds, i.e. if every continuous extensionf of j to W has a solution to f (x) = p, then deg(j) 6= 0. This alreadyindicates that the question of solvability is strongly related to theproblem of extending a mapping j to all of W. To be more precise, ifj : ∂W ⇢ Rn ! Rn\{p} has a continuous extension f : W! Rn\{p},then the boundary map j does not force solvability of f (x) = pfor all continuous extensions f , with f |

∂W = j. In this case j issaid to be inessential with respect to W. When j is not inessentialwith respect to W it is said to be essential with respect to W, whichimplies there are no continuous extension f : W ! Rn\{p}, andthus for continuous extension f takes values in Rn in general andthe equation f (x) = p has non-trivial solutions in W. A fundamentaltheorem by Hopf is used to prove that essential versus inessential iscompletely determined by the mapping degree of j.

The goal of this chapter is to broaden the above question to caseswhere the degree cannot decide between essential versus inessential,or when the mapping degree is not defined. An important case is formappings

f : W ⇢ Rn ! Rk,

where n is not necessarily equal to k. In this case the degree asintroduced in Chapter 2 is not defined. The question is whetherj = f |

∂W : ∂W ! Rk\{p} still determines the solvability if f (x) = p,for any continuous extension f of j.


3.1 Homotopy types and Hopf’s Theorem

3.1.1 Consider mappings y : ∂W ⇢ Rn ! Sn�1, where Sn�1 ⇢ Rn

is the standard unit sphere. In this equal dimension situation animportant version of the extension problem holds which can beregarded as a version of Hopf’s Theorem.

3.1.2 Theorem. Let W ⇢ Rn be a connected, bounded domain. Acontinuous mapping y : ∂W ⇢ Rn ! Sn�1 extends to a continuousmapping f : W ⇢ Rn ! Sn�1, with f |

∂W = y if and only if deg(y) =0.

Theorem 3.1.2 is also referred to the extension problem for map-pings y : ∂W ! Sn�1 ⇢ Rn and connected boundaries of boundedopen sets W ⇢ Rn. In the forthcoming sections this problem willbe put in a more general context. As explained above the extensionproblem is directly linked to the solvability problem, see Corollary3.1.15.

3.1.3 Remark. The connectivity condition Theorem 3.1.2 can be omit-ted by replacing the condition on the degree. Let y

i = y|∂Wi , where

Wi are the connected components of W, then the condition on thedegree becomes deg(yi) = 0 for all connected components Wi of W.The proof is obvious by applying Theorem 3.1.2 to each component.

3.1.4 Definition. A family of mappings yt = Y(·, t), with Y : ∂W⇥[0, 1] ! Sn�1 is continuous, is called a homotopy between y0, y1 :∂W! Sn�1. The mappings y0 and y1 are called homotopic.

Homotopy is an equivalence relation on C0(∂W; Sn�1) and itsequivalence classes are called homotopy types or homotopy classes.The homotopy type of a map y in C0(∂W; Sn�1) is denoted by [y] andthe collection of all homotopy types or equivalence classes is denotedby [∂W; Sn�1] =

�

[y] | y 2 C0(∂W; Sn�1)

.

3.1.5 Exercise. Show that homotopy type introduced above defines an equiva-lence relation on C0(∂W; Sn�1).

3.1.6 Theorem 3.1.2 can be proved by using the following fundamen-tal property of the generalized winding number (see Section 2.3),which is a special case of Hopf’s Theorem.

3.1.7 Lemma. A continuous mapping y : Sn�1 ! Sn�1, whereSn�1 ⇢ Rn is the standard unit sphere, has trivial homotopy type ifand only if deg(y) = 0.

Proof: The proof follows by combining Lemma 3.5.20 and Theo-rems 3.6.5 and 3.6.11.

elementary homotopy theory 63

3.1.8 Exercise. Give an elemtary proof of Lemma 3.1.7 (Hint: use an inductionargument in n).

Proof of Theorem 3.1.2. If there exists a continuous extension f : W ⇢Rn ! Sn�1 ⇢ Rn\{0}, then deg( f , W, 0) = 0, and thus by definitiondeg(y) = 0.

Now suppose deg(y) = 0. Let g : W ⇢ Rn ! Rn be a continuousextension (use Tietze’s Extension Theorem), with g|

∂W = y. Byconstruction g�1(0) ⇢ U ⇢ W, where U is compact. Moreover, gcan be chosen to be C1 on U, and such that 0 is a regular value (seeChapter 2). In this case g�1(0) is a finite set of points in U ⇢ W.Now connect the points xj 2 g�1(0) via a path g, such that g has noself-intersections.

Figure 3.1: The zeroes of g are con-tained in U ⇢ W and are connected by anon-intersecting path g [left]. The pathg ⇢ U can be covered by a union ofopen balls V ⇢ U [right].

Since g ⇢ U is a compact set it can be covered by finitely manysmall open ball Bi, which yields a compact set V ⇢ U which containsg, and has a piecewise smooth boundary ∂V. Moreover, V is homeo-morphic to the unit ball B1(0), with homeomorphism a : V ! B1(0).On the domain W0 = W�V it holds that

g : W0 ⇢ Rn ! Rn\{0},

and deg(g, W0, 0) = 0. Since ∂W0 = ∂W [ ∂V, Lemma (somewhere inSection 2.3) implies that

deg(y) = deg(g, ∂W, Sn�1) = deg(g, ∂V, Rn\{0}) = 0.

Now on ∂V, define y

0 = g/|g|, and y

0 : Sn�1 ! Sn�1 is continuous.By Lemma 3.1.7, y

0 is homotopic to a constant map, and thus alsog|

∂V is. Let h : ∂V ⇥ [0, 1] ! Rn\{0} be a homotopy between g|∂V

and a constant map. Invoking the homeomorphism introduced above,then k = h � a

�1 : Sn�1 ⇥ [0, 1] ! Rn\{0} is also a homotopy. Themap

p(tx) = k(x, t), x 2 Sn�1,


defines an extension to B1(0), and p � a is an extension of g|∂V to V,

and p � a : V ! Rn\{0}. Now adjust g with p � a on V to obtain anextension eg to W, which takes values in Rn\{0}. The normalizationf = eg/|eg| yields the desired extension that maps from W to Sn�1.

3.1.9 The following theorem due to E. Hopf shows that the homo-topy types in C0(∂W; Sn�1) are characterized by the mapping degree,which is therefore the only homotopy invariant on C0(∂W; Sn�1),which generalizes Lemma 3.1.7 and is a direct consequence of The-orem 3.1.2. Theorem 3.1.10 below is referred to as the classificationproblem and generalizations will be discussed in forthcoming sec-tions.

3.1.10 Theorem. Let ∂W ⇢ Rn be compact, connected, smooth hyper-surface.1 Then, two continuous mappings y0, y1 : ∂W ! Sn�1 are 1 A smooth hypersurface is the level

set H�1(0) of a smooth functionH : Rn ! R, where 0 a regular value.Such a hypersurface is an embeddedcodimension-1 submanifold of Rn.Moreover, ∂W is orientable.

homotopic if and only if deg(y0) = deg(y1).

Proof: Two mappings y0, y1 : ∂W ! Sn�1 are homotopic if andonly if there exists a homotopy yt = Y(·, t) between y0 and y1,where Y : ∂W ⇥ [0, 1] ⇢ Rn+1 ! Sn�1 is a continuous mapping.Let F : W⇥ [0, 1] ! Rn be an extension of Y (use Tietze’s ExtensionTheorem), then

deg( ft, W, 0) = deg( f0, W, 0) = deg( f1, W, 0),

and therefore deg(y0) = deg(y1).Now suppose deg(y0) = deg(y1). By assumption ∂W = H�1(0)

for some smooth function H : Rn ! R, with 0 a regular value.Therefore the interval [�e, e], e > 0 sufficiently small, consists ofregular values. The function is assumed to be negative on W, H < 0,and thus bounded from below. Define the domain

L :=�

x 2 Rn | � e < H(x) < 0

,

is connected with ∂L = ∂W � H�1(�e). The deformation lemmain Section 9.4 can be used now to show that there exists an isotopy2

2 An isotopy is a homotopy ht forwhich ht is a diffeomorphism for eacht 2 [0, 1].

from ∂W = H�1(0) to H�1(�e). Indeed, consider the normalizedgradient flow

dxdt

= � rH(x)|rH(x)|2 ,

The solution of the initial value problem for x 2 ∂W is given byx(x, t), with

x(x, 0) = x, H(x(x, t)) = H(x)� t.

For details see Section 9.4.

Figure 3.2: Deformation of ∂W via thenormalized gradient flow on H.

The mapping h(x, t) = x

�

x, t(H(x) + e)�

defines an isotopy from∂W to H�1(�e);

h : ∂W⇥ [0, 1]! Rn,


where each ht(·) = h(·, t) is diffeomorphism from ∂W to H�1(�et).Let y be a mapping from ∂L to Sn�1 defined as y0 on ∂W and y1 � h

�11

on H�1(�e). By Theorem 2.3.5

deg(y) =Z

∂Ly

⇤µ =Z

∂Wy

⇤0 µ�

Z

H�1(�e)

�

y1 � h

�11�⇤

µ.

Since h1 is a diffeomorphism, it holds thatR

H�1(�e)(y1 � h

�11 )⇤µ =

R

∂W y

⇤1 µ, and therefore deg(y) = 0. By Theorem 3.1.2 there exists a

continuous mapping f : L ⇢ Rn ! Sn�1 ⇢ Rn. Now define

Y(x, t) = f (h(x, t)) : ∂W⇥ [0, 1]! Sn�1,

which is a homotopy between y0 and y1, and therefore proves thetheorem.

3.1.11 Corollary. There exists a mapping y : ∂W ! Sn�1 of anydegree m 2 Z. In particular [∂W, Sn�1] ⇠= Z.

Proof: Under construction.

Theorem 3.1.10 is derived from the extension problem in Theorem3.1.2. On the other hand Theorem 3.1.2 can be derived from theclassification problem in Theorem 3.1.10 in the special case when∂W = Sn�1.

3.1.12 Remark. Hopf’s Theorem (Theorem 3.1.10) still holds in thecase that ∂W is a triangulable set. Recall that a set is triangulableif it is homeomorphic to n-dimensional simplicial complex Dn. Theresult also holds for maps y : X ! Sn�1, where X is a triangulabletopological space, with dim(X) = n� 1, see 3. In particular (abstract) 3 Sze-tsen Hu. Homotopy theory. Pure

and Applied Mathematics, Vol. VIII.Academic Press, New York, 1959

smooth manifolds M are triangulable, and therefore Hopf’s Theoremextends to maps y : M ! Sn�1. In Chapter ?? the notion of degreefor maps between smooth manifolds will be introduced.

3.1.13 Example. Consider the annulus W = {(x, y) 2 R2 | 1 x2 + y2 2}, and the mapping

f (x, y) =

�y/p

x2 + y2

x/p

x2 + y2

!

,

acting from W to R2\{0}. The boundary ∂W of the annulus is discon-nected and deg(y) = 0, where y = f |

∂W : ∂W ! S1. Clearly, [y] 6= 0,which shows that the connectivity condition in Hopf’s Theoremcannot be removed.

Connectivity of ∂W is not required for Theorem 3.1.2. The degreegives the proper invariant and and a straightforward calculationshows that deg(y) = 0, which is in compliance with the extension f .


3.1.14 The above theorem states that y is inessential with respectto W if and only if deg(y) = 0. The extension problem in Theorem3.1.2 can be rephrased into a solvability property for the equationf (x) = p, with f : W ⇢ Rn ! Rn, and j = f |

∂W. This problem will bereferred to as the solvability problem. In the latter case the boundarymapping is denoted by j : ∂W ⇢ Rn ! Rk\{p}, and the extensionproblem becomes; given j, does there exist a continuous extensionf : W ⇢ Rn ! Rn, with j = f |

∂W. This version of the extensionproblem is equivalent to the version in 3.1.2. Indeed, a normalizedmapping y = j�p

|j�p| : ∂W ⇢ Rn ! Sk�1 is inessential with respect

to W — there exists a continuous extension g : W ⇢ Rn ! Sk�1

— if and only if j : ∂W ⇢ Rn ! Rk\{p} is inessential — thereexists a continuous extension f : W ⇢ Rn ! Rk\{p}. Indeed, ifj is inessential then g = j�p

|j�p| gives the desired extension for y,and conversely, if y is inessential, then f = r · g + p is the desiredextension for j, where r : W ⇢ Rn ! R+ is a continuous extension ofr = |j� p| : ∂W ⇢ Rn ! R+ via Tietze’s Extension Theorem.

3.1.15 Corollary. Let W ⇢ Rn be a connected domain. A continuousmapping j : ∂W ⇢ Rn ! Rn\{p} is essential with respect to W if andonly if deg(j) 6= 0.

Proof: By the discussion above and Theorem 3.1.2 j is inessentialwith respect to W if and only if deg(j) = 0. Therefore, j is essentialwith respect to W if and only if deg(j) 6= 0.

3.1.16 Remark. If W is not necessarily connected, then the conditionon the degree has to be replaced with deg(j

i) 6= 0, for some i, wherej

i = f |∂Wi , and Wi are the connected components of W.

3.2 The extension problem for mappings on a ball

3.2.1 The main task in this section is to investigate the extension andclassification problems for n is not necessarily equal to k, in the spe-cial case when W is homoemorphic to a closed ball. As it turns outthe degree cannot be used as a invariant, but homotopy type playsa crucial role in this special case, as well the appropriate homotopyinvariants that will be discussed in Section 3.4. The generalization ofthe extension problem for mappings y : ∂W ⇠= Sn�1 ! Sk�1 can beformulated as follows.

3.2.2 Theorem. Let W be homeomorphic to B1(0). Then a continuousmapping y : ∂W ⇠= Sn�1 ! Sk�1 is inessential with respect to W — inother words there exists a continuous extension f : W ! Sk�1 withf |

∂W = y — if and only if y has trivial homotopy type.


3.2.3 In the previous section the degree was the appropriate invari-ant for extension problem. Just the homotopy type does not fullydescribe the problem as Example 3.1.13 shows. Her, in the case that∂W ⇠= Sn�1 homotopy type and degree contain the same informationby Hopf’s Theorem. Homotopy type makes sense when n 6= k, whichexplains the formulation of the above theorem using homotopy type.When ∂W is not homeomorphic to a sphere the situation becomesmore complicated.

Proof: Suppose y is inessential with respect to W, i.e. there existsa continuous extension f : W ⇢ Rn ! Sk�1. Let g : W ! B1(0) bea homeomorphism, then ef = f � g�1 : B1(0) ! Sk�1. Define thehomotopy

h(x, t) = ef (tx),

which, when restricted to Sn�1, becomes a homotopy between j � g�1

and the constant map x 7! ef (0). Via the homeomorphism g, thehomotopy k = h � g, which provides a homotopy between y and theconstant map. Therefore, y has trivial homotopy type.

Assume [y] = 0, then there exists a homotopy h : ∂W⇥ [0, 1] !Rk\{0} between y and a constant map. The map k = h � g�1 :Sn�1 ⇥ [0, 1] ! Sk�1 is then a homotopy between j � g�1 and aconstant map. Now define the continuous extension

ef : B1(0)! Sk�1, ef (tx) = k(x, t), x 2 Sn�1.

The map f = ef � g now is continuous extension of y to W.

3.2.4 As in the previous section above corollary is a characterizationof the solvability problem for domains homeomorphic to a ball.

3.2.5 Corollary. Let W be homeomorphic to B1(0). Then a continuousmapping j : ∂W ⇠= Sn�1 = ∂B1(0) ⇢ Rn ! Rk\{0} is essential withresepct to W — in other words any continuous extension f : W! Rk

of j has a nontrivial solution to the equation f (x) = 0 — if and onlyif the normalized boundary mapping y = j/|j| : Sn�1 ! Sk�1 hasnontrivial homotopy type.

3.2.6 In this section ∂W is homeomorphic to the standard unit sphereSn�1 ⇢ Rn, the homotopy classes [∂W; Sk�1] can be linked to thestandard homotopy groups pn�1(Sk�1).

3.2.7 Exercise. Show that there exists a canonical isomorphism [∂W; Sk�1] ⇠=pn�1(Sk�1), and give the group structure on [∂W; Sk�1].

3.2.8 Exercise. Prove that pn(Sn) ⇠= Z (Hint: construct the isomorphism).

3.2.9 Exercise. Show that pn�1(Sk�1) ⇠= 0 for all n < k.


3.2.10 In Section 3.1 the homotopy types [∂W; Sn�1] were charac-terized by the mapping degree. In this section the boundary ∂W isrestricted to the special case of a sphere, but is more general in thesense that k is not necessarily equal to n. The degree cannot be usedto classify the homotopy types [Sn�1, Sk�1]. The framed cobordismtheory of Pontryagin gives an satisfactory answer to this question.Framed cobordisms will be considered in Section 3.4 and will actu-ally apply to the general case of smooth boundaries ∂W.

For proper mappings f : Rn ! Rk the above theory can beused the give necessary and sufficient condition for the solvabilityproblem.

3.3 The general extension problem

3.3.1 Let f : W ⇢ Rn ! Rk, be a continuous mapping with j =

f |∂W 6= 0, and consider the equation

f (x) = 0.

In the case n = k the existence of a solution for any continuousextension f , with j = f |

∂W 6= 0, is equivalent to nontriviallity of[y] (assuming ∂W is connected). This a consequence of Hopf’s The-orem and a more general result for any n and k. Before formulatingthe general result and proving the above statement it is should bepointed out that solvability and homotopy type are not necessarilyequivalent when n 6= k, as the following example shows.

3.3.2 Example. Consider the mapping f : R3\{0} ! R2 and the solidtorus W ⇠= B2 ⇥ S1 both given by

f (x, y, z) =

�y/p

x2 + y2

x/p

x2 + y2

!

, (a, b, r) 7!

0

B

@

(2 + r cos(b)) cos(a)(2 + r cos(b)) sin(a)

r sin(b)

1

C

A

,

where (a, b, r) 2 (R/2pZ)2 ⇥ [0, 1).

Figure 3.3: The map j = f |∂W maps to

the unit circle in R2\{0}, and can beviewed as constant vector field alongthe torus. Clearly the homotopy type ofj is non-trivial.

Figure 3.3 clearly shows that f is a continuous extension to Wwhich is nowhere vanishing. This explains that non-trivial homotopytype of a map j on a boundary ∂W which is not homeomorphic tosphere, and n 6= k, does not necessarily imply that every continuousextension has a zero.

3.3.3 In order to give criteria for the general solvability problem,the problem will be reformulated in terms of the general extensionproblem. As before consider mappings y : ∂W ! Sk�1. An if andonly if criterion for extendibility to all of W also provides an if andonly if criterion for solvability of f (x) = 0 in terms of j restricted


to ∂W. Therefore, a criterion for the extension problem is formulatedfirst.

3.3.4 Lemma. The extension problem for y only depends the homo-topy type of y. To be more precise, given an extension f : W ! Sk�1,then for every if for every (partial) homotopy ht = h(·, t) : ∂W ⇥[0, 1] ! Sk�1, for which h0 = y, there exists an homotopy extensionft = f (·, t) : W[0, 1] ! Sk�1. In case the pair (W, ∂W) is said to satisfythe homotopy extension property (HEP).


3.3.5 Remark. The homotopy extension property holds for an arbi-trary pair (X, A), with X a metric space, and A a closed subspace,with respect to Sk�1. The latter can also be relaxed to be a finitelytriangulable space Y. If (X, A) is a finitely triangulable pair, then thehomotopy extension property is satisfied for any topological space Y,in which case (X, A) said to satisfy the absolute homotopy extensionproperty (AHEP).

3.3.6 The the sets of homotopy classes [W; Sk�1] and [∂W; Sk�1] donot a priori have an algebraic structure. For example if ∂W = Sn�1,then [Sn�1; Sk�1] can be identified with the homotopy groupspn�1(Sk�1). Some functorial properties of [·; ·] can be easily derived.Consider topological spaces X and Y and continuous mappingsg : X ! Y, then by considering the set of homotopy classes [X; Sk�1]

and [Y; Sk�1], there is an induced morphism, or mapping

g⇤ : [Y; Sk�1]! [X; Sk�1],

defined as follows. A class q 2 [Y; Sk�1] is represented by a mappingh : Y ! Sk�1, and h � g : X ! Sk�1 represents a the homotopy class

g⇤(q) := [h � g] 2 [X; Sk�1].

3.3.7 Exercise. Show that [·, Sk�1] defines a contravariant functor from thecategory of topological spaces to the category of sets.

3.3.8 Let i : A ⇢ X be the inclusion map, and y : A ! Sk�1, whichhas an extension f : X ! Sk�1, such that f |A = f � i = y. Clearly,y

⇤ : [Sk�1; Sk�1] ! [A; Sk�1], and [Sk�1; Sk�1] ⇠= pk�1(Sk�1) ⇠= Z. Let1 2 pk�1(Sk�1) ⇠= Z correspond to the identity Id in [Sk�1; Sk�1], theny

⇤(1) = [y]. The assumption that there exists an extension f thenyields

i⇤( f ⇤(1)) = y

⇤(1), f ⇤(1) 2 [X; Sk�1],

which shows that y

⇤(1) 2 i⇤�

[X; Sk�1]�

. The converse also holds,which gives the following result.


3.3.9 Theorem. Let (X, A) be a topological pair and y : A ! Sk�1 acontinuous mapping. There exists a continuous extension f : X !Sk�1 if and only if

[y] = y

⇤(1) 2 i⇤�

[X; Sk�1]�

, (3.1)

where 1 = [Id], and Id : Sk�1 ! Sk�1.

Proof: The necessity of (3.1) was shown above. As for sufficiencythe following holds. Since y

⇤(1) 2 i⇤�

[X; Sk�1]�

, there exists a classa 2 [X; Sk�1], generated by g : X ! Sk�1, such that y

⇤(1) = i⇤(a).Now

[g � i] = i⇤(a) = y

⇤(1) = [y],

and thus g � i ⇠= y. By Lemma 3.3.4 and Remark 3.3.5, the homotopyextension property, there exists an extension f : X ! Sk�1 such thatf � i = y.

3.3.10 This theorem can be applied to the pair (W, ∂W) in particular.Recall that solvability of f (x) = 0 is equivalent to j being essentialwith respect to W. Indeed, given a mapping j : ∂W ! Rk\{0}, thenj can be extended to a mapping f : W ! Rk\{0} if and only ify

⇤(1) 2 i⇤�

[X; Sk�1]�

, where y = j/|j|. This criterion can also beformulated as [j] = j

⇤(1) 2 i⇤�

[X; Rk\{0}]�

.

3.3.11 Theorem. A continuous mapping j : ∂W ⇢ Rn ! Rn\{0} isessential with respect to W if and only if

[y] =⇥

j/|j|⇤

62 i⇤�

[X; Sk�1]�

,

or in other words if and only if [j] 62 i⇤�

[X; Rk\{0}]�

.

Proof: Theorem 3.3.9 shows that j is inessential with respect to Wif and only if

⇥

j/|j|⇤

2 i⇤�

[X; Sk�1]�

. Clearly, j is the essential withrespect to W if and only if

⇥

j/|j|⇤

is not in i⇤�

[X; Rk\{0}]�

.

3.3.12 As pointed out Theorem 3.3.11 generalizes the results forW ⇠= Bn, and k = n. Without any algebraic structure on the homotopyclasses the criterion in Theorem 3.3.11 may be hard to verify. Inthe next section we will discuss various algebraic structures andapplications to a number of spacial cases. For example when n = k itfollows from Hopf’s theorem that the isomorphism

deg : [∂W; Sn�1]! Z,

can be used to further simplify (3.1). From Theorem 2.4.4 it followsthat deg

�

i⇤�

[X; Rk\{0}]��

= 0, where i : ∂W ! W. Theorem 3.3.9then gives the criterion deg(y) 6= 0, which yields Corollary 3.1.15. Inthe next section algebraic structures needed to understand (3.1) arediscussed and a part of algebraic topology dealing with extensionproblems, called obstruction theory, is introduced.


3.4 Framed cobordisms

This section gives an introduction to Pontryagin’s theory of framedcobordisms. The treatment of framed cobordisms here is reminiscentof the homotopy principle in Subsection 1.2.

3.4.1 In order to give a sufficient introduction to the theory of cobor-disms the notion of smooth (embedded) n-dimensional manifold inM ⇢ R` is required. In the appendix a detailed account of elemen-tary facts about manifolds can be found. From this point on M willbe referred to as an n-dimensional manifold.

Let M be n-dimensional manifold and N ⇢ M a smooth, closed4

4 A closed submanifold is compact andits relative boundary is the emptyset.submanifold of codimension k. For brevity N will be referred to as

a codimension k submanifold of M. It is important to point out thatN is not necessarily connected. This plays an important role for thegroup structure of cobordisms. When n = k, then N is a finite set ofpoints.

3.4.2 Definition. Two codimension k submanifolds N, N0 ⇢ M, aresaid to be cobordant, if there exists a smooth, compact manifoldP ⇢ M⇥ [0, 1] such that

∂P =�

N ⇥ {0}�

[�

N0 ⇥ {1}�

.

Notation: N ⇡ N0. The manifold M is called cobordism between Nand N0, and the equivalence classes are called cobordism classes.

3.4.3 Exercise. Prove that cobordism defines an equivalence relation onsmooth, closed, codimension k submanifolds of W (Hint: Show that by‘gluing’ two cobordisms it is possible to construct a smooth cobordism).

3.4.4 If Tx N denotes the tangent at a point x 2 N, then Tx N ⇢Tx M ⇠= Rn. The latter allows a decomposition

Tx M = Tx N ��

Tx N�?,

where the orthogonal complement is taken with respect to the stan-dard inner product in Rn. Clearly Tx N ⇠= Rn�k and

�

Tx N�? ⇠= Rk.

3.4.5 Definition. A framing of a codimension k submanifold N is asmooth function v on N, defined by

x 7! v(x) =�

v1(x), · · · vk(x)�

2�

Tx N�? ⇥ · · ·⇥

�

Tx N�?,

such that v(x) is a basis for�

Tx N�? for all x 2 N. This function is

called a framing of N. Together with the framing v, (N, v) is called aframed submanifold.


3.4.6 A submanifold does not necessarily allow a framing. For ex-ample, the Möbius strip is not orientable and does not allow anyframing!

Of special importance are submanifolds given as level sets ofsmooth functions f : M ! M0, where M0 is a k-dimensional mani-fold. By Sard’s Theorem most value are regular, and by the ImplicitFunction Theorem

N = f�1(p),

is a codimension k submanifold (closed and smooth) contained in M,k n.

3.4.7 Example. Consider f : R2 ! R, defined by f (x) = x21 + x2

2 � 1.Then 0 is a regular value and N = S1 = f�1(0) is the unit circle inR2. The tangent space at a point x is given by TxS1 = {(x1, x2) 2R2 | x1x1 + x2x2 = 0}. This gives a ‘bundle’ of straight lines tangent toS1. The gradient r f gives a vector field in

�

TxS1�? at each x. This isan example of a framing of S1.

3.4.8 For submanifolds given as regular level sets the tangent spacesare defined as follows. For x 2 N = f�1(p) set

Tx N :=�

x 2 R` | d f (x)x = 0

.

Example 3.4.7 gives a framing via r f . The differential

d f (x) : Tx M ⇠= Rn ! Tf (x)M0 ⇠= Rk,

is a mapping whose null space at x is the tangent space Tx N, andmaps

�

Tx N�? isomorphically onto Rk. Therefore, vj(x) = (d f (x))�1yj,

with y = {yj} ⇢ Tq M0 a basis for Rk, provides a framing ofN = f�1(p). Notation

f ⇤y = (d f (x))�1y,

which gives a framing of N. A basis y can also be regarded as aelement in Gl(Rk). Therefore, for a given framing v(x), the functionx 7! d f (x)v(x) = y(x) can be interpreted as smooth path in eitherGl+(Rk), or Gl�(Rk). The notation y may indicate a fixed choiceof a basis of Rk, or a path in Gl±(Rk). The framed submanifold( f�1(p), f ⇤y) is called a Pontryagin framed manifold associated withf , or Pontryagin manifold for short.

Framing can also be incorporated within the definition of cobor-dism.


3.4.9 Definition. Two framed submanifolds (N, v) and (N0, v

0) of Mare said to be framed cobordant if there exists a cobordism P, and aframing w = w(x, t) of P, such that

pxw(x, 0) = v(x), pxw(x, 1) = v

0(x),

where px is the projection onto the first n components.

Cobordism is the smooth regular analogue of homotopy, whichallows for additional structures to be carries across, such as framing.

3.4.10 Remark. In some definitions of cobordism P has the propertythat it contains

�

N ⇥ [0, e)�

[�

N0 ⇥ (1� e, 1]�

. Such a cobordismcan be obtained by adding a collar at ∂P, and will be referred as acobordism with collar. See appendix for existence of collars. One caneasily prove that both definitions are equivalent.

3.4.11 Exercise. Show that N, N0 are cobordant if and only if there exists acobordism P with collar.

3.4.12 Lemma. Framed cobordism defines an equivalence relation onthe set of smooth, oriented, codimension k framed submanifolds ofM. The equivalence classes are called framed cobordism classes, andare denoted by Pk(M).

Proof: Clearly N is cobordant to itself by the trivial cobordism, andN ⇡ N0 implies N0 ⇡ N by reflecting the t-coordinate; t 7! 1� t.Transitivity can be obtained as follows. Say N ⇡ N0 with cobordismP, and N0 ⇡ N00 cobordism P0. Both P and P0 can be assumed to becobordisms with collar. Define P#P0 as the cobordism between Nand N0 by gluing the interval [0, 1] and [1, 2] and rescaling t. Clearly,P#P0 is a cobordism with collar between N and N00. Since two collarsare glued at N0 the two framings v and v

0 automatically glue to aframing w = v#v

0.

In the case of a manifold M with boundary ∂M the framed cobor-dism classes are denoted by Pk(M, ∂M).

3.5 Pontryagin manifolds

As explained in the previous section sublevel sets of smooth func-tions are particular cases of framed submanifolds, or Pontryaginmanifolds. As pointed out before cobordism is a smooth analogueof homotopy. When dealing with Pontryagin manifolds it becomesapparent that cobordisms are strongly related to smooth homotopies.Pontryagin manifolds are a different perspective on the solvalbil-ity problem f (x) = p. For this reason in this section M is either abounded domain W or its boundary ∂W, and mappings that map Minto Rk.


Pontryagin manifolds of bounded domains

3.5.1 Let M = W ⇢ Rn a bounded domain in Rn, with boundary∂W. As open set, W is a smooth n-dimensional manifold. Considersmooth mappings

f : W ⇢ Rn ! Rk, k n.

Consider Pontryagin manifolds of the following form. Let p 2Rk\ f (∂W) be a regular then

(N, v) =�

f�1(p), f ⇤y�

, y 2 Gl(Rk),

is a framed submanifold of W. Such Pontryagin manifolds lie incertain cobordism classes in Pk(W, ∂W), and may be able to givemore insight into the structure of Pk(W, ∂W) and its relation tomappings f as described above.

The lemmas about framed cobordism classes below follow exactalong the lines as the proof of the homotopy principle of Lemma1.2.13.

3.5.2 Lemma. For any two bases y, y

0 of TqRk of the same orien-tation (positively, or negatively oriented5), the framed submani- 5 This condition is equivalent to

det(y) = det(y0).folds (N, f ⇤y) and (N, f ⇤y0) are framed cobordant, i.e.⇥

(N, f ⇤y)⇤

=⇥

(N, f ⇤y0)⇤

2 Pk(W, ∂W).

Proof: By assumption y and y

0 are of the orientation and there-fore as matrix, det y) = det(y0), and y, y

0 lie in the same connectedcomponent of Gl(Rk). Let yt be a smooth path in Gl±(Rk).6 Define 6 A smooth path can be found because

Gl(Rk) is a smooth manifold.F(x, t) = f (x), and w(x, t) = f ⇤yt, then P = N ⇥ [0, 1] is a framedcobordism between (N, f ⇤y) and (N, f ⇤y0), with framing w.

3.5.3 Lemma. Let f , f 0 : W ⇢ Rn ! Rk be smooth mappings such thatk f � f 0kC1 < e, and p is a regular value for both f and f 0. Let N, N0

be smooth submanifolds of W defined by N = f�1(p) and N0 =( f 0)�1(p). If e > 0 is small enough, then (N, f ⇤y) and (N0, ( f 0)⇤y) areframed cobordant, i.e.

⇥

(N, f ⇤y)⇤

=⇥

(N0, ( f 0)⇤y)⇤

2 Pk(W, ∂W).

Proof: As in Section 2.1 define a smooth homotopy ft(x) =

F(x, t) = (1� t) f (x) + t f 0(x). If e > 0 is chosen suffciently smallthen p 2 Rk\ ft(∂W), and p is a regular value for ft, for all t 2 [0, 1].By definition P = F�1(p) is a cobordism between N and N0. Indeed,dF(x, t)(x, t) = d f (x)x + t(d f � d f 0)(x)x � ( f � f 0)(x)t, and sincee > 0 is small, dF is onto Rk, and therefore p is a regular value of F.Since p is regular F⇤y gives a framing for P, which proves the lemma.


As in Chapter 2 denote the connected components of Rk\ f (∂W)

by D. The next step is to prove that the cobordism class of N isindependent of the chosen regular value p 2 D.

3.5.4 Lemma. Let p, p0 2 D be regular values. Then, the framedsubmanifolds (N, f ⇤y) and (N0, f ⇤y), given by N = f�1(p), andN0 = f�1(p0), are framed cobordant, i.e.

⇥

(N, f ⇤y)⇤

=⇥

(N0, f ⇤y)⇤

2Pk(W, ∂W).

Proof: Since D ⇢ Rk\ f (∂W) is open there exists a smooth pathpt 2 D connecting p and p0. Define the smooth homotopy F(x, t) =f (x)� pt. By Sard’s Theorem there exists a regular value 00 arbitrarilyclose to 0. The level set M = F�1(00) is a framed cobordism betweeneN = f�1(p + 00) and eN0 = f�1(p0 + 00), and therefore

⇥

( eN, f ⇤y)⇤

=⇥

( eN0, f ⇤y)⇤

.

By Lemma 3.5.4 it holds that when |0� 00| < e, sufficiently small, then⇥

(N, f ⇤y)⇤

=⇥

( eN, f ⇤y)⇤

and⇥

(N0, f ⇤y)⇤

=⇥

( eN0, f ⇤y)⇤

. Conclusion⇥

(N, f ⇤y)⇤

=⇥

( eN, f ⇤y)⇤

=⇥

( eN0, f ⇤y)⇤

=⇥

(N0, f ⇤y)⇤

,


Combining the above lemmas yields the following theorem onframed cobordisms.

3.5.5 Theorem. Let f : W ⇢ Rn ! Rk be a smooth mapping, andp 2 D regular value. Then, for any regular value p0 2 D, andany oriented basis y

0 with the same orientation as y, it holds that(N0, f ⇤y0) is framed cobordant to (N, f ⇤y), where N = f�1(p) andN0 = f�1(p0).

Proof: Combine Lemmas 3.5.2, 3.5.3, and 3.5.4.

3.5.6 Homotopies of mappings f discussed here need to have theproperty that a distinguished point p is not contained in f (∂W).The proper way of discussing homotopies in this setting and theirhomotopy types is via homotopy of pairs. A topolgical pair (X, A)

consists of a topological space X and a subspace A ⇢ X. In general, acontinuous mapping f between topological pairs (X, A) and (Y, B),

f : (X, A)! (Y, B),

is a continuous mapping f : X ! Y, with the additonal property thatf (A) ⇢ B.

Here, consider topological pairs (W, ∂W) and (Rk, Rk\{p}), andcontinuous mappings f : (W, ∂W) ! (Rk, Rk\{p}). By definition thismeans that mappings f : W ! Rk, have the additonal property thatf (∂W) ⇢ Rk\{p}, which is equivalent to saying that p 62 f (∂W), orp 2 Rk\ f (∂W).


3.5.7 Definition. A family of mappings ft = F(·, t), with F : (W, ∂W)⇥[0, 1] ! (Rk, Rk\{p}) continuous, is called a homotopy betweenf , g : (W, ∂W) ! (Rk, Rk\{p}). The mappings f and g are calledhomotopic as mappings of topological pairs.

3.5.8 Homotopy is an equivalence relation and its equivalence classesare called homotopy types or homotopy classes. The homotopy typeof a map f is denoted by [ f ] and the collection of all homotopy typesis denoted by

⇥

(W, ∂W); (Rk, Rk\{p})⇤

=n

[ f ] | f : (W, ∂W)! (Rk, Rk\{p})o

.

3.5.9 Exercise. Show that the homotopy type introduced above defines anequivalence relation.

3.5.10 Theorem. Let f , f 0 : (W, ∂W)! (Rk, Rk\{p}) be a smooth map-pings, with p 2 Rk a regular value, which are smoothly homotopicwith respect to a homotopy F : (W, ∂W)⇥ [0, 1]! (Rk, Rk\{p}). Then,the submanifolds (N, f ⇤y) and (N0, ( f 0)⇤y) are framed cobordant,where N = f�1(p) and N0 = ( f 0)�1(p).

Proof: Let p0 2 Rk be a regular value for f , f 0 and F (use Sard’sTheorem), such that F : (W, ∂W) ⇥ [0, 1] ! (Rk, Rk\{p0}) is againa smooth homotopy. By definition P = F�1(p0) is a framed cobor-dism between the Pontryagin manifolds f�1(p0) and ( f 0)�1(p0). Thetheorem now follows from Lemma 3.5.4 (or Theorem 3.5.5).

3.5.11 The Theorems 3.5.5 and 3.5.10 reveal that the cobordism class⇥

( f�1(p), f ⇤y)⇤

2 Pk(W, ∂W),

is invariant under homotopy in f , p and y, and thus a homotopyinvariant of Pontryagin manifolds ( f�1(p), f ⇤y). The classes ofPontryagin manifolds are denoted by Pontk(W, ∂W) form a subset ofcobordism classes in Pk�W, ∂W; Rk, Rk\{p}

�

.

Pontryagin manifolds of smooth boundaries

3.5.12 Let M = ∂W ⇢ Rn be the smooth boundary of a compactdomain W ⇢ Rn. The boundary ∂W is a smooth hypersurface in Rn

and therefore a smooth, compact (n� 1)-dimensional manifold. Assuch ∂W is a 1-framed submanifold of Rn, which framing correspond-ing with the inward, or outward pointing normal. Consider smoothmappings

j : ∂W ⇢ Rn ! Rk\{0}, k n,


which are restrictions to ∂W of smooth mappings f : U ⇢ Rn !Rk\{0}, where U is a tubular neighborhood with coordinates (x, u) 2∂W⇥ (�e, e). The mapping f is called a tubular extension.

3.5.13 Remark. By definition j is the restriction to ∂W of smooth map-ping f : U ! Rk\{0}. By Teitze’s Extension Theorem, g extends to acontinuous mapping ef : Rn ! Rk. Let V be an open neighborhoodof Rn\U, such that ∂W 6⇢ V. Now smoothen ef on V (use Lemma ??),leaving ef = f unchanged on a neighborhood of ∂W. This map is nowthe desired extension and is again denoted by f .

3.5.14 Exercise. Carry out the smoothing procedure in the above remark (Hint:Use a covering of open balls for V).

3.5.15 A value p 2 Rk�1\{0} is called regular if it is a regular valuefor some tubular extension f . Consider Pontryagin manifolds of thefollowing form. Let p 2 Rk\{0} be a regular, then

(N, v) =�

j

�1(p), j

⇤y

�

, y 2 Gl(Rk),

is a k-framed, or codimension k framed submanifold of ∂W. SuchPontryagin manifolds lie in the cobordism classes in Pk�1(∂W).

The Lemmas 3.5.2, 3.5.3, and 3.5.4 are still true in this case and theproof are almost identical, and will therefore be omitted. The mainresults can be phrased as follows.

3.5.16 Theorem. Let j : ∂W ⇢ Rn ! Rk\{0} be a smooth mapping.Then, for any pair of regular values p, p0 2 Rk\{0}, and any orientedbases y, y

0 with the same orientation, it holds that�

j

�1(p), j

⇤y

�

isframed cobordant to

�

j

�1(p0), j

⇤y

0�.

Homotopies of mappings j were discussed in Section 3.1. Thehomotopy type of a map j is denoted by [j] and the collection of allhomotopy types is denoted by

⇥

∂W; Rk\{0}⇤

=n

[j] | j : ∂W! Rk\{0}o

.

3.5.17 Theorem. Let j, j

0 : ∂W ! Rk\{0} be a smooth mappings,which are smoothly homotopic with respect to a homotopy F :∂W⇥ [0, 1]! Rk\{0}. Then, the Pontryagin manifolds

�

j

�1(p), j

⇤y

�

and�

(j

0)�1(p0), (j

0)⇤y0�

are framed cobordant, for any regular valuep 2 Rk\{0}.

3.5.18 The Theorems 3.5.16 and 3.5.17 reveal that the cobordism class⇥

(j

�1(p), j

⇤y)⇤

2 Pk�1(∂W; Rk\{0}),

is invariant under homotopy in j, p and y, and thus a homotopyinvariant of Pontryagin manifolds (j

�1(p), j

⇤y). The classes of


Pontryagin manifolds are denoted by Pontk�1(∂W; Rk\{0}) and theyform a subset of Pk�1(∂W).

Homotopy types

3.5.19 The results of the previous subsections imply that if twomappings j, j

0 lie in the same homotopy class in⇥

∂W; Rk\{0}⇤

, thenthe associated Pontryagin manifolds are framed cobordant. Thisdefines a homomorphism

ak�1 :⇥

∂W; Rk\{0}⇤

! Pontk�1(∂W).

It actually holds that ak�1 is onto and one-to-one, and thus an isomor-phism. This is the subject in the next section. A first step toward thisresult is:

3.5.20 Lemma. The homotopy classes of mappings j : ∂W ! Rk\{0}and y : ∂W! Sk�1 are isomorphic;

⇥

∂W; Rk\{0}⇤ ⇠=

⇥

∂W; Sk�1⇤,

where the isomorphism is given by [j] 7!⇥

j/|j|⇤

.

Proof: The inclusion map i : Sk�1 ,! Rk\{0} shows that eachmapping y yields a mapping j = i � y, and thus the homotopy classes⇥

∂W; Sk�1⇤ are contained in⇥

∂W; Rk\{0}⇤

.Let F be a homotopy between mapping j and j

0, then Y = F/|F|defines a homotopy between the mappings j/|j|, j

0/|j0| : ∂W !Sk�1, which, by the previous, shows that j 7! j/|j| defines aisomorphism bewteen

⇥

∂W; Rk\{0}⇤

and⇥

∂W; Sk�1⇤.

3.5.21 A similar correspondence holds for mappings f : W! Rk andthe homotopy types of pairs. The considerations in Subsection 3.5yield a homomorphism

bk :h

�

W, ∂W�

;�

Rk, Rk\{p}�

i

! Pontk�W, ∂W; Rk, Rk\{p}�

.

The homotopy type can be described as homotopy types of therestrictions to the boundary.

3.5.22 Lemma. The homotopy classes of mappings f : (W, ∂W) !(Rk, Rk\{p}) and y : ∂W! Sk�1 are isomorphic;

h

�

W, ∂W�

;�

Rk, Rk\{p}�

i

⇠=⇥

∂W; Sk�1⇤,

where the isomorphism is given by [j] 7!⇥

j/|j|⇤

, with j = f |∂W.


Proof: Clearly, if two mappings f , f 0 : (W, ∂W) ! (Rk, Rk\{p})are homotopic via F, then Y = F/|F|, with F = F|

∂W, defines ahomotopy between the boundary restrictions.

On the other if two mappings j, j

0 : ∂W! Rk\{p} are homotopicvia F, then by Tietze’s Extension Theorem there exists an exten-sion F to W, which defines a homotopy between mappings of pairs.Combining this with Lemma 3.5.20 then proves the lemma.

3.6 Framed cobordism classes and homotopy types

The objective of this sectionis to characterize homotopy types byframed cobordism classes, which turn can be equipped with analgebraic structure (Subsection 3.6). The homotopy classes will thusbe classified by algebraic invariant. In the case k = n this recoversthe Brouwer degree. The first result concerns the relation betweenframed cobordism classes and Pontryagin manifolds. After that thePontryagin manifolds will be linked to homotopy classes.

Framed cobordism classes as Pontryagin manifolds

3.6.1 Let M = W ⇢ Rn a bounded domain in Rn, with boundary∂W. As open set, W is a smooth n-dimensional manifold. Considersmooth mappings

j : ∂W! Rk\{0},

as defined in the previous subsection. The following lemma showsthat a framed submanifold in ∂W is always a Pontryagin manifold forsome mapping j.

3.6.2 Theorem. Any framed submanifold (N, v) in ∂W is a Pontryaginmanifold (j

�1(p), j

⇤y) for some smooth mapping j : ∂W ! Rk\{0}

and regular value p 2 Rk\{0}. Consequently,

Pontk�1(∂W; Rk\{0}) ⇠= Pk�1(∂W).

Proof: Consider the mapping h : N ⇥Rk ! Rn, defined by

h(x, y) := x + y1v1(x) + · · · ykvk(x), x 2 N, y = (y1, · · · , yk) 2 Rk.

The set N ⇥ {0} plays a special role, and the derivative is given by

dh(x, 0)(x, h) = t

1(x)x1 + · · ·+ t

n�kxn�k + v1(x)h1 + · · · vk(x)hk,

where the vectors t

1(x), · · · , t

n�k(x) span Tx N. This shows thatdh(x, 0) is invertible for all x 2 N, and therefore, h maps Ux ⇥ B

e

(0),


Ux 3 x an open neighborhood of x, diffeomorphically onto an openset Vx ⇢ Rn. By the compact of N, e > 0 can be chosen uniformly forall x 2 N. Denote the image of N ⇥ B

e

(0) by V = [xVx. It remainsto show that h(x, y) = h(x0, y0) if and only if (x, y) = (x0, y0). Assume(x, y) 6= (x0, y0), then since y, y0 2 B

e

(0), it holds that |x � x0| < Ce,uniformly for x 2 N. If e > 0 is sufficiently small local behavior of hyields a contradiction.

Compose h with the mapping yi 7! e

2yi/(e2 � |y|2), and denoteit by eh. Then eh maps N ⇥Rk diffeomorphically onto V. When e > 0is sufficiently small, then V ⇢ W ⇢ Rn. The relation p(h(x, y)) = ydefines a smooth mapping p : V ⇢ W ! Rk. It holds that 0 is aregular value and p

�1(0) = N.Let sq : Rk ! Sk\{q} be the inverse stereographic projection, then

s(y) = sq�

x/w(|x|)�

, with supp(w) ⇢ B1(0), maps smoothly from Rk

to Sk. Now define j : W ! Sk as s � p for points in V and j = q forpoints in W\V.

3.6.3 Consider smooth mappings

g : (W, ∂W)! (Sk, q), q 2 Sk,

where Sk ⇢ Rk+1 is the unit sphere in Rk+1. Then the framed cobor-dism classes in Pk(W, ∂W) are related to Pontryagin manifolds inPontk�W, ∂W; Sk, q

�

.

3.6.4 Theorem. Any framed submanifold (N, v) in W is a Pontryaginmanifold (g�1(p, q), g⇤y) for some smooth mapping g : (W, ∂W) !(Sk, q), q 2 Sk and regular value p 2 Rk\{0}. Consequently,

Pontk�W, ∂W; Sk, q� ⇠= Pk(W, ∂W).

Proof: The proof follows along the same lines as the proof ofTheorem 3.6.2.

Pontryagin manifolds and homotopy types

The final step towards the main theorem is proving a correspondencebetween the classes of Pontryagin manifolds and homotopy types.

3.6.5 Theorem. Two mappings j, j

0 : ∂W ! Rk\{0} are smoothlyhomotopic if and only if their associated Pontryagin manifolds areframed cobordant. In other words

Pontk�1(∂W; Rk\{0}) ⇠=⇥

∂W; Rk\{0}⇤

.


3.6.6 Corollary. For smooth mappings j : ∂W! Rk\{0} it holds that

⇥

∂W; Rk\{0}⇤ ⇠= Pk�1(∂W).

3.6.7 Corollary. For mappings f : (W, ∂W) ! (Rk, Rk\{p}) it holdsthat

h

�

W, ∂W�

;�

Rk, Rk\{p}�

i

⇠= Pk�1(∂W).

One can even prove that Pontk�W, ∂W; Rk, Rk\{p}�

is in fact iso-

morphic to Pontk�1(∂W; Rk\{0}) and thus toh

�

W, ∂W�

;�

Rk, Rk\{p}�

i

.

The degree isomorphism for n-framed submanifolds

3.6.8 In the case of differentiable mapping the cobordism classes⇥

(N, v)⇤

2 Pk(W) provide a generalization of the C1-mappingdegree as introduced in Section 1.2. To compare, set n = k, andconsider the framed cobordism classes Pn(W). In that case a framedsubmanifold (N, f ⇤y) consists of finitely many points xj 2 W, andd f (x) : Rn ! Rn is an isomorphism for all x 2 N = f�1(p).Consequently, the determinant det( f ⇤y) is either positive or negative.For two framed cobordant submanifolds (N, f ⇤y) and (N0, f ⇤y) onlypoints of the same sign in N and N0 respectively, can be connectedby a component of a cobordism M, or points in N (or in N0) withopposite signs.

3.6.9 Example. Consider the homotopy F(x, t) = x2 + t2 � 14 between

the maps f (x) = x2 � 14 and f 0(x) = x2 + 3

4 . The submanifoldN consists of the points x = ± 1

2 , and d f (x) = 2x. By choosingy = 1 2 R as the positive basis, the associated framing for N is givenby: f ⇤(1) = 1 at x = 1

2 , and f ⇤(1) = �1 at x = � 12 , see Figure 3.4.

The framing f ⇤(1) is given by the Jacobian.Now consider the cobordism M = F�1(0). The tangent space is

given by the relation dF(x, t)x = 2xx1 + 2tx2 = 0, and

T(x,t)M = span

(

�tx

!)

, (T(x,t)M)? = span

(

xt

!)

.

It holds that dF(x, t) is an isomorphism between (T(x,t)M)? and R,and The calculation at the points (x, t) 2 M

dF(x, t)l

xt

!

= 2l(x2 + t2) =l

2= 1,


yields (dF(x, t))�1(1) =

2x2t

!

. The first component of (dF(x, 0))�1(1)

is equal to f ⇤(1) at x = ± 12 , which proves that w = (dF(x, t))�1(1) is

the right framing and M a framed cobordism. This example showsthat opposite signs can be connected in M.

hbt

Figure 3.4: The semi-circle is a framedcobordism between N and N0 = ?. Theframing given by rF.

3.6.10 Lemma. Let y = {e1, · · · , en} be the standard basis for Rn, thenf ⇤y = d f (x), and det( f ⇤y) = J f (x).

Proof: By definition d f (x) f ⇤y = Id, and since d f (x) is invertible, itfollows that f ⇤y = (d f (x))�1

y. Clearly, det( f ⇤y) = J f (x).

3.6.11 Theorem. Let n � 2, then the set of n-framed cobordismclasses Pn(W) is isomorphic to Z. In particular, deg(N, f ⇤y) =

ÂN sign det( f ⇤y) can be regarded as an isomorphism

deg : Pn(W)! Z,

in the sense that deg(N, f ⇤y) = deg( f , W, p), where N = f�1(p).

Proof: By Lemma 3.6.10 deg(N, f ⇤y) = deg( f , W, p), and the mapdeg is onto by virtue of Corollary 3.1.11 and the fact that deg( f , W, p)is given by the degree of the restriction of f to ∂W. This proves thatthe map deg is onto Z.

Injectivity can be proved as follows. Given a mapping f : W! Rn

of degree m � 0 can be chosen to have exactly m zeroes of positiveorientation. Let g : W ! Rn any other admissible function withdeg(g, W, p) = m. By the definition of degree g has m + m0 positivelyoriented zeroes and m0 negatively oriented zeroes, see Figure 3.5.

Figure 3.5: Connecting the m positivelyoriented zeroes across and the con-nect the remaining m0 positively andnegatively oriented zeroes of g.

Now connects the m positively oriented zeroes of f with m pos-itively oriented zeroes of f 0 via m smooth, non-intersection curves.The remaining m0 positively and negatively oriented zeroes of g arepairwise connected by m0 smooth, non-intersecting curves, which alsoavoid the first m curves. These m + m0 non-intersecting, smooth curveform M = F�1(p), for some smooth homotopy F, and is a framedcobordism between N = f�1(p) and N0 = g�1(p). The case m 0follows along the smae lines.

For n = 1, there are three framed cobordism classes, characterizedby the functions f (x) = x, f (x) = �x, and f (x) = 1 respectively, anddeg is an isomorphism from P1(W) to the set {�1, 0, 1}.

3.6.12 Exercise. Prove the above statement for n = 1.

3.6.13 Using the characterization of the mapping degree in terms ofthe boundary restriction, Theorem 3.6.11 shows that if two mappingsy, y

0 : ∂W ! Sn�1 have the same the degree, then their extensionto W yield the same framed cobordism class, and therefore the samehomotopy type. This proves Lemma 3.1.7.


The group structure of framed cobordism classes and cohomotopy groups

3.6.14 Define the group operation on Pk�1 and explain the link tocohomotopy groups. In the case of k = n the degree is rediscovered.If computable these algebraic invariants are useful for studying theextension problem. The Pontryagin manifolds and framed cobor-disms are a differentiable tool for studying cohomotopy. Comparethe proof in the previous subsection and the cobordism proof of thehomotopy invariance of degree.

3.7 Algebraic structures and obstruction theory

3.7.1 Remark. Recall from Example 3.1.13 that f is a nowhere vanish-ing mapping while [j] = [y] 6= 0, which indicates that connectivitycannot be removed. Removing the latter leads to a more complicatedquestion.

The main result for general domains W and any n and k is weakerthan Theorem ?? as was already suggested in Example 3.3.2. The lastpart of this theorem can also be proved using the following theorem(see Corollary 3.7.5 and Theorem ??).

3.7.2 Theorem. A continuous mapping j : ∂W ⇢ Rn ! Rk\{0} isinessential if and only if [y] = 0.


Proof of Theorem 3.3.11. If [y] 6= 0, then by Theorem 3.1.10 deg(y) =deg(j, ∂W, Rn\{0}) = deg( f , W, 0) 6= 0, for any continuous extensionf to W. Since deg( f , W, 0) 6= 0, the equation f (x) = 0 has a zero in W,and therefore j is essential with respect to W.

Let j is essential with respect to W. Suppose [y] = 0, then by The-orem 3.7.2 the mapping j is inessential, which means in particularthat there exists a continuous extension f to W, such that f 6= 0, acontradiction. Therefore, [y] 6= 0.

Theorem 3.7.2 can also be formulated in terms of j being essential.If j : ∂W ⇢ Rn ! Rk\{0} is not inessential, then j is essential withrespect to Rn\∂W.

3.7.3 Corollary. A continuous mapping j : ∂W ⇢ Rn ! Rk\{0} isessential with respect to Rn\∂W, or simply essential, if and only if[y] 6= 0.

3.7.4 The question of characterizing mappings that are essentialwith respect to W, like in Theorems 3.3.11 and ?? does not havean immediate answer as Example 3.3.2 shows. As a consequence


of Theorem 3.7.2, part of this question can be answered as in theTheorems 3.3.11 and ?? and holds in general.

3.7.5 Corollary. If a continuous mapping j : ∂W ⇢ Rn ! Rk\{0} isessential with respct to W, then [y] 6= 0.

Proof: See the proof of Theorem ??.

4The Leray-Schauder degree

A natural question to ask is if there exists a degree theory for map-pings on infinite dimensional spaces? The answer to this question isnot so straightforward as the following example will show. Considerthe space of sequences defined by `2 := {x = (x1, x2, · · · ) | Âi x2

i <

•}. The space `2 ⇠= R• has a natural norm kxk2`2 := Âi x2

i and innerproduct hx, yi := Âi xiyi and is a complete normed space, called aHilbert space. Let B• = {x 2 `2 | kxk`2 1} and define a mapping fas follows:

f (x) =⇣

q

1� kxk2`2 , x1, x2, · · ·

⌘

,

which is a continuous mapping from B• to ∂B• =: S•. The mappingf has no fixed points in B•. Indeed, for x 2 S• it holds that f (x) =(0, x1, x2, · · · ) 6= x. On the other hand f (B•) ⇢ B• and the mappingf satisfies the requirements of the Brouwer fixed point theorem,which therefore does not holds for continuous mappings of on `2. If adegree theory for continuous maps on `2 exists so does the Brouwerfixed point theorem. This is already an indication that a degreetheory for continuous mappings on infinite dimensional spaces doesnot exist. More precisely, following the proof of the Brouwer fixedpoint theorem, define r(x) = x + l�(x)( f (x)� x), where l�(x) 0.Since r|S• = id the homotopy property of the degree yields that

deg(r, B•, 0) = deg(id, B•, 0) = 1,

which implies that r(x) = 0 has a non-trivial solution. On the otherhand, since f has no fixed points it holds that r(B•) = S•, which is acontradiction with the existence of a degree theory,

Another consequence of the above construction is that any twomappings g1, g2 from S• to S• are homotopic. Indeed, h(x, t) =

r�

(1� t)g1(x) + tg2(x)�

1 gives a homotopy bewteen g1 and g2. If 1 One easily verifies that k(1� t)g1 +tg2k2

`2 1 provided kgik2`2 = 1, i = 1, 2.g1 = id then deg(g1) = 1 and if g2 = f then deg(g2) = 0, which

implies degree cannot characterize homotopy classes. In particularS• is contractible. This is far from the situation in finite dimensions.


The goal of this chapter is to find an adequate degree theory forthe infinite dimensional setting and to extend the theory of homotopyclasses of maps from Rn to Rn to homotopy classes of maps oninfinite dimensional spaces.

4.1 Notation

4.1.1 The infinite dimensional spaces under consideration in thischapter are normed linear vector spaces are their subsets. Extensionswill be discussed in the next chapter. Let X be a (real) linear vectorspace. On X we define a norm k · kX, or k · k for short which satisfiesthe following hypotheses:

(i) kx + yk kxk+ kyk, for all x, y 2 X,

(ii) klxk = |l|kxk, for all x 2 X, and for all l 2 R,

(iii) kxk = 0 if and only if x = 0.

If there is no ambiguity about the space involved we simply write k · k.The combination (X, k · k) is called a normed linear vector space.

If in addition X is complete it is called a Banach space. A normedlinear space is complete if every Cauchy sequence has a limit in X;{xn} ⇢ X, with kxn � xmk ! 0, as n, m! •, implies that there existsa x 2 X such that kxn � xk ! 0, as n! •. Normed vector spaces andBanach spaces are examples of metric and complete metric spacesrespectively, where the metric is given by

d(x, y) := kx� yk.

For the remainder of this chapter X is assumed to be complete, i.e. aBanach space.

As in the previous chapter W ⇢ X denotes an open and boundedsubset of X. The closure in X is denoted by W and the boundary isgiven by ∂W = W\W.

Continuity

Throughout this section X, and Y are (real) Banach spaces withnorms k · kX and k · kY respectively. We omit the subscripts is there isno ambiguity about the notation.

4.1.2 Definition. A mapping f : X ! Y is continuous if xn ! x (inX) implies that f (xn) ! f (x) (in Y). A map is uniformly continuouson X, if for for any e > 0 there exists a d

e

> 0 such that kx� yk < d

implies that k f (x)� f (y)k < e. The latter can also be defined withrespect to a closed subset A ⇢ X.

the leray-schauder degree 87

4.1.3 A continuous function f : X ! Y is bounded if f (W) ⇢ Y isbounded for any bounded subset W ⇢ X. Continuous mappings onRn are necessarily bounded, i.e. bounded sets in Rn are mapped tobounded set under f . This is however not the case in general Banachspaces.

4.1.4 Exercise. Give an example of continuous map between Banach spacesthat is not bounded.

4.1.5 Lemma. A uniformly continuous map is bounded.

Proof: We need to show that for any bounded set A ⇢ X the imagef (A) ⇢ Y is also bounded. Choose R > 0 such that A ⇢ BR(0),and let n > 2R

d

. Then for any two points x, y 2 A it holds thatkx� yk 2R, and one can define the line-segment xt = x + t(y� x),t 2 [0, 1], in BR(0). For ti = i

n we obtain point xti ⇢ BR(0), withkxti � xti+1k < d, by the choice of n. Since f is uniformly continuousit follows that k f (xti ) � f (xti+1)k < e, for all i. From the triangleinequality we then get

k f (x)� f (y)k Âik f (xti )� f (xti+1)k < ne,

which proves the boundedness of f .

4.1.6 The space of continuous mappings from X to Y is denotedby C0(X, Y) and the bounded continuous maps by C0

b(X, Y). Formappings on bounded domains f : W ⇢ X ! Y we write f 2C0(W; Y) or C0(W) in the case that X = Y. The space of boundedcontinuous functions on W is denoted by C0

b(W; Y), or C0b(W). On C0

bthe following norm is defined

k f kC0b

:= supx2Wk f (x)kY,

which makes C0b(W) a normed linear space.

Differentiability

4.1.7 Definition. A mapping f 2 C(X, Y) is called Fréchet differ-entiable at a point x0 2 X, if there exists a bounded linear mapA : X ! Y such that

k f (x)� f (x0)� A(x� x0)k = o(kx� x0k),

in a neighborhood N of x0.

We use the notation A = f 0(x0) for the Fréchet derivative. If themap x 7! f 0(x) is continuous as a map from X to B(X, Y), then f is ofclass C1; notation f 2 C1(X, Y).


4.1.8 Definition. A mapping f 2 C(X, Y) is called Gateaux differen-tiable in the direction h 2 X, at a point x0, if there exists a y 2 Y suchthat

limt!0k f (x0 + th)� f (x0)� tyk = 0,

with x0 + th defined in a neighborhood N of x0.

The Gateaux derivative at at point is usually denoted by d f (x0, h)and is commonly referred to as the directional derivative in thedirection h. In Rn is notion is known as partial derivative and is aweaker notion of differentiability.

4.1.9 Exercise. Give an example of a function that is Gateaux differentiable ina point, but is not Fréchet differentiable.

4.1.10 For functions on Rn there is an important relation betweenthe two notions of differentiability, i.e. the partial derivatives existand are continuous, then the function is differentiable. In the Banachspace setting the same result holds.

4.1.11 Theorem. If f 2 C(X, Y) is Fréchet differentiable at a pointx0, then f is Gateaux differentiable at x0. Conversely, if a functionf 2 C(X, Y) is Gateaux differentiable at x0 for all directions h 2 X,and the mapping x 7! d f (x, ·) 2 B(X, Y) is continuous at x0, then f isFréchet differentiable at x0.

In the latter case we write, by the linearity of d f in h, that d f (x, h) =d f (x0)h = f 0(x)h.

Proof: The first claim of the theorem simply follows from thedefinition of the Fréchet derivative. For the converse we argue asfollows. By assumption the map f (x0 + th) is differentiable in t(sufficiently small), and d f (x0 + th, h) = d f (x0 + th)h is continuous int. Therefore,

f (x0 + h)� f (x) =Z 1

0d f (x0 + th)hdt.

Using this identity we find the following estimate:

k f (x0 + h)� f (x0)� d f (x0)hk = kZ 1

0

�

d f (x0 + th)� d f (x0), h�

dtk

Z 1

0k�

d f (x0 + th)� d f (x0), h�

kdt

Z 1

0kd f (x0 + th)� d f (x0)kB(X,Y)khkXdt

= o(khk),

by the continuity of d f (x0 + th).


4.1.12 The notions differentiability can be further extended to higherderivatives and we leave this to the reader. Furthermore, one caneasily prove various basic properties of derivatives:

4.1.13 Exercise. Prove the chain rule: Let f : X ! Y, g : Y ! Z, and f andg are differentiable at x0 and y0 = f (x0) respectively. Then

�

g( f (x0))�0

=

g0( f (x0)) · f 0(x0).

4.1.14 Exercise. Prove the product rule: Let f : X ! R, and g : X ! Ybe differentiable at x0, then f · g is differentiable at x0, and ( f · g)0(x0)h =

f 0(x0)h · g(x0) + f (x0) · g0(x0)h.

4.1.15 A value p 2 Y is called a regular value if f 0(x) 2 B(X, Y) issurjective for all x 2 f�1(p) and p is singular, or critical if it is nota regular value. A point x 2 X is called regular if f 0(x) is surjectiveand otherwise a point is called singular, or critical point.

Fredholm mappings and proper mappings

Let f be a C2 proper Fredholm mapping of index 0.Important! Mention the Kuiper result

4.2 Compact and finite rank maps

4.2.1 An important subspace of continuous mappings are the com-pact mappings from f : X ! X. A mapping f : X ! X is compactif f (W0) is compact for any bounded subset W0 ⇢ X. Compactmappings are bounded since f (W0) is bounded for any boundedset W0 ⇢ X. The space of compact mappings on X is denoted byK(X). This definition of compact mappings also holds for continuousmappings on subsets of X.

4.2.2 Definition. A continuous map f : W ⇢ X ! X is called compactif f (W) is compact.

In particular for any W0 ⇢ W it holds that f (W0) is compact sincef (W0) ⇢ f (W). The space of compact mappings on W is denoted byK(W) ⇢ C0

b(W). Compact maps are examples of mappings which areclose to mappings in finite dimensional Euclidean space.

4.2.3 The following lemma explains how compact maps can be ap-proximated by maps of finite rank. To be more precise, a finite rankmap is a mapping whose range is contained in a finite dimensionalsubspace of X. The subspace of finite ranks mappings is denoted byF(W; X) ⇢ C0(W; X).


4.2.4 Lemma. Let f 2 K(W), then for any e > 0, there exists a finiteranf map f e 2 F(W) such that k f � f ekC0

b< e. i.e. f e 2 F(W) \ C0

b(W).

Proof: We follow the proof given by Berger2. Since f (W) is compact2 Melvin S. Berger. Nonlinearity andfunctional analysis. Academic Press[Harcourt Brace Jovanovich Publishers],New York, 1977. Lectures on nonlinearproblems in mathematical analysis,Pure and Applied Mathematics

it can be covered by finitely many balls Be

(xi), with xi 2 f (W).Define

µ

i(x) =l

i(x)Âj l

j(x),

where l

i(x) = max(0, e � k f (x) � xik). This maximum is zerowhenever f (x) 62 B

e

(xi) and therefore µ

i(x) = 0, unless k f (x)� xik <e. Set

f e(x) = Âi

µ

i(x) · xi.

Now f e(W) ⇢ span(xi). As for the approximation we obtain

k f � f ekC0b=�

� f �Âi

µ

i · xi��

C0b=�

�

�Âi

µ

i · ( f � xi)�

�

�

C0b

,

using the fact that Âi µ

i = 1. By construction kµi(x)( f (x)� xi)k <

µ

i(x)e and thus

k f � f ekC0b

= supx2W

�

�

�Âi

µ

i(x)( f (x)� xi)�

�

�

supx2W

Âi

�

�

�

µ

i(x)( f (x)� xi)�

�

�

< Âi

µ

i(x)e = e,


The converse of this lemma can be formulated as follows:

4.2.5 Lemma.

3 For any sequence { f e} ⇢ F(W) \ C0b(W), with f e ! f 3 Melvin S. Berger. Nonlinearity and

functional analysis. Academic Press[Harcourt Brace Jovanovich Publishers],New York, 1977. Lectures on nonlinearproblems in mathematical analysis,Pure and Applied Mathematics

in C0b(W), as e! 0, it holds that f 2 K(W).

4.2.6 Finally, using the above characterization of compact mappings,it is worth mentioning a version of Tietze’s extension theorem forcompact mappings.

4.2.7 Proposition. Any compact mapping f 2 K(W) extends to acompact mapping f 2 K(X).

4.2.8 Proposition. Let f 2 K(W) \ C1(W), then for any x 2 W thelinear operator f 0(x) : X ! X is compact.



4.3 Definition of the Leray-Schauder degree

The problem with degree theory in infinite dimensional spaces isthat homotopy invariance, a basic property of the degree, preventsthe existence of a non-trivial degree theory (compare the axioms fordegree theory, Section 2.2). We can alter the notion of homotopy in-variance in order to build a degree theory, or limit the types of mapsfor which a degree is well-defined. The Leray-Schauder degree doesboth by considering specific types of mappings, namely mappings ofthe form

f = Id� k,

where Id is the identity map on X and k 2 K(W). Homotopies areconsidered in the same class. Denote the function class by C0

Id(W) =

{ f = Id� k | k 2 K(W)} and by C0Id(X) for mapping defined on X.

These classes are affine subspaces of C0b(W) and C0

b(X) respectively.

4.3.1 Lemma. Let W ⇢ X be a bounded set. Then, ∂W is a closed andbounded set in X. Due to the specific form of f the set f (∂W) is alsoclosed and bounded.

Proof: Indeed, let xn 2 ∂W such that f (xn)! x⇤. Since k is compactwe have that k(xn) has a convergent subsequence and k(xnk )! x⇤⇤.4 4 This uses the boundedness of ∂W.

Therefore, xnk = f (xnk ) + k(xnk ) ! x⇤ + x⇤⇤ = x 2 ∂W,5 which, by 5 Due to the closedness of ∂W.

continuity, implies that x⇤ = x� k(x) = f (x), proving the closednessof f (∂W). For the boundedness we argue as follows: k(∂W) is pre-compact and thus f (∂W) is bounded.

4.3.2 Lemma. For p 62 f (∂W) we have

infx2∂W

kp� f (x)k = infy2 f (∂W)

kp� yk = d > 0.

Proof: Indeed, if not, there exists a minimizing sequence xn 2 ∂Wsuch that f (xn) ! p. By the closedness of f (∂W) then p 2 f (∂W), acontradiction.

4.3.3 Definition. Let f be a continuous map of the form f = Id� k,with k 2 K(W), and let p 62 f (∂W). Let ke be a finite rank perturbationwith kk � kekC0

b< e and e < d/2 (d as given above) and with

ke(W) ⇢ Ye ⇢ X (subspace). Then for any finite dimensionalsubspace Xe containing both Ye and p, define the Leray-Schauderdegree as

degLS( f , W, p) := deg( f e, W \ Xe, p),

where f e = id� ke.


4.3.4 The remainder of this section is devoted to showing that theLeray-Schauder degree is well-defined. By the choice of domainW \ Xe it follows that f e : W \ Xe ! Xe and p 2 Xe. Moreover, ifp 62 f (∂W), Lemma 4.3.2 implies

infx2∂W

kp� f e(x)k = infx2∂W

kp� x + ke(x)k

� infx2∂W

kp� x + k(x)k � d/2 > d/2 > 0,

which proves that p 62 f e(∂W) and in particular p 62 f e(∂W \ Xe).6 6 For a linear subspace Xe ⇢ X it holdsthat ∂(W \ Xe) = ∂W \ Xe.Consequently, the degree deg( f e, W \ Xe, p) is well-defined. We show

now that the definition is independent of the chosen subspace Xe andapproximation ke.

4.3.5 Let y = Id� f, where f : W ⇢ Rn ! F ⇢ Rn, and F a linearsubspace of dimension m n. The restriction to of y to F is denotedby yF : W \ F ! F. We define deg(yF, W \ F, p) as the degree ofq � y � q�1, where q(F) = Rm � 0 is a linear change of variables, see1.3.7 and 2.2.17.

4.3.6 Lemma. Let p 2 F \ y(∂W), then deg(y, W, p) = deg(yF, W \F, p).

Proof: Let q(x) = y be a linear change of variables on Rn such thatq(F) = Rm � 0. The transformed function q � y � q�1 = Id� q � f � q�1

is a again denoted by y = Id � f, and we prove the lemma forF = Rm � 0. Identify Rm � 0 with Rm. On Rn we use coordinates(x, h), with x 2 Rm and h 2 Rn�m. For p 2 Rm we have thatx 2 y

�1(p) satisfies: x = p + f(x) 2 Rm and thus x 2 W \Rm. Thisshows that y

�1(p) = y

�1F (p).

Figure 4.1: Zeroes for p-values re-stricted to the subspace F.

In order to compute and compare the degree we assume withoutloss of generality that y is C1 and p is regular. The mapping y isgiven by y(x, h) = (x + f(x, h), h) and the derivative at zeroesx = (x, 0) is

y

0(x, h) =

IRm + dx

f(x, 0) dh

f(x, 0)0 IRn�m

!

=

y

0F(x, 0) ⇤

0 IRn�m

!

This expression shows that the signs of the determinants are the ze-roes are the same for both y

0(x, 0) and y

0F(x, 0), and since y

�1(p) =y

�1F (p) we obtain the desired identity.

4.3.7 Let W ⇢ X be a bounded set and let f = Id� f, where f : W ⇢X ! E ⇢ X, with E ⇢ X a finite dimensional subspace of X. Supposep 62 f (∂W), then define F ⇢ X to be a finite dimensional subspacethat contains both E and p. The restriction fF = f |W\F = Id� f|W\Fis a mapping fF : W \ F ⇢ F ! F. We may think of F as Rn via a


linear isomorphism. The degree is then defined for the transformedmapping and the degree does not depend on the particular choice ofthe linear isomorphism. We have

4.3.8 Lemma. The degree deg( fF, W \ F, p) is independent of thechoice of subspace F ⇢ X.

Proof: Let eF be another subspace that contains both E and p, thenalso F \ eF suffices. Note that f(W) ⇢ F \ eF and p 2 (F \ eF) \ f (∂W).Then by Lemma 4.3.6 we have that

deg( fF, W \ F, p) = deg( f |W\F\eF, W \ F \ eF, p).

The same identity holds for the degree deg( feF, W \ eF, p), which proves

the lemma.

4.3.9 We return to the mapping f e : W ⇢ X ! X, with p 62 f e(∂W),which follows from the assumptions on ke with e d/2.

4.3.10 Lemma. Let Xe, eXe ⇢ X be finite dimensional subspacessuch that Ye ⇢ Xe, eXe and p 2 Xe, eXe. Then deg( f e, W \ eXe, p) =

deg( f e, W \ Xe, p).

Proof: Apply Lemma 4.3.8 to f = ke, fF = f e and F = Xe.

4.3.11 The final step is to show that Definition 4.3.3 is independent ofthe chosen approximation ke.

4.3.12 Lemma. Let ke and eke both be finite rank approximations for kwith kk� kekC0

b< e , kk� ekekC0

b< e and e < d/2. Then

deg( f e, W \ Xe, p) = deg( ef e, W \ eXe, p),

for any subspaces Xe and eXe containing both p and the ranges of ke

and eke respectively.

Proof: Let Ze ⇢ X be a finite dimensional linear subspace contain-ing both Xe and eXe. From Lemma 4.3.10 it follows that

deg( f e, W \ Xe, p) = deg( f e, W \ Ze, p),

deg( ef e, W \ eXe, p) = deg( ef e, W \ Ze, p).

Consider the compact homotopy ke

t = (1� t)ke + teke, which yields ahomotopy f e

t = Id� ke

t . By the choices of ke

and eke

, the homotopy ke

tis a proper homotopy.7 By Property (ii) of Section 2.2 then deg( f e, W \ 7 Estimate k f e

t (x)� pk, x 2 ∂W by sametoken as in 4.3.4.Ze, p) = deg( ef e, W \ Ze, p) which then proves that

deg( f e, W \ Xe, p) = deg( ef e, W \ eXe, p).

The Leray-Schauder degree is well-defined.


4.4 Properties of the Leray-Schauder degree

The properties of the (Brouwer) degree listed in Section 2.2 holdequally well for the Leray-Schauder degree. For this list we refer toSection 2.2. For proving these properties for the Leray-Schauder de-gree one has to make sure that approximations are constructed suchthat the conditions for the Brouwer degree are met. An importantproperty is the validity property.

4.4.1 Proposition. If p 62 f (W), then degLS( f , W, p) = 0.

Proof: By the same token as Lemma 4.3.1 the set f (W) is closed andsince p 62 f (W) we have that infy2 f (W) kp � yk � d > 0. Let f e bean approximation for f as described in the definition of the Leray-Schauder degree (Definition 4.3.3) and with Xe such that f e(W) ⇢ Xe

and p 2 Xe. If we choose e > 0 small enough, i.e. e d/2, then forall k f � f ekCb

0< e it holds that infy2 f e(W\Xe) kp� yk � infy2 f (W) kp�

yk � d/2 � d/2 > 0 and p 62 f e(W \ Xe). From the properties ofthe Brouwer degree we now have that deg( f e, W \ Xe, p) = 0. As animmediate consequence it now holds that

degLS( f , W, p) 6= 0,

implies that f�1(p) 6= ?. Indeed, if f�1(p) = ?, then dLS( f , W, p) =0, a contradiction.

4.4.2 Another way to treat the Leray-Schauder degree is to show thatthe axioms of a degree theory are satisfied and derive the propertiesfrom that. We start with the Leray-Schauder degree theory andexplain the axioms in a more general context later on.

Consider triples ( f , W, p) with W ⇢ X a bounded and open set in aBanach space (X, k · k), f 2 C0

Id(W) and p 2 X\ f (∂W). For such tripleswe assign the Leray-Schauder degree

( f , W, p) 7! degLS( f , W, p).

4.4.3 Theorem. For Leray-Schauder degree we have the followingproperties:

(A1) if p 2 W, then degLS(Id, W, p) = 1;

(A2) for W1, W2 ⇢ W, disjoint open subsets of W, and p 62 f�

W\(W1 [W2)

�

, it holds that degLS( f , W, p) = degLS( f , W1, p)+degLS( f , W2, p);(A3) for any continuous paths t 7! ft = Id� kt, kt 2 K(W) and t 7! pt,

with pt 62 ft(∂W), it holds that degLS( ft, W, pt) is independent oft 2 [0, 1];

and degLS is called a degree theory.



As in the case of the Brouwer degree the essential properties ofthe Leray-Schauder degree follow from (A1)-(A3). In Section 2.2 wechoose to prove these properties of the degree using only the axioms.Therefore most properties hold also for the Leray-Schauder degreewith the same proofs. There are some differences though. Let us gothrough the list in Section 2.2 and point out the differences.

4.4.4 (Validity of the degree, Property 2.2.5) If p 62 f (W), thendegLS( f , W, p) = 0. Conversely, if degLS( f , W, p) 6= 0, then thereexists a x 2 W, such that f (x) = p.

4.4.5 (Continuity of the degree, Property 2.2.6) The degree degLS( f , W, p)is continuous in f = Id� k, i.e. there exists a d = d(p, f ) > 0, such thatfor all g = Id� k satisfying kk� kkC0

b< d, it holds that p 62 g(∂W) and

deg(g, W, p) = deg( f , W, p).

4.4.6 (Dependence on path components, Property 2.2.7) The degreeonly depends on the path components D ⇢ X\ f (∂W), i.e. for anytwo points p, q 2 D ⇢ X\ f (∂W) it holds that degLS( f , W, p) =

degLS( f , W, q). For any path component D ⇢ X\ f (∂W) this justifiesthe notation degLS( f , W, D).

4.4.7 (Translation invariance, Property 2.2.8) The degree is invariantunder translation, i.e. for any q 2 X it holds that degLS( f � q, W, p�q) = degLS( f , W, p).

4.4.8 (Excision, Property 2.2.9) Let L ⇢ W be a closed subset in Wand p 62 f (L). Then, degLS( f , W, p) = degLS( f , W\L, p).

4.4.9 (Additivity, Property 2.2.10) Suppose that Wi ⇢ W, i =

1, · · · , k, are disjoint open subsets of W, and p 62 f�

W\([iWi)�

, thendegLS( f , W, p) = Âi degLS( f , Wi, p).

As for the Brouwer degree the Leray-Schauder degree can alsobe defined in the C1-case. Let p 2 X\ f (∂W) be a regular valuethen by the Inverse Function Theorem the set f�1(p) consists ofisolated points. Let xn 2 f�1(p), then xn = p + k(xn) which hasa convergent subsequence by the compactness of k and thereforef�1(p) is compact. Combined with isolation this yields that f�1(p)is a finite set. Using the excision and additivity Properties 4.4.8and 4.4.9 we derive that degLS( f , W, p) = Âj degLS( f , N

e

(xj), p),xj 2 f�1(p). It holds that

degLS( f , Ne

(xj), p) = degLS( f , Be

0(xj), p) =: i( f , xj),

for all for any 0 < e

0 sufficiently small. The integer i( f , xj) is calledthe index of an isolated zero. For the Brouwer degree the index is


given by the sign of the Jacobian at xj. Since for each x 2 f�1(p)the operator f 0(x) = I � k0(x) 2 B(X)8 is invertible de index given 8 The identity operator x 7! x is

denoted by Id and it linearization by I.by the following spectral formula. Let l > 1 be an eigenvalue ofA = I � k0(x), i.e. Ax = lx, X 3 x 6= 0, then

nl

= dim

•[

k=1ker(lI � A)k

!

< •,

by the compactness of k0(x).

4.4.10 Lemma. Let x 2 W be a regular point of f = C0Id(W), then

i( f , x) = (�1)b,

where b = Âl>1 n

l

.


4.4.11 Remark. The formula for the index can also be given for theBrouwer degree because (�1)b = sign(J f (x)) is the finite dimen-sional case. The above consideration also show how the Leray-Schauder degree is defined axiomatically and leads to a similarexpression as a sum of indices in the C1-case. The latter can also beused as a first definition.

4.5 Compact homotopies

4.5.1 Corollary 3.2.5 relates the existence of a zero of any extensionf : B1(0) ! Rk of f |

∂B1 to the homotopy type of the y = f /| f | :∂B1 = Sn�1 ! Sk�1, i.e. a non-trivial zero exists if and only if[y] 2 pn�1(Sk�1) is non-trivial. The example in the introductionshowed that the infinite dimensional version of this result doesnot hold true without an appropriate notion of homotopy type. Amapping f : ∂B1 ! Rk\{0} that admits an extension f : B1 ! Rk

for which f 6= 0 is called inessential with repect to B1. Otherwisef : ∂B1(0) ! Rk\{0} is called essential with respect to B1, i.e. everycontinuous extension f : B1 ! Rk has the property that f�1(0) 6= ?.

Let W ⇢ X be a closed and bounded subset and F : X ! Y acontinuous mapping.

4.5.2 Definition. Two mappings f , g : W ⇢ X ! Y are said to becompactly homotopic relative to F is there exists a family of compactmappings k(t, ·) : W ⇢ X ! Y, t 2 [0, 1], such that(i) h(t, x) = F(x) + k(t, x);(ii) h(0, x) = F(x) + k(0, x) = f (x);(iii) h(1, x) = F(x) + k(1, x) = g(x).The associated compact homotopy classes are denoted by [ f ]c.


4.5.3 Using the Leray-Schauder degree we have the following ana-logue of Theorem 3.1.10 and Corollary 3.1.15 in the case that W ⇢ Xis a bounded and convex domain and Y = X. As before the Leray-Schauder degree only depends on the boundary behavior of a map-ping and for a mapping j 2 C0

Id(∂W; X\{0}) we define

degLS(j) := degLS( f , W, 0),

for any continuous extension f : W ⇢ X ! X.

4.5.4 Theorem. Let W ⇢ X be a bounded and convex domain.

(i) Two mappings j0, j1 2 C0Id(∂W; X\{0}) are compactly homotopy

if and only if degLS(j0) = degLS(j1).

(ii) A mapping j 2 C0Id(∂W; X\{0}) is essential with respect to W if

and only if degLS(j) 6= 0.

In the latter case f (x) = 0 has a non-empty solution set for anycontinuous extension f : W ⇢ X ! X of j.


4.5.5 The Leray-Schauder degree classifies the homotopy classesand gives necessary and sufficient consitions for the extension, orsolvability problem (compare Section 3.1). The restriction here isthat W is convex. To extend the Leray-Schauder theory and theresults in Section 3.2 we start with mappings f : W ⇢ X ! Y,where Y ⇢ X is a closed linear subspace of finite codimension. Thefunction class with compact perturbations of a fixed inear Fredholmoperator A : X ! Y is denoted by C0

A. Let j 2 C0A(∂W; \{0}),

where range(j) ⇢ Y ⇢ Y, then considered as map into Y\{0} f ishomotopically trivial. Indeed, degLS(j) = degLS( f , W, 0) = 0. Thelatter follows since Y\ f (∂W) = Y\j(∂W) is path connected. In thecase that W = B1(0) the following result by Svarc generalizes thetheory in Section 3.2.

In order to explain this extension we first discuss stable homotopytypes. We start with the Freudenthal suspension for maps betweenunit spheres Sn ⇢ Rn+1. Let y : Sn ! Sk be a continuous then thesuspension operator defines an operator Sy : Sn+1 ! Sk+1. Letf : Dn+1 ⇢ Rn+1 ! Rk+1 be any continuous extension of j. thendefine f : Dn+2 ⇢ Rn+2 ! Rk+2 by f (x, xn+2) := ( f (x), xn+2).Now set Sy = f /| f | by restricting to Sn+1 and Sy : Sn+1 ! Sk+1.It follows from the construction that [Sy] only depends on [y] andS : pn(Sk) ! pn+1(Sk+1) defines an isomorphism. This constructionallows to define the notion of stable homotopy type.


4.5.6 Theorem. Let W = B1(0) and let A 2 B(X, Y) be a Fredholmoperator of index ` � 0. Then the homotopy classes of mappings j 2C0

A(∂W; Y\{0}) are given by the stable homotopy groups pn+`(Sn),n > `+ 1.


In the case that Y = X and A = Id the stable homotopy is given bythe Leray-Schauder degree.

4.5.7 Theorem. Let W = B1(0) and let A 2 B(X, Y) be a Fredholmoperator of index ` � 0. Then the mapping j 2 C0

A(∂W; Y\{0})is essential with respect to W if and only if [j]c corresponds to anon-trivial stable homotopy class in pn+`(Sn), n > `+ 1.


4.6 Stable cohomotopy

Describe the stable cohomotopy theory and the infinte dimensionalframed cobordism theory. Explain some of the theory in Nirenberg,Berger and the papers by Elworthy and Tromba.

4.7 Semi-linear elliptic equations and a priori estimates

4.7.1 In this section we will give a application of the Leray-Schauderdegree in the context of nonlinear elliptic equations. We follow thenotes by L. Nirenberg. The methods that we discuss apply in generalfor elliptic differential operator of any order. In order to simplifymatter here we will restrict ourselves to the Laplace operator withDirichlet boundary data. Let D ⇢ Rn be a bounded domain withsmooth boundary ∂D. Consider the problem

�Du = g(x, u,ru), u = 0, x 2 ∂D.

For the nonlinearity g we assume that C•-function of arguments, i.e.g 2 C•(D⇥R⇥Rn), and

|g(x, u,ru)| C + C|ru|g, g < 1,

uniformly in x 2 D, and u 2 R. Under these conditions we can provethe following result.


4.7.2 Theorem. Under the assumptions on g the above elliptic equa-tion has a solution u 2 C•(D). Moreover, if g(x, 0, 0) 6⌘ 0, then thesolution u is not identically zero.

Proof: The idea behind the proof is the formulate the above ellipticequation as a problem of finding zeroes of an appropriate function fon a (infinite dimensional) Banach space. Let us start with choosingan appropriate space in which to work. Define X = H2 \ H1

0(D) to bethe intersection of two Sobolev spaces. For details on Sobolev spacewe refer to the next chapter. We will use the implications of thischoice with respect to the well-defined of the elliptic equation, andpostpone to proofs to the next chapter. The space H2 \ H1

0 is a Hilbertspace with norm kukX =

R

D |Du|2dx. Due to the Dirichlet boundaryconditions the Laplace operator �D : H2 \ H1

0(D) ⇢ L2(D) ! L2(D)

has a compact inverse (�D)�1 : L2(D) ! L2(D). We rewrite theelliptic equation as

u� (�D)�1g(x, u,ru) = 0. (4.1)

The above equation can be regarded as a seeking zeroes of the (Ne-mytskii) mapping f (u) = u� (�D)�1g(x, u,ru) on H2 \ H1

0(D). Bythe estimate on g we have that

Z

D|g(x, u(x),ru(x)|2dx C

Z

D

h

1 + |ru(x)|2g

i

dx

C0⇣

Z

D

h

1 + |ru(x)|2i

dx⌘

g

C⇣

1 + kuk2H1

0

⌘

g

,

which proves that for u 2 X, g(x, u,ru)(x) is an L2-function. Con-sequently, the composition (�D)�1⇥g(x, u,ru)

⇤

2 X, proving thatf : X ! X is well-defined. The latter follows from the fact thatR�

(�D)�1� = H2 \ H10(D). As a map from L2 to H2 \ H1

0 , the inverseLaplacian is an isometry. Concerning the continuity of this substi-tution map we refer to the next section. If we define Y = H1

0(W)

then f is a map from Y to Y, and f = id� k, where k : Y ! Y isa compact map. Indeed, k is a composition of the Nemytskii mapu 7! g(x, u,ru) (from Y to L2), the inverse Laplacian (�D)�1 (fromL2 to X), and the compact embedding X ,! Y, which proves thecompactness of k. This brings us into the realm of the Leray-Schauderdegree.

Suppose u 2 X is a solution of the equation (4.1), then the estimateon g(x, u,ru) can be used now to obtain an a priori estimate on thesolutions:

kuk2Y Ckuk2

X = Ckg(x, u,ru)k2L2

C⇣

1 + kuk2Y

⌘

g

,


which, since g < 1, implies that kxkY R.

4.7.3 Exercise. Prove the inequalities kukL2 CkukX, and kukH10 CkukX, for all

u 2 X.

Define the domain W = B2R(0) ⇢ Y. Clearly, f is a continuous mapfrom W into Y, which is of the form identity minus compact. Dueto the above a priori estimate f�1(0) ⇢ BR(0) ⇢ Y, and therefore0 62 f (∂B2R(0)). Consequently, the Leray-Schauder degree

degLS( f , W, 0),

is well-defined.In order to compute this degree we consider the following homo-

topy:

ft(u) = u� t(�D)�1[g(x, u,ru)], t 2 [0, 1].

Notice, for t 2 [0, 1] we have via the same a priori estimates, thatf�1t (0) ⇢ BR(0), and therefore 0 62 ft(∂W) for all t 2 [0, 1]. Homotopy

invariance of the Leray-Schauder degree then yields

degLS( f , W, 0) = degLS(id, W, 0) = 1,

which implies, by validity property of the Leray-Schauder degree,that f�1(0) 6= ?. Equation (4.1) thus has a solution u 2 Y. Theequation yields u = (�D)�1[g(x, u,ru)] 2 X, which implies that thesolution also lies in X.

To prove regularity we use a bootstrapping argument. The integralestimates on g can be adjusted to Lp-estimates. This gives, by theSobolev embeddings that:

u 2 H1,p =) g(x, u,ru) 2 Lp =) u 2 H2,p =) u 2 H1,p0 ,

where 1p0 =

1p �

1n , provided n > p. This yields the recurrence relation

1pk+1

=1pk� 1

n.

We can repeat these recurrent steps until k times until 2(k + 1) >

n > 2k, and then u 2 H2,pk , where pk = 2nn�2k . Again by the Sobolev

embeddings, we have that

H2,pk (D) ,! C1,a(D),

where a = 1� npk

, since npk

= n2 � k, and k + 1 > n

2 > k, it holds that0 < a < 1. We now repeat the bootstrapping in the Hölder space:

u 2 C1,a =) g(x, u,ru) 2 C0,ga =) u 2 C2,a0 ,


where a

0 = ga. The idea now is the use the elliptic regularity theoryfor the Laplacian by differentiation the equation. Let vi =

∂u∂xi

, then

�Dvi = ∂xi g + (∂ug)vi + Âj

∂vj g∂vj

∂xi.

Since g is a C•-function of its arguments, and u 2 C2,a0 , the righthand side is in C0,a0 , implying that vi 2 C2,a0 , and thus u 2 C3,a0 . Wecan repeat this process indefinitely, which proves that u 2 C•(D).

If g(x, 0, 0) 6⌘ 0, then u = 0 cannot be a solution, and thus u 6⌘ 0.

4.8 Positive solutions of elliptic PDE’s

Write something on the application of degree theory to positivesolutions of elliptic PDE’s of the form �Du = f (x, u) and f (x, u) ⇠|u|p�1u. Use the result by Brezis, Lions, Nussbaum including thePohozaev identity.

5Dynamical Systems

The content of this chapter is basedon notes with W.D. Kalies and K.Mischaikow, entailed ‘Conley Theory’.

Prototypical examples of dynamical systems are systems of ordinarydifferential equations. But also iterations of mappings or homeo-morphisms define dynamical systems (discrete time). In this part wewill consider a topological theory of dynamical systems that allowsone to study the global behavior of of dynamical systems. There arealso links with degree theory. For example the problem of findingsolutions for equation f (x) = 0, where f : Rn ! Rn, can be reformu-lated in terms of dynamical systems. The mapping f is regarded asa vector field on Rn and integrating the vector field yields the differ-ential equation x0 = f (x), x(0) = x0. The solution operator x(t; x0)

is denoted by j(t, x) and satisfies the properties that: j(0, x) = xand j(t + s, x) = j(t, j(s, x)), for all x 2 Rn and all s, t 2 R. We doassume here that f is for example Lipschitz and bounded (otherwiseconsider f /(1 + | f |2) as vector field). The mapping j is called a flowon Rn and fixed points of j correspond to zeroes of f .

We will be interested here more than just fixed points of dynam-ical systems. We also do not assume that dynamical systems arenecessarily defined by differential equations. Initially, we start with(invertible) dynamical systems with continuous time t 2 R. At a laterstage extension to semi-dynamical systems and discrete time willbe made. We do not assume that the phase space of the dynamicalsystem is finite dimensional, so that we can also apply the theory topartial differential equations.

5.1 Preliminaries on Dynamical systems

5.1.1 Let (X, d) be a compact metric space X with metric d and thespace of time variables is R.

5.1.2 Definition. A (continuous time) dynamical system on X is acontinuous mapping j : R⇥ X ! X that satisfies the following twoproperties:


(i) j(0, x) = x for all x 2 X, and

(ii) j(t, j(s, x)) = j(t + s, x) for all s, t 2 R and all x 2 X.

The latter is referred to as the group property for j.

From the group property it follows that a dynamical system isa one-parameter family of homeomorphisms of X and j

�1(t, ·) =

j(�t, ·). A continuous time dynamical is also referred to as a (global)flow on X. A vector field generates a local flow (local in the timevariable). By renormalizing the vector field one obtains a global flow(reparametrizing time).

5.1.3 Exercise. Let f : Rn ! Rn be a Lipschitz continuous and bounded onRn. Show that x0 = f (x) generates a dynamical system on X = Rn, satisfyingDefinition 5.1.2.

5.1.4 Definition. An orbit through x 2 X is the image of the functiont 7! j(t, x) and is denoted by gx.

5.1.5 Remark. An orbit is also referred to as trajectory and for sim-plicity of notation, gx will be used both to denote the functiongx : R ! X which defines a trajectory through x and its image{gx(t) 2 X | t 2 R } which is the trajectory.

5.1.6 Complete orbits are examples of subsets of X that have theproperty that the set is invariant under the action of j. In otherwords, complete orbits are invariant under the dynamics of j. Thisleads to the following definition.

5.1.7 Definition. A set S ⇢ X is invariant if j(t, S) = S for all t 2 R.

The set of all invariant sets of a dynamical system is denote by

Invset(X, j) := {S ⇢ X | j(t, S) = S, 8t 2 R} ,

If there is no ambiguity about the dynamical system j we writeInvset(X). Observe that for any dynamical system, the empty set is aninvariant set, i.e ? 2 Invset(X).

5.1.8 Exercise. Show that S is an invariant set for j if and only if j(t, S) = Sfor all t 2 R+.

5.1.9 If S ⇢ X is an invariant set, then the restriction of j to S isdefined as jkS(t, x) = j(t, x) for all x 2 S, t 2 R, and the mappingjkS : R⇥ S! S is a dynamical system on S.

5.1.10 Exercise. Show that S is an invariant set if and only S ⇢ j(t, S), for allt 2 R. Equivalently, show that S is invariant if and only if j(t, S) ⇢ S for allt 2 R.

dynamical systems 105

5.1.11 A set S is forward invariant if j(t, S) ⇢ S for all t 2 R+

and the set of forward invariant sets is denoted by Invset+(X, j). Aset S is backward invariant if j(t, S) ⇢ S for all t 2 R� and setof backward invariant sets is denoted by Invset�(X, j). For flows itholds that a sets S ⇢ X is invariant if and only if it is both forwardand backward invariant, Invset(X) = Invset+(X) \ Invset�(X). Thefollowing lemma is additional characterizations of invariant sets.

5.1.12 Proposition. The following statements are equivalent:(i) S is invariant;(ii) S =

T

t2R j(t, S) and jkS is surjective;(iii) for all x 2 S there exists an orbit gx ⇢ S;(iv) j(t, S) = S for all t 2 (0, t], for some t > 0.Similar characterizations can be given for forward/backward invari-ant sets.

5.1.13 Exercise. Prove Proposition 5.1.12.

5.1.14 Set inclusion ✓ induces a partial order on the set of (for-ward/backward) invariant sets (see also Appendix A.5). From thedefinition of invariant set one derives that unions and intersectionsof invariant sets are again invariant. As a consequence Invset(X) is adistributive lattice.

5.1.15 Theorem. The set�

Invset,_,^�

with

S _ S0 = S [ S0, S ^ S0 = S \ S, (5.1)

is a bounded distributive lattice. The sets ? and X are the neutralelements.

5.1.16 Exercise. Show that the sets Invset+(X) and Invset�(X) are bounded,distributive lattices with respect to \ and [, and ? and X are the neutralelements.

The mapping S 7! Sc is a lattice anti-isomorphism betweenInvset+(X) and Invset�(X).

5.1.17 Corollary. The set invariant sets Invset(X) is a Boolean algebra.

The above described algebraic structures become interesting ifconsider special invariant sets such as attractors and repeller, whichwill be discussed in the next chapter.

Asymptotic limit sets


5.1.18 A point x 2 X need not have a limit under j as t ! •. Forexample points y = j(t, x) approaching a periodic orbit. In order todescribe the limiting behavior of a point x one can consider limitingbehaviors for arbitrary time sequences tn ! •.

5.1.19 Definition. A point y is called a omega limit point of a setU ⇢ X under j if there exist times tn ! • and points xn 2 U suchthat limn!• j(tn, xn) = y. The set of all omega limit points y is calledthe omega limit set of U and is denoted by w(U, j).

5.1.20 Exercise. Construct an example to show that in general w(U) 6=S

x2U w(x).

If there is not ambiguity about the dynamical system j, the shorthand notation w(U) is used. The following lemma gives a convenientcharacterization of omega limit sets and which is sometimes used asa definition.

5.1.21 Proposition. Let U ⇢ X be a non-empty set. Then

w(U) =\

t�0cl�

j([t, •), U)�

. (5.2)

The omega limit set w(U) is non-empty, compact, invariant, andcontained in cl(j(, R+, U)). If U ⇢ X is forward invariant, then

w(U) =\

t�0cl�

j(t, U)�

, (5.3)

and w(U) = cl(U). If U is connected, then also w(U) is connected.

5.1.22 Let U ⇢ X, then the maximal invariant set in U is defined as

Inv(U, j) := {x 2 U | 9gx ⇢ U}.

The omega limit set of U has the property that w(U) = Inv�

cl(j(R+, U))�

.


The following lemma provides an additional list of useful proper-ties of omega limit sets.

5.1.24 Proposition. Let U, V ⇢ X, then the omega limit sets satisfy thefollowing list of properties:(i) if V ⇢ U, then w(V) ⇢ w(U);(ii) w(U [V) = w(U) [w(V) and w(U \V) ⇢ w(U) \w(V);(iii) if V ⇢ w(U), then w(V) ⇢ w(U);(iv) w(U) = w(cl(U)), i.e. cl(w(U)) = w(cl(U));(v) w(U) = w

�

j(t, U)�

for all t 2 T.(vi) if there exists a backward orbit g

�x ⇢ U, then x 2 w(U);

Moreover, if x 2 U, then d(j(t, x), w(U))! 0 as t! •.



Describing the limiting behavior of j as t ! �• is identical tot! • by simply reversing time t! �t.

5.1.26 Definition. A point y is called a alpha limit point of a setU ⇢ X under j if there exist times tn ! �• and points xn 2 U,yn 2 j(tn, xn) such that limn!• yn = y. The set of all alpha limitpoints y is called the alpha limit set of U and is denoted by a(U, j).

5.1.27 As for omega limit sets we have a similar characterization foralpha limit sets.

5.1.28 Lemma. Let U ⇢ X be a non-empty set. Then,

a(U) =\

t0cl�

j((�•, t], U)�

. (5.4)

The alpha limit set a(U) is non-empty, compact, invariant, and con-tained in cl(j(, R�, U)). If U ⇢ X is backward invariant, then

a(U) =\

t0cl�

j(t, U)�

, (5.5)

and a(U) = cl(U). If U is connected, then also a(U) is connected.

5.1.29 Exercise. Prove Lemma 5.1.28.

It follows from the invertibility of j that a(U, j) = w(U, j

�1).This way the analogue of Proposition 5.1.24 holds for alpha limit sets.

Stable and unstable sets and connecting orbits

5.1.30 Related to the idea of a limit set is the notion of a stable orunstable set of an equilibrium point or more generally of an invariantset.

5.1.31 Definition. Let S ⇢ X be an invariant set, then

Ws(S, j) = {x 2 X | limt!•

d�

j(t, x), S�

= 0}

Wu(S, j) = {x 2 X | limt!�•

d�

j(t, x), S�

= 0},

are called the stable and unstable sets for S respectively.

By definition, S is always included in Ws(S) and Wu(S), and it canhappen that both Ws(S) = Wu(S) = S. In the case that S = {x} isa hyperbolic fixed point for a smooth dynamical system on Rn, thenthe stable and unstable sets are immersed submanifolds. If x is anattracting fixed point, then Wu(S) = S. If x is a repelling fixed point,then Ws(S) = S.


5.1.32 Proposition. Let S ⇢ X be an invariant set for j, then bothWs(S) and Wu(S) are invariant.

5.1.33 Exercise. Prove Proposition 5.1.32

5.1.34 The space of connecting orbits between two invariant sets Sand S0 are points that lie in the stable set of S and the unstable set ofS0. More precisely:

5.1.35 Definition. For two invariant sets S and S0 the set of connect-ing orbits from S0 to S is defined as C(S0, S) := Ws(S) \Wu(S0, j).

For each x 2 C(S0, S) there exists an orbit gx ⇢ C(S0, S) such that

both limt!• d(gx(t), S) = 0 and limt!�• d(gx(t), S0) = 0. Such anorbit is called a connecting orbit from S0 to S.

5.1.36 Proposition. Let S, S0 ⇢ X be invariant sets, then

(i) C(S0, S) = Ws(S) \Wu(S0) is invariant;

(ii) S \ S0 = ?, implies that C(S0, S) \ S0 = ? and C(S0, S) \ S = ?;

(iii) if the sets S and S0 are compact, then the connecting orbits arecharacterized by C(S0, S) = {x | w(x) ⇢ S, and a(x) ⇢ S0}.

This yields the decomposition X = S t S0 t C(S0, S) t C(S, S0).


5.2 Attractors and repellers

5.2.1 In Chapter 5 we introduced invariant sets. Since every completeorbit of a dynamical system is invariant, studying the lattice Invset(X)

is like understanding every single orbit. It turns out that in generaldynamical systems have global characteristics that can be describedin terms of special types of invariant sets, which will play the role ofbuilding blocks. A class of such invariant sets are attractors. Thinkof the physical system describing the damped pendulum: q

00 � kq

0 +

sin(q) = 0. Except from the unstable position at q = p, all statesconverge to q = 0, which acts as an attractor. Next to attractors onecan also define there dual, which repel as t ! • and which arecalled repellers.

5.2.2 Definition. A trapping region is a forward invariant set U 2Invset+(X), such that j(T, cl(U)) ⇢ int (U) for some T > 0. A setA ⇢ X is called an attractor if there exists a trapping region U suchthat A = Inv(U).


5.2.3 Proposition. An attractor A is a compact, invariant set. If U is atrapping region for A, then A = Inv(U) = Inv(cl(U)) = w(U) andA ⇢ int (U). Moreover, U 6= ?, then A 6= ?.

Proof: Let U be a trapping region for A. If A0 = Inv(cl(U)), thenA0 = j(T, A0) ⇢ j(T, cl(U)) ⇢ int (U) (choose T > 0 from thedefinition), and thus A0 ⇢ Inv(U) = A. On the other hand A =

Inv(U) ⇢ Inv(cl(U)) = A0, which proves that A = A0 = Inv(cl(U))

and A ⇢ int (U). The closure cl(A) is compact, invariant and cl(A) ⇢cl(U) and therefore cl(A) ⇢ A, which proves that A = cl(A) iscompact. Since A is compact, invariant, A = w(A) ⇢ w(U) ⇢ cl(U)

(Proposition 5.1.21), and thus A ⇢ w(U) ⇢ Inv(cl(U)) = A, whichproves that A = w(U). The latter also implies that if U 6= ?, thenA 6= ?.

5.2.4 Definition. A repelling region is a backward invariant set U 2Invset�(X), such that j(T, cl(U)) ⇢ int (U) for some T < 0. A setR ⇢ X is called an repeller if there exists a repelling region U suchthat R = Inv(U).

5.2.5 Proposition. A repeller R is a compact, invariant set. If U is arepelling region for R, then R = Inv(U) = Inv(cl(U)) = a(U) andR ⇢ int (U). Moreover, U 6= ?, then R 6= ?.

Proof: Use the fact that R is the attractor in U for j

�1.

The set of attractors and repellers in X are denoted by Att(X) andRep(X) respectively.

Attracting and repelling neighborhoods

5.2.6 A neighborhood U ⇢ X is an attracting neighborhood ifw(U) ⇢ int (U). Similarly, a neighborhood U ⇢ X is an repellingneighborhood if a(U) ⇢ int (U). Via attracting and repelling neigh-borhoods we obtain the following characterization of attractors andrepellers.

5.2.7 Theorem. A set A ⇢ X is an attractor if and only if there exists aneighborhood U of A such that A = w(U). A set R ⇢ X is an repellerif and only if there exists a neighborhood U of R such that R = a(U).In particular, U is an attracting/repelling neighborhood and for everyU there exists a trapping/repelling region U0 ⇢ U.

Before proving this theorem we establish the following technicallemma.


5.2.8 Lemma. A compact subset N ⇢ X is an attracting neighborhoodif and only if there exists T > 0 such that j(t, N) ⇢ int (N) for allt � T.

Proof: Let N be a compact attracting neighborhood, then bydefinition cl(j([T, •), N)) ⇢ int (N) for some T > 0, and thusj(t, N) ⇢ int (N) for all t � T.

Suppose there exists T > 0 such that j(t, N) ⇢ int (N) for allt � T. Fix x 2 w(N). Since w(N) is invariant, there exists an orbitgx : T ! w(N). Then x = j(T, gx(�T)) 2 int (N) and thusw(N) ⇢ int (N).

Proof of Theorem 5.2.7. If A is an attractor, then any trapping region Uis an attracting neighborhood.

Now suppose U is an attracting neighborhood, then also N =

cl(U) is an attracting neighborhood with w(N) = w(U) = A. SinceA = w(N) is compact, there exists e0 > 0 such that cl(B

e

(A)) ⇢int (U) for all 0 < e < e0. Fix 0 < e < e0. By the properties of omegalimit sets,

w(N) = w(w(N)) = w(A) ⇢ w(cl(Be

(A))) ⇢ w(N),

which further implies that w(cl(Be

(A))) = A ⇢ Be

(A). Consequently,cl(B

e

(A)) is an attracting neighborhood. By Lemma 5.2.8, there existsT

e

> 0 such that j(t, cl(Be

(A))) ⇢ Be

(A) for all t � Te

. Define

N0e

= j([0, Te

], cl(Be

(A))).

By definition N0e

is forward invariant and compact, but N0e

may notbe a subset of N. We claim that e can be chosen small enough so thatN0

e

⇢ N. Suppose not and choose en ! 0 such that N0en 6⇢ N. Then

there exists xn 2 Ben(A) and tn T

en such that xn ! x 2 A andj(tn, xn) /2 N with j(tn, xn) ! z /2 int (N) since S \ int (N) is closed.Passing to a subsequence either tn ! t < • or tn ! •. In the formercase j(t, x) = z, which contradicts the invariance of A. In the lattercase z 2 w(N) ⇢ int (N), a contradiction. Choose e > 0 so thatN0

e

⇢ N. By construction,

j(Te

, N0e

) = j

⇣

Te

,[

t2[0,Te

]

j

�

t, cl(Be

(A))�

⌘

=[

t2[0,Te

]

j

�

Te

, j

�

t, cl(Be

(A))��

=[

t2[0,Te

]

j

�

Te

+ t, cl(Be

(A))�

⇢ Be

(A) ⇢ int (N0e

),

so that N0e

is a trapping region, and

A = w(A) ⇢ w(N0e

) ⇢ w(N) = A.

Since w(N) is the maximal invariant set in N0e

we have that A =

Inv(N0e

), so that A is an attractor. If N 6= ?, then A 6= ?.


5.2.9 Attracting and repelling neighborhoods are more flexible thantrapping/repelling regions and therefore the appropriate neighbor-hoods to study attractors and repellers. The attracting neighborhoodsin X are denoted by ANbhd(X) and the repelling neighborhoods byRNbhd(X).

Lattice structures

5.2.10 The sets of attracting and repelling neighborhoods ANbhd(X)

and RNbhd(X) have additional structure as posets and distributivelattices.

5.2.11 Theorem. The sets ANbhd(X) and RNbhd(X) are bounded,distributive lattices with respect to the binary operations _ = [ and^ = \. The neutral elements are 0 = ? and 1 = X.

Proof: The elements ? and X are trivial attracting neighborhoods.Recall that a subset U ⇢ X is an attracting neighborhood if Inv(U) =

w(U) ⇢ int (U). Let U, U0 be attracting neighborhoods for attractorsA, A0 respectively. Then U [ U0 and U \ U0 are neighborhoods ofA [ A0 and A \ A0 respectively. For omega limit sets it holds that

w(U[U0) = w(U)[w(U0) = A[A0 ⇢ int (U)[ int (U0) ⇢ int (U[U0),

which shows that U [U0 2 ANbhd(X). For the intersection we argueas follows. It holds that A \ A0 ⇢ U \U0 is invariant, and thereforeA \ A0 = w(A \ A0) ⇢ w(U \U0). Combining this with Proposition5.1.24 we obtain

A \ A0 ⇢ w(U \U0) ⇢ w(U) \w(U0) = A \ A0,

which proves that w(U \ U0) = A \ A0 ⇢ int (U) \ int (U0) =

int (U \U0) and U \U0 2 ANbhd(X) is an attracting neighborhood.The distributivity follows since ANbhd(X) is a (0, 1)-sublattice ofP(X) and therefore a lattice of sets.

The proof for RNbhd(X) is identical if we replace omega limitsets by alpha limit sets. For repelling neighborhoods U, U0 ⇢ Xwe obtain the identities a(U [ U0) = R [ R0 ⇢ int (U [ U0) anda(U \U0) = R \ R0 ⇢ int (U \U0).

5.2.12 Theorem. The sets Att(X) and Rep(X) are (0, 1)-sublattices ofInvset(X) and w : ANbhd(X) ! Att(X) and a : RNbhd(X) ! Rep(X)

are surjective (0, 1)-homomorphisms.

Proof: Let A = w(U) and A0 = w(U0) be attractors with attractingneighborhoods U and U0 respectively. From Theorem 5.2.11 we havethe identities A [ A0 = w(U [U0) and A \ A0 = w(U \U0), which


shows that Att(X) is a sublattice of Invset(X). Also, w(X) is themaximal element in Invset(X) and w(?) = ? is the minimal elementin Invset(X), which proves that Att(X) is a (0, 1)-sublattice and w a(0, 1)-homomorphism. The proof for Rep(X) is identical.

5.3 Attractor-repeller pairs

5.3.1 The mapping U 7! Uc = X \ U is a lattice anti-isomorphism.Therefore, trapping regions are mapped to repelling regions and viceversa. Let U be a trapping region, then

j

�

T, (cl(U))c� = j(T, cl(U))c � (int (U))c,

and thus (cl(U))c � j

�

�T, (int (U))c�. Using the fact that (cl(U))c =

int (Uc) and (int (U))c = cl(Uc) we obtain

j(�T, cl(Uc)) ⇢ int (Uc),

which proves that Uc is a repelling region. The same holds for re-pelling regions U.

For an attractor A 2 Att(X), with trapping region U, the dualrepeller of A is defined by A⇤ = Inv(Uc) = a(Uc), and for a repellerR 2 Rep(X), with repelling region U, the dual attractor of R isdefined by R⇤ = Inv(Uc) = w(Uc).

By the above, dual attractors and repellers are well-defined. More-over, a dual repeller is a repeller and a dual attractor is an attractor.

5.3.2 Proposition. For an attractor A, the dual repeller A⇤ is character-ized by

A⇤ = {x 2 X |w(x) \ A = ?} .

For a repeller R, the dual attractor R⇤ is characterized by

R⇤ = {x 2 X | a(x) \ R = ?} .

A pair (A, A⇤), or equivalently (R⇤, R), is called an attractor-repellerpair in X.


5.3.4 The following proposition is an easy consequence of the defini-tion of dual attractor and repeller.

5.3.5 Proposition. For an attractor (A⇤)⇤ = A and a for repeller(R⇤)⇤ = R.


5.3.6 Proposition 5.3.5 shows that every repeller is the dual repellerof some attractor, and every attractor is the dual attractor of somerepeller. Attractor-repeller pairs play a fundamental role in thedecomposition of dynamical systems. The follow result gives animportant characterization of attractor-repeller pairs and reducesdynamics to a simple partial order on the attractor-repeller pairs.

5.3.7 Theorem. Let A, R ⇢ X be subsets, then the following state-ments are equivalent.(i) (A, R) is an attractor-repeller pair.(ii) A and R are disjoint, compact invariant sets, such that for every

x 2 X \ (A [ R) we have a(x) ⇢ R and w(x) ⇢ A.The set X \ (A [ R) are the connecting orbits between A and R.

Proof: Suppose (A, R) is an attractor-repeller pair. Proposition5.3.2 implies that A, R are compact, invariant and disjoint. By The-orem 5.2.7 there exists e0 > 0 such that w(B

e

(A)) = A for all0 < e < e0. By Proposition 5.3.2 if x /2 A [ R, then w(x) \ A 6= ?,which implies that there exists T > 0 such that d(j(T, x), A) < e0.Hence w(x) = w(j(T, x)) ⇢ w(B

e0(A)) = A. By Theorem 5.2.7,there exists a compact trapping region N = cl(U) for A withN ⇢ Bmin{e0,d(x,A)}(A). Since N is forward invariant, Nc is back-ward invariant, and hence x 2 Nc implies a(x) ⇢ Inv(Nc) = R, usingthe fact that dual repeller is well-defined and independent of thechoice of trapping region (see Proposition 5.3.2), which proves that (i)implies (ii).

Suppose A and R are disjoint, compact invariant sets in X as instatement (ii). Note that the conditions on A, R imply that X =

A [ R [�

Ws(A) \Wu(R)�

. We show that R = A⇤. Indeed, if x 2 R,then by invariance w(x) ⇢ w(R) ⇢ R, so that w(x) \ A = ?, whichimplies R ⇢ A⇤. Conversely, if x 2 A⇤, then x /2 Ws(A) = S \ R, andthus x 2 R and A⇤ ⇢ R. If A is an attractor, then R = A⇤ is its dualrepeller, and hence (A, R) is an attractor-repeller pair. It remains toshow that A is an attractor.

Fix a compact neighborhood U of A such that A ⇢ int (U) andU \ R = ?. We now argue that U is an attracting neighborhood ofA. Since R is invariant, Corollary 5.1.17 implies that X \ R is alsoinvariant and therefore, w(U) ⇢ w(X \ R) = cl(X \ R). We dividethe remainder of the proof into two steps. First we first show thatw(U) \ R = ? and second we show that w(U) = A.

Suppose y 2 w(U) \ R. By definition, there exists a sequence(tn, xn) with xn 2 U such that yn = j(tn, xn) ! y and tn ! • asn! •. Let Tn = sup {t 2 [0, tn] | j(t, xn) 2 U }. Define zn = j(Tn, xn)

and, along a subsequence if necessary, zn ! z 2 cl(U \ A). Theuniform continuity of j implies j(t, zn) ! j(t, z) for all t � 0,


and the convergence is uniform on compact intervals [0, T]. Definetn = tn � Tn > 0. We claim that tn ! • as n ! •. Suppose not,then there exists a subsequence such that tnk ! t as k ! •, suchthat j(tnk , znk ) = j(tnk , xnk )! j(t, z) = y 2 R, which contradicts theinvariance of X \ R. Along a subsequence we may assume that tn isincreasing. By definition of Tn, we have j(t, xn) 2 Uc for t 2 (Tn, tn],and hence j(t, zn) 2 Uc for t 2 [0, tn]. Therefore j(ti, zn) 2 Uc forall n � i, which implies that j(ti, z) = limn!• j(ti, zn) 2 cl(Uc) forall i � 1. Along a subsequence j(tik , z) ! w 2 cl(Uc) as k ! • andtherefore w 2 w(z). By assumption w(z) ⇢ A for any point z 2 X \ R,and hence w(z) \ cl(Uc) = ?, which is a contradiction. This provesthe first claim that w(U) ⇢ X \ R.

The second step uses the invariance of w(U). Let x 2 w(U) \ (U \A), then by the previous step x 2 X \ (A [ R). Invariance of w(U)

implies that w(x), a(x) ⇢ w(U). On the other hand, since x 2 X \ (A[R), we have that a(x) ⇢ R, which shows that w(U) \ (U \ A) = ?,and thus w(U) = A.

5.3.8 The fundamental theorem of attractor-repeller pairs yields aprecise characterization of attracting and repelling neighborhoods.

5.3.9 Theorem. Let A 2 Att(X) be an attractor. A subset U ⇢ X isan attracting neighborhood for A if and only if U is a neighborhoodof A and cl(U) \ A⇤ = ?. If R 2 Rep(X) is a repeller, then a sub-set U ⇢ X is a repelling neighborhood for R if and only if U is aneighborhood of R and cl(U) \ R⇤ = ?.

Proof: The choice of the neighborhood U in the proof of Theorem5.3.7 was arbitrary, which shows that U is an attracting neighborhood.Obviously, if U � A is an attracting neighborhood, then cl(U) ⇢X \ A⇤ by Theorem 5.3.7.

The sets U in Theorem 5.3.9 are contained in Ws(A) or Wu(R),which we refer to as the basin of attraction or basin of repulsionrespectively.

5.4 Some properties of attractors and repellers

5.4.1 An important property of attractors concerns relative attractorsinside an attractor.

5.4.2 Theorem. Let A ⇢ X be an attractor and A0 ⇢ A is an attractorfor jkA, then A0 is an attractor for j on X. Similarly, if R⇢ X be arepeller and R0 ⇢ R is a repeller for jkR, then R0 is a repeller for j onX.


5.4.3 Lemma. Suppose N is a compact set with S = Inv(N) ⇢ int (N).For T > 0 define N+

T = {x 2 N | j([0, T], x) ⇢ N }. Then there existsdT > 0 such that B

dT (S) ⇢ N+T , i.e. j([0, T], B

dT (S)) ⇢ N.

Proof: By continuity and the invariance of S, for each x 2 S thereexists dx > 0 such that j([0, T], B

dx (x)) ⇢ N. By compactness thereexists finitely many such balls B

dxi(xi) such that S ⇢ [iBdxi

(xi). Sincethis union is open, there exists d > 0 such that B

d

(S) ⇢ [iBdxi(xi) ⇢

N+T .

5.4.4 Lemma. Suppose N is a compact set such that Inv(N) ⇢ int (N)

with the property that there are no backward orbits g

�x : T� ! N

through x 2 N \ Inv(N). Then for every e > 0 there exists T > 0 suchthat there are no backward orbit segments g

�x : [�T, 0]! N through

x 2 N \ Be

(Inv(N)).

Proof: Define N�T = {x 2 N | 9 a backward orbit segment g

�x : [�T, 0]! N }

for T > 0, and let e

�T = supx2N�T

{d(x, S)}. First we show that e

�T ! 0

as T ! •. Suppose not. Then there exists a sequences Tn ! •and xn 2 N with xn ! x 2 N \ Inv(N) and g

�xn : [�Tn, 0] ! N.

By compactness there exists a backward orbit g

�x ⇢ N, which is a

contradiction. Finally choose T > 0 large enough so that e

�T < e.

5.4.5 Lemma. Let j : T+ ⇥ X ! X be a dynamical system. Aninvariant set A is an attractor if and only if there exists a compactneighborhood N of A such that there are no backward orbits g

�x :

T� ! N through x 2 N \ A.

Proof: The necessity of the condition is straightforward. Suppose Ais an attractor and N is a compact trapping region so that A = w(N).If there is a backward orbit g

�x ⇢ N with x 2 N \ A, then the complete

orbit gx lies in N, which implies x 2 Inv(N) = A, a contradiction.Now suppose N is a neighborhood with the stated property. Since

there are no backward orbits g

�x : T� ! N with x 2 N \ A and N is a

neighborhood of A, we have A = Inv(N) ⇢ int (N). By Lemma 5.4.3,there exists d1 > 0 such that B

d1(A) ⇢ N and j([0, 1], Bd1(A)) ⇢ N.

Fix e < d1 and consider the neighborhood Be

(A). By Lemma 5.4.4,there exists T > 0 such that there are no backward orbit segmentsg

�x : [�T, 0]! N for x 2 N \ B

e

(A).Now we construct an attracting neighborhood for A. By Lemma

5.4.3, there exists d2T > 0 such that the neighborhood U = B2T(A)

satisfies j([0, 2T], U) ⇢ N. We show that U is an attracting neighbor-hood. Let V0 = j(T, U). For each x 2 V0 there exists a backward orbitsegment g

�x : [�T, 0]! N. The definition of T from Lemma 5.4.4 im-

plies that that V0 ⇢ Be

(A). Moreover, by our choice of e < d1 we havej([0, 1], B

e

(A)) ⇢ N so that V1 = j([0, 1], V0) ⇢ N. Thus for each


x 2 V1 there exists a backward orbit segment g

�x : [�T � 1, 0] ! N

which implies that g

�x ([�T, 0]) ⇢ N so that V1 ⇢ B

e

(A). We canrepeat this argument inductively to prove that Vk = j([k� 1, k], V0) ⇢B

e

(A) ⇢ N for all k > 0. Therefore j([0, •), U) ⇢ N. Finally, A ⇢ Uimplies that A = w(A) ⇢ w(U), and j([0, •), U) ⇢ N implies thatw(U) ⇢ Inv(N) = A. Therefore A = w(U), and A is an attractor byTheorem 5.2.7.

Proof of Theorem 5.4.2. Let N be a (compact) attracting neighborhoodfor A in X. Choose a neighborhood N0 ⇢ N of A0 in X such thatN0 \ A is an attracting neighborhood for A0 in A. Suppose that g

�x

is a backward orbit through x 2 N0 \ A0 such that g

�x ⇢ N0. Then

g

�x ⇢ N so that x 2 w(N) = A. Indeed the entire backward orbit g

�x

is contained in w(N) = A by a similar argument. Thus g

�x ⇢ N0 \ A,

and therefore x 2 w(N0 \ A) = A0, a contradiction. The criterion inLemma 5.4.5 yields that A0 is an attractor for j in X.

5.4.6 Theorem. Let j : T+ ⇥ X ! X a dynamical system on a compactmetric space X. Given an attractor-repeller pair (A, A⇤) the followingstatements hold.(i) A and A⇤ are disjoint.(ii) A = Inv(X, j) = w(X) if and only if A⇤ = ?, and A = ? if and

only if A⇤ = X.(iii) If x 2 X such that w(x) \ A 6= ?, then w(x) ⇢ A.(iv) If x 2 X and a(x) \ A⇤ 6= ?, then a(x) ⇢ A⇤.(v) If A0 is an attractor with A ( A0, then A0 \ A⇤ 6= ?.The same statements hold if we replace attractors by repellers anduse the appropriate duality.

Proof: (i) This was proved in Theorem 5.3.7.

(ii) Proposition 5.3.2 implies that if A = Inv(X, j) = w(X), thenA⇤ = ?. If A⇤ = ?, then ? is a repelling region for A⇤ and A =

Inv(X \?, j) = Inv(X, j). Likewise if A = ?, then ? is a trappingregion for A and hence A⇤ = Inv(X \?, j) = Inv(X, j) = X since Xis forward invariant. If A⇤ = X, then w(x) \ A = ? for all x 2 X byProposition 5.3.2 so that A = ?.

(iii) and (iv) This follows from Proposition 5.3.2 and Theorem 5.3.7.

(v) Let x 2 A0 \ A. Let gx be a (complete) orbit through x. Since x /2 A,Theorem 5.3.7 implies that a(x) ⇢ A⇤. Also, ? 6= a(x) ⇢ A0, since A0

is a compact invariant set. Thus A0 \ A⇤ 6= ?.

5.4.7 From the previous considerations we define the mappings

⇤ : Att(X) ! Rep(X)

A = w(U) 7! A⇤ = a(Uc),


and

⇤ : Rep(X) ! Att(X)

R = a(U) 7! R⇤ = w(Uc).

5.4.8 Theorem. The mappings ⇤ : Att(X)! Rep(X) and ⇤ : Rep(X)!Att(X) are (0, 1)-lattice anti-isomorphisms, and (A⇤)⇤ = A and(R⇤)⇤ = R.

Proof: From Theorem 5.2.11 it follows that

(A [ A0)⇤ =�

w(U) [w(U0)�⇤

=�

w(U [U0)�⇤

= a(Uc \U0c) = a(Uc) \ a(U0c) = A⇤ \ A0⇤,

and similarly

(A \ A0)⇤ =�

w(U) \w(U0)�⇤

=�

w(U \U0)�⇤

= a(Uc [U0c) = a(Uc) [ a(U0c) = A⇤ [ A0⇤,

which proves that ⇤ is a (0, 1)-lattice anti-homomorphisms. Finally,Proposition 5.3.5 shows that ⇤ is an anti-isomorphism.

5.4.9 The above propositions are summarized in the following com-mutative diagram:

ANbhd(X) RNbhd(X)

Att(X) Rep(X)

-c

?

w

?

a

-⇤

5.4.10 Example. Let (x, y) 2 R2. Consider the gradient flow generatedby

x = x� x2 (5.6)

y = y� y2

where the potential is given by V(x, y) = 13 �

12 x2 � 1

2 y2 + 13 x3 + 1

3 y3.Since this system is an uncoupled system of first order equations

its dynamics can be described explicitly. First label the four equilib-rium points by e1 = (1, 1), e2 = (0, 1), e3 = (1, 0) and e4 = (0, 0).

The set of bounded orbits is the unit square X = [0, 1] ⇥ [0, 1].Since X is bounded, it is a compact invariant set for the system ofdifferential equations, as shown in Figure 5.1. We define j as theinduced flow restricted to X. Because this system is a gradient flow,we can completely characterize the dynamics on X by an acyclic


��

s

s

s

s

x

y

�✓

�

�

-

-

�

�

? ?

6 6

? ?

e4

e2

e3

e1

✓

✓

✓

✓

(a)

ce1

@@@@

@@R

ce2 ��

��

c e3

c@@@@

@@R

��

��

e4

(b)

Figure 5.1: (a) Phase portrait forExample 5.4.10. The set of boundedorbits is the unit square S whoseinterior is filled with connecting orbitsfrom e4 to e1. (b) Acyclic directed graphrepresentation of the dynamics in X.

directed graph whose vertices are the four equilibrium points. FromFigure 5.1 there are four connecting orbits e4 ! e2, e4 ! e3, e2 ! e1and e3 ! e1 on the boundary of the square, and the interior of thesquare is filled with connecting orbits e4 ! e1. Thus there are fiveedges in the acyclic directed graph representation of the dynamics.

The non-trivial attractor-repeller pairs are given by

(A1, A⇤1) :=⇣

{e1} , {e2, e3, e4} [ C�

{e4}, {e3}�

[ C�

{e4}, {e2}�

⌘

,

(A2, A⇤2) :=⇣

{e1, e3} [ C�

{e3}, {e1}�

, {e2, e4} [ C�

{e4}, {e2}�

⌘

,

(A3, A⇤3) :=⇣

{e1, e2} [ C�

{e2}, {e1}�

, {e3, e4} [ C�

{e4}, {e3}�

⌘

,

(A4, A⇤4) :=⇣

{e1, e2, e3} [ C�

{e2}, {e1}�

[ C�

{e3}, {e1}�

, {e4}⌘

,

c?

cA1@

@@cA2

��

c A3

c@@@

��

A4

cX

cX⇤

cA⇤4@

@@cA⇤2

��

c A⇤3

c@@@

��

A⇤1

c?⇤ Figure 5.2: Posets (Att,✓) and (Rep,✓)for Example 5.4.10.

The set of all attractors Att(X) consists of

A0 = ?, A1, A2, A3, A4, A5 = X.


Similarly, the set of repellers Rep(X) consists of the dual repellers

A⇤0 = X, A⇤1, A⇤2, A⇤3, A⇤4, A⇤5 = ?.

This establishes the sets Att(X) and Rep(X), both consisting of sixelements. In this example Att(X) and Rep(X) are finite. This is nottrue in general. The sets of attractors and repellers as posets areillustrated in Figure 5.2.

The set sets Att(X) and Rep(X) are in fact bounded (finite in thisexample) distributive lattices.

6Decompositions of Dynamical Systems

The content of this chapter is basedon notes with W.D. Kalies and K.Mischaikow, entailed ‘Conley Theory’.

A non-trivial attractor-repeller pair yields a decomposition of Xconsisting of three invariant sets A, A⇤ and C(A⇤, A). For any x 2C(A⇤, A) = X \ (A [ A⇤), w(x) ⇢ A and a(x) ⇢ A⇤. This concept canbe generalized to more than three invariant sets.

6.1 Morse decompositions

The following definition generalizes the notion of attractor-repellerpair decomposition to more than two compact invariant sets.

6.1.1 Definition. Let (M,) be a finite poset consisting of non-empty,compact, pairwise disjoint, invariant sets M, such that for everyx 2 X \ (SM2M M), there exists M < M0 such that

a(x) ⇢ M0 and w(x) ⇢ M.

A Morse decomposition for j in X is an order-embedding i : P !M. The sets M 2 M are called Morse sets. The set of all Morsedecompositions of X under j is denoted by MD(X).

6.1.2 Morse decompositions are posets and are intimately relatedto sublattices of attractors. The goal of this section is to show thatthe structure present in a Morse decomposition is equivalent to thestructure present in a finite (0, 1)-sublattice of attractors. The setof equilibria {e1, · · · , e4} in Example 5.4.10 is a Morse decomposi-tion. On fact it is the finest Morse decomposition. We will use thisexample to further illustrate the theory of Morse decompositions.

6.1.3 Example. The collection E = {e1, · · · , e4} in Example 5.4.10

is a Morse decomposition. Since E consists of equilibrium pointsthere are no finer Morse decompositions. Before explaining thisprocedure let us investigate the Morse decompositions of Equation(5.7) consisting of three Morse sets. In all five cases we have a total


order M1 < M2 < M3. Upon inspection we have

M1 : M1 = {e1}, M2 = {e2, e3}, M3 = {e4};

M2 : M1 = {e1, e3} [ C�

{e3}, {e1}�

, M2 = {e2}, M3 = {e4};

M3 : M1 = {e1}, M2 = {e3}, M3 = {e2, e4} [ C�

{e4}, {e2}�

;

M4 : M1 = {e1, e2} [ C�

{e2}, {e1}�

, M2 = {e3}, M3 = {e4};

M5 : M1 = {e1}, M2 = {e2}, M3 = {e3, e4} [ C�

{e4}, {e3}�

;

6.1.4 Exercise. Show that M1, · · · ,M5 are Morse decompositions for X inExample 5.4.10, with total order.

If we inspect the diagram of the partial order for the smallestMorse decompositions in Figure 5.1 we can ‘coarsen’ the partial orderin five different ways by every time identifying two conseccutiveelements. Figure 6.1 explains how to obtain M2.

c1

@@@@

@@R

c2�

��

��

c 3

c@@@@

@@R

��

��

4

c{1, 3}

c2

c4?

?

��

s

s

s

s

✓⌘◆⇣

✓⌘◆⇣

x

y

�✓

�

�

-

-

�

�

? ?

6 6

? ? $

%

'

&

M1

e4

M3

e2

M2

e3

e1

Figure 6.1: Coarsening of the partialorder by identifying two consecutiveelements by a double line (a), the newpartial order (b), and the Morse sets inS for M2 (c).

Note that if we have a totally ordered Morse decomposition con-sisting of the four equilibrium points, then only three Morse decom-positions with three elements can be found via coarsening: M1,M4

and M5. The coarsening procedure we described relies on the factthat we group together certain elements in the partial order. It isnot allowed to have ‘holes’ in between. Such a grouping together ofelements is called a convex set for the partial order. A further obser-vation with the coarsening and convex sets in posets is how a convexset is related to a Morse set. By inspection we observe that

I = {e1, e3} �! M1 = M1 = {e1, e3} [ C�

{e3}, {e1}�

= Wu(e1) [Wu(e3) =[

ep2IWu(ep).

The Morse set M1 is an attractor. If we for example consider M3, then

decompositions of dynamical systems 123

I = {e2, e4} �! M3 is an repeller and

M3 = {e2, e4} [ C�

{e4}, {e2}�

=[

ep2IWs(ep).

This construction suggest that convex sets I that contain a minimalelement in P correspond to Morse sets that are built up form unsta-ble sets and are attractors. Similarly, for convex sets that contain amaximal element we obtain Morse sets containing of stable sets andare repellers. Consider M1 and I = I1 \ I2, with I1 = {e1, e2, e3} andI2 = {e2, e3, e4}, then

I = {e2, e3} �! M2 =⇣

[

ep2I1

Wu(ep)⌘

\⇣

[

eq2I2

Ws(eq)⌘

,

which is a Morse set that is the intersection of an attractor and arepeller.

With Morse decompositions we can also go the other direction, i.e.building finer decompositions. Let start with Morse decompositionM = M2. The Morse sets M2 and M3 are equilibrium points and cantherefore not be decomposed further, see Figure 6.1. The set M1 isnot an equilibrium point and we can ask whether M1 allows a non-trivial attractor-repeller pair decomposition. Figure 6.1 shows that M1indeed has a non-trivial attractor-repeller pair decomposition, withA = {e1} an attractor in M1 and A⇤ = {e3} the dual repeller in M1.Note that e3 is not a repeller in X! We have that E = {e1, · · · , e4} isa Morse decomposition. In general, the refining procedure yields afiner Morse decomposition. We now compare Morse decompositionswith attractors and repellers in the system, and we will reconsiderthese constructions again in terms of just attractors and repellers.This provides a systematic way to retrieve the order.

In Example 5.4.10 all attractors and repellers were identified andthe both lattices Att(X) and Rep(X) consist of six elements. Figure5.2 depicts the lattices Att(X) and Rep(X). By examining Figure 5.2we observe that the elements A1, A2, A3 and X have a unique pre-decessor in the order. These are called the join-irreducible elementsof the lattice. In a similar fashion one find the elements that have aunique successor, which are called the meet-irreducible elements. Theirreducible element form a poset. There is a fundamental theoremby Birkhoff that allows us to relate the join-irreducible attractors orrepellers (similarly for meet-irreducible attractors or repellers) to theMorse sets in a Morse decomposition. In Figure 6.2 we have depictedthe correspondence of Birkhoff’s theorem:

P = {1, 2, 3, 4},

O(P) =n

?, {1}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 2, 3, 4}o

.


s1@

@@s2

��

s3

s@@@

��

4

c?

s{1}@@@s{1, 2}

��

s{1, 3}

c@@@

��

{1, 2, 3}

s{1, 2, 3, 4}Figure 6.2: The poset P = {1, 2, 3, 4}and the lattice of attracting intervalO(P). The join-irreducible elements aresolid dots.

Label the set of join-irreducible attractors as the diagram in Figure6.2[left], then by Birkhoff’s theorem the lattice is retrieved by considerthe lower sets, or attracting intervals in the poset.

If we let the attractors in A 2 Att(X) correspond to the elementsin I 2 O(P), then for each join-irreducible A 2 Att(X), the uniquepredecessor A0 ( A yields a dual repeller A0⇤ in A. In O(P) let J_(O)be the join-irreducible elements, then this correspondence yields:I0 is the unique predecessor of I 2 J_(O) and I0|⇤I = I\I0 is therepelling set, or element in the poset (I,). Clearly, I0|⇤I = {p}, p 2 P

and we label A0⇤ accordingly: A0⇤ = Mp. This yields the Morse setsM = {Mp}p2P from Att(X), via the correspondence between P andO(P). Conversely, the correspondence that builds O(P) from P yields

AI =[

Mp2IWu(Mp), I 2 O(M).

Here the poset is M instead of labeling via P.

6.1.7 Exercise. Show that A⇤I =S

Mp 62I Ws(Mp), I 2 O(M).

6.2 Attractor lattices and Morse decompositions

The theory od Morse decompositions is best explained in termsof lattice and poset theory. In Example 6.1.3 we already gave anidea how the two concepts are intimately related. At the core of therelation between Morse decompositions and lattices of attractors isBirkhoff’s theorem.

Birkhoff’s Representation Theorem

6.2.1 Consider a poset (P,). A subset A ⇢ P is attracting if x 2 Aand y x implies that y 2 A. Under the operations of intersectionand union the collection of attracting subsets of (P,) defines thelattice of attracting sets denoted by O(P). Observe that ?,P 2 O(P)

and act as the 0 and 1 elements, respectively.


6.2.2 Example. Consider the poset P = {1, 2, 3, 4} with the orderindicated in Figure 6.2. Then the lattice of attracting sets is given by

O(P) =n

?, {1}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 2, 3, 4}o

and can be visualized as in Figure 6.2.

6.2.3 Exercise. Prove that O(P) is a distributive (0, 1)-lattice with operationsgiven by intersection and union.

6.2.4 Notice that given any x 2 P, the set #x := {y 2 P | y x} 2O(P). To see that not every element of O(P) has this form, considerthe element {1, 2, 3} 2 O(P) from Example 6.2.2. As the followingproposition indicates in the setting of finite posets these sets describethe join-irreducible elements of O(P).

6.2.5 Theorem. Let P be a finite poset. The function

# : (P,) !�

J_(O(P)),✓�

p 7! #p = {q 2 P | q p}

is a poset isomorphism.

Before beginning the proof we state the following lemma which isof interest in its own right.

6.2.6 Lemma. If I 2 O(P), then I =W

p2max(I) #(p).

6.2.7 Exercise. Prove Lemma 6.2.6.

Proof of Theorem 6.2.5. We first show that #(P) ⇢ J_(O(P)). Letp 2 P and assume #(p) = B _ C = B [ C where B,C 2 O(P). Observethat if B 6= #(p), then B ⇢ #(p). Thus there exists x 2 #(p) \ B. Thefact that B is attracting implies that p 62 B. Since p 2 #(p), eitherB = #(p) or C = #(p) and hence #(p) 2 J_(O(P)). The oppositeinclusion, J_(O(P)) ⇢ #(P), follows from Lemma 6.2.6.

6.2.8 Notice that given any p 2 P, the set #p := {q 2 P | q p} 2O(P). To see that not every element of O(P) has this form, considerthe element {1, 2, 3} 2 O(P) from Example 6.2.2. In O(P) sets of theform #p are examples of join-irreducible elements defined as follows.An element c 2 L is join-irreducible if(i) c 6= 0 and(ii) c = a _ b implies c = a or c = b for all a, b 2 L.The set of join-irreducible elements in L is denoted by J_(L). Simi-larly, an element x 2 L is meet-irreducible if (i)-(ii) are satisfied with _replaced by ^. The set of meet-irreducible elements in L is denotedby J^(L). The sets J_(L) and J^(L) are posets as subsets of L. An


element b 2 L is join-irreducible if and only if it has a unique prede-cessor a in the covering relation, i.e. a ⌃ b, and hence we can definethe unique predecessor map pred : J_(L) ! L. An element a 2 L ismeet-irreducible if and only if it has a unique successor b in the cov-ering relation which leads the unique successor map succ : J^(L)! L.A representation a =

W

i ci is redundant if a =W

i 6=j ci for some j,otherwise the the representation is irredundant.

6.2.9 Theorem. Let L be a finite distributive lattice. For every a 2 L

there exists a unique set of irredundant join-irreducible elementsi

_(a) ⇢ J_(L) and a unique set of irredundant meet-irreducibleelements i

^(a) ⇢ J^(L) such that

a =_

b2i

_(a)b =

^

c2i

^(a)c.

Proof: See Roman 1, p. 124, Theorem 4.30. 1

6.2.10 When P is a finite poset, then the irredundant meet- andjoin-irreducible representations of elements in the lattice O(P) arecharacterized as follows. Let I 2 O(P), then I =

W

p #p, where thejoin is taken over all maximal p 2 I. Similarly, I =

V

p("p)c, wherethe meet is taken over all minimal p 2 Ic. Fundamental in the theoryof finite distributive lattices is Birkhoff’s Representation Theorem,which states that all finite distributive lattices are of the form O(P)

for some finite poset P.Now consider a finite distributive lattice (L,_,^). Using the

order relation on L the set of join-irreducible elements J_(L) is afinite poset. This in turn can be used to define the lattice O(J_(L). Inanalogy with the case of posets we can ask what is the relationshipbetween L and O(J_(L)?

6.2.11 Theorem. (Birkhoff’s Representation Theorem) Let L be a finitedistributive lattice and let P be finite partially ordered set. Then

# : P! J_ (O(P))

is a poset isomorphism and

#_ : L ! O�

J_(L)�

x 7! #_(x) :=�

y 2 J_(L) | y x

is a lattice isomorphism.

Proof: To be typed.


6.2.12 We will often have need move between posets and lattices, andso to maintain clarity we adopt the language of category theory. LetLatF

D denote the category of finite distributive lattices, whose mor-phisms are (0, 1)-homomorphisms, and PosetF denote the categoryof finite posets, whose morphisms are order-preserving mappings.The following two results follow from heorem 8.24!2. If L and K are 2

objects in LatFD and f : L! K is a lattice homomorphism. Then

J_( f ) : J_(K) ! J_(L)

a 7! min�

b 2 J_(L) | a f (b)

is an order-preserving. If P and Q are objects in PosetF and y : P! Q

is an order-preserving mapping. Then

O(y) : O(Q) ! O(P)

I 7! y

�1(I) =[

p2Iy

�1(p)

is a lattice homomorphism. Consequently we have the followingtheorem, see .

Theorem 6.2.13 The mappings J_ : LatFD ! PosetF and O : PosetF !

LatFD define contravariant functors.

In purely combinatorial setting, Birkhoff’s Representation Theoremprovides a precise description of the relation between posets andlattices. Our interest in this theorem is that it provides a tool bywhich we can relate objects of dynamical interest such as Morsedecompositions and attractors.

Building lattices of attractors from Morse decompositions

6.2.14 The goal of this section is to demonstrate two fundamentalproperties of Morse decompositions. First, Morse sets and connectingorbits can be used to construct attractors, and second, this construc-tion is essentially determined by the combinatorial poset structureof the Morse decomposition. Recall that Morse decompositions areposets and attractors form lattices. In light of Birkhoff’s Represen-tation Theorem 6.2.11 consider down-sets of Morse sets. Define themapping

n : O(M) ! Invset(X)

I 7! n(I) =[

M2IWu(M).


6.2.15 Theorem. Let (M,) be a Morse decomposition in X. Thenn : O(M)! Att(X) is a (0, 1)-lattice embedding.

6.2.16 Since n is a (0, 1)-lattice embedding, the image n(O(M)) is afinite (0, 1)-sublattice of Att(X). Theorem 6.2.15 associates a particu-lar finite (0, 1)-sublattice of attractors to a Morse decomposition. Welabel the associated lattice of attractors as follows:

A(M) = {A = n(I) | I 2 O(M)} 2 subF0,1Att(X),

where subF0,1Att(X) is the set of (0, 1)-sublattices of Att(X).

6.2.17 Exercise. Show that subF0,1Att(X) is a lattice with ^ set intersection and

_ defined by A_ A0 =T{A00 2 subF

0,1Att(X) | [ A0 ⇢ A00}.

The following lemma is needed to prove Theorem 6.2.15.

6.2.18 Lemma. If M0 is a maximal element in M, then the pair (A, R)defined by

A =[

M 6=M0

Wu(M) and R = M0

is an attractor-repeller pair in X.

Proof: The set A and R are invariant since stable/unstable setsare invariant for all M. To show that (A, R) is an attractor-repellerpair in X we apply Theorem 5.3.7. Let x 2 X \ (A [ R), then x 2X \ (SM2M M). Since M is a Morse decomposition, it follows fromDefinition 6.1.1 that w(x) ⇢ M and a(x) ⇢ M0 for some M < M0 M0. Therefore, w(x) ⇢ M for some M 2 M \ {M0} and thus w(x) ⇢ A.Since a(x) ⇢ M0, if follows that x 2Wu(M0), which is a contradictionsince x 62 A, and therefore M0 = M0 and a(x) ⇢ M0 = R.

If x 2 A \ R, then w(x) ⇢ M0 and a(x) ⇢ M for some M 6= M0.This implies that x 2 X \ (SM2M M) and thus M0 < M. This isimpossible since M0 is maximal and therefore A \ R = ?.

It remains to show that A is closed and thus compact. SupposeA is not closed, then there exists a sequence xn 2 A such that xn !x 2 X \ A, i.e. x /2 A. Since x /2 A, x 2 Wu(M0), which implies thata(x) ⇢ M0. Because the sets M 2 M are compact and disjoint, we canchoose an open neighborhood U of M0 such that cl(U) \M = ? forall M 6= M0. By the definition of alpha limit set there exists a t0 0such that j(tn, x) 2 U for all t t0. The compactness of X impliesthat there exists a sequence tn ! �• such that j(t, xn) 2 U for tn <

t t0 and j(tn, xn) 3 U. By compactness there exists a subsequencej(tnk , xnk )! y 2 cl(U). We conclude that j(t, y) 2 cl(U) for all t � 0,and therefore, w(y) ⇢ M0 ⇢ cl(U). As before, since y \ (SM2M M),this contradicts the maximality of M0. This proves that A is closed.


6.2.19 Exercise. If M0 is a minimal element of M, then (A, R), with A = M0

and R =S

M 6=M0 Ws(M) is an attractor-repeller pair in X. The proof of thisversion follows along the same line as Lemma 6.2.18.

Proof of Theorem 6.2.15. To see that n is a (0, 1)-mapping observe thatby definition n(?) = ? and n(P) =

S

M2M Wu(M) = X.Let I1, I2 2 O(M), and suppose that n(I1) and n(I2) are attractors in

Att(X), then n(I1) _ n(I2) = n(I1 _ I2) and n(I1) ^ n(I2) = n(I1 ^ I2)

are attractors in Att(X). Finally, since M 6= ? for all M 2 M, I1 6= I2

implies that n(I1) 6= n(I2) so that n : O(M) ! Att(X) is a latticeembedding.

It remains to show that n(I) 2 Att(X) for all I 2 O(M). Theposet of meet-irreducible elements J^(O(M)) is isomorphic to M.For any maximal element I 2 J^(O(M)) we have I = ("M0)c =

{M 2 M | M 6= M0}, where M0 is the corresponding maximalelement M0 2 M. From Lemma 6.2.18, n(I) is an attractor in X. SetM0 = M \ {M0}, then M0 = {M | M 2 M0} is a Morse decompositionfor n(I) and the argument can be repeated. This yields an attractorn(I0) in n(I). By Theorem 5.4.2, n(I0) is an attractor in X and thusn(I) 2 Att(X) for all I 2 J^(O(M)). By Theorem 6.2.9 all elementsin O(M) can be uniquely represented as an irredundant meet, whichestablises that A(M) = n

�

O(M)�

is a (0, 1)-sublattice of Att(X).

6.2.20 The following commuting diagram summarizes the aboveconstruction.

O(M) A(M)

J_(O (M)) J_(A(M))

-n

?J_

?J_

�J_(n)

Constructing a Morse decomposition from a lattice of attractors

6.2.21 In this section concerns the construction of a Morse decompo-sition from a lattice of attractors. In this setting the above commutingdiagram is not a satisfactory answer. However, the heuristics stillapply, to produce Morse sets from attractors we need to removethe appropriate connecting orbits. For the purposes of the currentdiscussion we restrict our attention to a finite lattice of attractorsA 2 subF

0,1Att(X). Birkhoff’s Representation Theorem 6.2.11 motivatesrestricting our attention further to J_(A). Recall that one of the prin-ciple properties of a join-irreducible element is that it has a uniquepredecessor, which leads the definition of the mapping


µ : J_(A) ! Invset(X)

A 7! µ(A) = pred(A)⇤ \ A.

6.2.22 Theorem. Let A be a (0, 1)-sublattice of Att(X). Then µ mapsthe join-irreducible attractors in A to Morse sets. Denote the image ofµ by

M(A) = µ

�

J_(A)�

=�

µ(A)�

� A 2 J_(A)

,

with a partial order given by: µ(A) µ(A0) if and only if A ✓ A0.Then, (M(A),) is a Morse decomposition in X and µ : J_(A) !M(A) is an order-isomorphism.

6.2.23 Lemma. Given an orbit gx ⇢ X define A1 = min {A 2 A | gx ⇢ A}and A0 = min {A 2 A | w(x) ⇢ A}. Then, A0, A1 2 J_(A) andA0 ⇢ A1. Furthermore,

ao(g�x ) ⇢ µ(A1) and w(x) ⇢ µ(A0). (6.1)

Proof: Since Ai 2 Att(X), i = 0, 1 they are compact forwardinvariant sets. In particular, if x 2 Ai, then w(x) ⇢ Ai. Thus, A0 ⇢ A1.To show that A0, A1 2 J_(A), it suffices to show that if Ai = A1

i ^ A2i

for Aji 2 A, j = 1, 2 then either A1

i = Ai or A2i = Ai. Observe that

Aji ⇢ Ai. Consider the case of i = 1. Since Aj

1, j = 1, 2 are invariant ifgx(s) 2 Aj

1, then gx(t) 2 Aj1 for all t � s. However, gx ⇢ A1

1 [ A21 and

thus without loss of generality we can assume that gx ⇢ A11. Since

A1 is the minimal element in A with this property, A1 ⇢ A11 which

implies A1 = A11. Consider the case i = 0. Since w(x) ⇢ A0, it can

without loss of generality be assumed that w(x) \ A10 6= ? and thus

w(x) ⇢ A10. The latter follows from Proposition 5.3.2 and Theorem

5.3.7. Therefore, A0 ⇢ A10 which implies A0 = A1

0.To characterize the alpha and omega limit sets observe that

(pred(A1), µ(A1)) is an attractor-repeller pair in A1. Assuming thatgx ⇢ pred(A1) contradicts the definition of A1 since pred(A1) ⇢ A1.If w(x) ⇢ µ(A1), then w(x) \ p(A1) = ?. This implies that A0 = A1and hence (6.1) is satisfied. Thus without loss of generality it canbe assumed that x 2 A1 \ (pred(A1) [ µ(A1)) in which case Theo-rem 5.3.7 guarantees (6.1).

Proof of Theorem 6.2.22. Proving that M(A) is Morse decompositionrequires demonstrating that the sets µ(A) are disjoint, nonemptycompact invariant sets, and that order on M(A) induced by J_(A)

satisfies the conditions in Definition 6.1.1. The latter follows fromLemma 6.2.23.


As indicated in the definition of µ, µ(A) is a nonempty compactinvariant set. Consider Ai 2 J_(A), i = 1, 2, with A1 6= A2. It needsto be shown that µ(A1) \ µ(A2) = ?. So assume the contrary. ByTheorem ??, µ(A1) \ µ(A2) is forward invariant. Since it is compact? 6= Inv (µ(A1) \ µ(A2)). Observe that Inv (µ(A1) \ µ(A2)) ⇢Inv(A1 \ A2) = A1 ^ A2 2 A, which follows from the fact that A isa lattice. Since A1 and A2 are join-irreducible A1 ^ A2 ⇢ pred(A1).Recall that µ(A1) = pred(A1)⇤ with respect to j|A and thus

Inv�

µ(A1)\µ(A2)�

\µ(A1) ⇢ (A1^A2)\µ(A1) ⇢ pred(A1)\pred(A1)⇤ = ?.

This is the desired contradiction.

6.2.24 Heuristically, given an attractor, if one strips away the appro-priate families of connecting orbits then one is left with a collectionof Morse sets. The following commuting diagram summarizes theabove construction.

J_ (A) M(A)

O�

J_ (A)�

O (M(A))

-µ

?O

?O

�O(µ)

Morse sets

beginproperty The lattice U(M) is anti-isomorphic to O(M) via thelattice anti-isomorphism I 7! Ic = M \ I, and this mapping is aninvolution. We have the following commuting diagram

O(M) U(M)

Att(X) Rep(X)?

n

-c

?n

-⇤

Define the mapping n : U(M) ! Rep(X), via I 7! n(I) = n(M \ I)⇤,by chasing the commutative diagram. The set n(M \ I)⇤ is the dualrepeller of n(M \ I) in X.

6.2.25 Theorem. Let M be a Morse decomposition in X. Then themapping n : U(M)! Rep(X) is a (0, 1)-lattice embedding. The imageA⇤(M) := n(U(M)), is a finite sublattice dual to L(M).

Proof: Since n : A(P) ! Att(X) is a lattice-embedding, then alson(P \ ·)⇤ is also a lattice-embedding, since the maps ⇤ are lattice


anti-isomorphisms. The composition is a isomorphism because

I [ I0 ! (P \ I) \ (P \ I0) ! n

�

(P \ I) \ (P \ I0)�

= n(P \ I) ^ n(P \ I0)

n(P \ I) ^ n(P \ I0) !�

n(P \ I) ^ n(P \ I0)�⇤

= n(P \ I)⇤ _ n(P \ I0)⇤.

The same holds for I \ I0, which completes the proof.

6.2.26 Remark. As for the definition of n on O(M) we can prove asimilar characterization for n on U(M) in terms of stable sets. LetI 2 U(M), then n(I) =

S

M2I Ws(M), which which characterizes allrepellers induced by U(M).

6.2.27 Theorem. Let M bea Morse decomposition and M0 2 M is aMorse set, then there exist an attractor A 2 Att(X) and a repellerR 2 Rep(X) such that M0 = A \ R.

Proof: By Theorem 6.2.15, A = n(#M0) is an attractor and byTheorem 6.2.25, R = n("M0) = n(#M0 \ M0)⇤ is repeller, and M0 ⇢A \ R. Let x 2 (A \ R) \ M0, then x 2 X \SM2M M, and for any orbitgx, ao(g�x ) ⇢ M0 and w(x) ⇢ M0. This contradicts the fact that M isa Morse decomposition and thus A \ R ⇢ M0.

6.2.28 Define Morse(X) =�

M = A \ R | A 2 Att(X), R 2 Rep(X)

,which called the set of Morse sets in X. As the lattices of attractorsand repellers Att(X) and Rep(X) also the Morse sets Morse(X) form alattice.


Morse(X),^�

is a bounded meet-semilatticewith the binary relation M ^M0 = Inv(M \M0). The neutral elements? and w(X).

Proof: Set inclusion ✓ is a partial order on Morse(X) and thereforeinf(M, M0) = Inv(M \ M0). By Theorem 6.2.27 there are attractorsA, A0 2 Att(X) and repellers R, R0 2 Rep(X) such that M = A \ R andM0 = A0 \ R0, and

M ^M0 = Inv(M \M0) = Inv(A \ R \ A0 \ R0)

and by Lemma ??

= Inv(A \ A0 \ R \ R0) = Inv�

Inv(A \ A0) \ Inv(R \ R0)�

= Inv�

(A ^ A0) \ (R \ R0)�

= (A ^ A0) \ (R \ R0),

which is a Morse set. Boundedness is immediate by construction,which proves that Morse(X) is a bounded meet-semilattice.


6.2.30 Let I = #I \ "I 2 C(M) be a convex set. This representation isobviously not unique. Define n : C(M)! Morse(X), via

I 7! n(I) = n(#I) \ n("I).

6.2.31 Theorem. The mapping n : C(M) ! Morse(X) is well-defined,i.e. independent of the representation. Moreover, the mapping n :C(P)! Morse(X) is a (0, 1)-semilattice embedding.

Proof: Let I1, I2 2 C(P), then

n(I1 ^ I2) = n

�

#(I1 ^ I2)�

\ n

�

"(I1 ^ I2)�

= n

�

#I1 ^ #I2�

\ n

�

"I1 ^ "I2�

=⇥

n

�

#I1�

^ n

�

#I2�⇤

\⇥

n

�

"I1�

^ n

�

"I2�⇤

=⇥

n

�

#I1�

^ n

�

#I2�⇤

\⇥

n

�

"I1�

\ n

�

"I2�⇤

Let A = n

�

#I1�

, A0 = n

�

#I2�

, R = n

�

"I1�

and R0 = n

�

"I2�

, thenby (A ^ A0) \ (R \ R0) = (A \ R) ^ (A0 \ R0) (compare Theorem6.2.29). This shows that n(I1 ^ I2) = n(I1) ^ n(I2) and it shows that n iswell-defined.

In particular, it follows that n is order-preserving. If I1 6= I2, thenby the expression for n in Theorem 6.2.31, n(I1) 6= n(I2), which showsthat n is an order-embedding and a bijective map onto its image. Thisimplies that n is a lattice-embedding.

6.2.32 Remark. The mapping n : C(M) ! Morse(X) characterizedin terms of connecting orbits between the Morse sets. Let I 2 C(M),then

n(I) =

[

M2IM

!

[

0

B

@

[

M,M02IM<M0

C�

M0, M�

1

C

A

,

which characterizes all Morse sets induced by C(M).

6.3 A Dynamical version of Birkhoff’s Representation Theorem

Birkhoff’s Representation Theorem as presented in Section ??, Theo-rem 6.2.11, allows us to prove that Morse decompositions and finite(0, 1)-lattices of attractors are equivalent.

6.3.1 Theorem. If M 2 MD(X) is a Morse decomposition in X, then


the following diagram commutes:

M

O (M) A(M)

J_ (O (M)) J_(A(M)) M

?O

?J_

-n

?J_

�J_(n) -µ

and µ � J_ � n �O = idM.

6.3.2 Lemma. It holds that µ � n � # = idM, and in particular µ

�1 =

n � #.

Proof: Let M0 2 M, then #M0 2 J_(O(M)) and n

�

#M0�

is a join-irreducible attractor J_(A(M)) by Theorem 6.2.15. By the commutingdiagram in §§6.2.12 and the fact that n : C(M) ! Morse(X) is asemilattice homomorphism we obtain

µ

�

n(#M0�

= pred�

n(#M0)�⇤ \ n(#M0) = n

�

pred(#M0)�⇤ \ n(#M0)

= n

�

#M0 \ M0�⇤ \ n(#M0) = n("M0) \ n(#M0) = M0,


Proof of Theorem 6.3.1. By Theorem 6.2.15, mapping n : O(M) ! A(M)

is a (0, 1)-lattice isomorphism and therefore by the functoriality ofJ_ (see Theorem ??), J_(n) : J_ (O (M)) ! J_ (A (M)) is a posetisomorphism with J_(n)�1 = n. By Birkhoff’s Theorem 6.2.11 wehave that J_ �O = # is a poset isomorphism. Combining these factsgives

µ � J_ � n �O = µ � J_(n)�1 � J_ �O = µ � n � J_ �O = µ � n � # = idM,

by Lemma 6.3.2, which proves the theorem.

6.3.3 Theorem. If A 2 subF0,1Att(X) then the following diagram

commutes,

A

J_ (A) M(A)

O�

J_ (A)�

O (M(A)) A

?J_

?O

-µ

?O

�O(µ) -n


and n �O � µ � J_ = idA.

6.3.4 Lemma. It holds that n � µ � #_ = idA, and in particular n

�1 =

µ � #_.

Proof: Let A 2 J_(A) be a join-irreducible attractor, then µ(#_A) =

#µ(A), since M(A) and J_(A) are order-isomorphic. Let J_(A) =

{A = n(#µ(A)) | A 2 J_(A)} and let where A be the associated lattice.Since n : O(M) ! Att(X) is a lattice embedding it follows that A isisomorphic to A. By Lemma 6.3.2 and the definition of µ we have thatfor any A 2 J_(A), µ(A) = µ

�

n(#µ(A))�

= µ(A). Now apply Lemma?? to f = n � µ � #_, which proves the lemma.

Proof Theorem 6.3.3. By Theorem 6.2.22, mapping µ : J_(A) !M(A) is a poset isomorphism and therefore by the functoriality ofO (see Theorem ??), O(µ) : O (J_ (A)) ! O (M (A)) is a (0, 1)-latticeisomorphism with O(µ)�1 = µ. By Birkhoff’s Theorem 6.2.11 we havethat O � J_ = #_ is a lattice isomorphism. Combining these facts gives

n �O � µ � J_ = n �O(µ)�1 �O � J_ = n � µ �O � J_ = n � µ � #_ = idA,

by Lemma 6.3.4, which proves the theorem.

6.3.5 So far we considered (0, 1)-sublattices and their equivalence toMorse decompositions of X. If we broaden the notion of sublatticewe may consider sublattices of Att(X) which contain ?, and theset of such sublattices will be denoted by subF

0Att(X). Similarly,one defines subF

1Att(X) as the sets of sublattices containing X andsubFAtt(X) as the set of all finite sublattices of Att(X). All these setsare sublattices of subFAtt(X). By the same procedures as describedabove A 2 subF

0Att(X) corresponds to Morse decompositions of anattractor A = supA, and A 2 subF

1Att(X) corresponds to Morsedecompositions of an repeller R = infA. An arbitrary A 2 subFAtt(X)

corresponds to a Morse decomposition of a Morse set

S = supA\ (infA)⇤.

The proof of these facts follow from restricting j to invariant set S.The idea of using more general sublattices is used to derive Morserelations in Section 7.4. For example sublattices A 2 subFAtt(X)

consisting of two attractors, i.e. A = {A0, A}, A0 ⇢ A, corresponds toa single Morse set M = A \ A0⇤.

6.4 Morse tilings

In this section we apply the theory of Section 6.1 to attracting neigh-borhood lattices as opposed to lattices of attractors. Analogous


to attracting and repelling neighborhoods we define the notionof isolating neighborhood. A neighborhood U ⇢ X is an isolat-ing neighborhood if Inv(cl(U)) ⇢ int (U), and the set of isolatingneighborhoods is denoted by INbhd(X). For isolating neighbor-hood U, the set Inv(U) is the associated isolated invariant set. Notethat Inv(U) = Inv(cl(U)). An invariant set S ⇢ X is an isolatedinvariant set if the there exists an isolating neighborhood U suchthat S = Inv(U). The set of isolated invariant sets is denoted byIsol(X),which is a poset with respect to set inclusion. The meetoperation on the lattice of invariant sets also restricts to Isol(X),i.e. for any two isolated invariant sets S, S0 2 Isol(X) it holds thatS ^ S0 = Inv(S \ S0, j) 2 Isol(X).


Isol(X, j),^�

is a bounded meet-semilattice.The elements ? and Inv(X) = w(X) are the neutral elements 0 and 1respectively.

Proof: The set Isol(X) is an ordered set under the ordering given byinclusion ✓. Let S, S0 2 Isol, then, as before, inf(S, S0) = Inv(S \ S0) =S ^ S0. It holds that

Inv(S \ S0) ⇢ S \ S0 ⇢ int (U) \ int (U0) = int (U \U0),

where U and U0 are isolating neighborhoods for S and S0 respectively.The intersection U \U0 has the property that Inv(S \ S0) = Inv(U \U0), since S and S0 are maximal. This proves that S ^ S0 2 Isol(X).Since inf(S, S0) exist for all S, S0 2 Isol, the ordered set Isol is a semi-lattice with the binary operation defined by inf. Boundedness isimmediate by construction.

From the above proof it follows that the intersection of two isolat-ing neighborhoods is again an isolating neighborhood.

6.4.2 Corollary. The set�

INbhd(X),\�

is a bounded meet-semilattice.The elements ? and X are the neutral elements 0 and 1 respec-tively. The mapping Inv : INbhd(X) ! Isol(X) is a surjective (0, 1)-semilattice homomorphism.

6.4.3 Let N ⇢ ANbhd(X) be a finite (0, 1)-sublattice, and for U 2J_(N) define the mapping U 7! µ(U) = U \ pred(U) = pred(U)c \U. Since pred(U), U 2 ANbhd(X) and pred(U)c 2 RNbhd(X),itfollows from Corollary 6.4.2 that µ(U) 2 INbhd(X), and thereforeµ : J_(N) ! INbhd(X) is well-defined. In the following definition weintroduce the analogue of a Morse decomposition.

6.4.4 Definition. Let N be a (0, 1)-sublattice of ANbhd(X). Denote theimage of µ by

T(N) = µ

�

J_(N)�

=�

C = µ(U)�

�U 2 J_(N)

,


with a partial order given by: µ(U) µ(U0) if and only if U ✓U0. Then, (T(N),) is called a Morse tiling of X induced by N andµ : J_(N)! T(N) is an order-isomorphism.

6.4.5 A Morse tiling consists of tiles C 2 INbhd(X). We start weestablishing a number of fundamental properties of Morse tiling.

6.4.6 Theorem. Let T(N) be a Morse tiling and let I 2 O(T(N)), thenthe mapping I 7! (I) =

S

C2I C takes values in ANbhd(X) and: O(T(N)) ! ANbhd(X) is a (0, 1)-lattice embedding. Moreover, thefollowing diagram commutes

N

J_ (N) T(N)

O�

J_ (N)�

O (T(N)) N

?J_

?O

-µ

?O

�O(µ) -

and �O � µ � J_ = idLatFD

.

Proof: By the Birkhoff Representation Theorem 6.2.11, O � J_ = #_is a lattice isomorphism and (#_)�1 = _ is set union. The mappingµ : J_(N) ! T(N) is an order-isomorphism and by functoriality of O(see §??) O(µ) = µ

�1 acting on O(T(N)). This gives most of the abovediagram, except for the action of . Chasing the diagram we have that#_ = O � J_ = µ

�1 �O � µ � J_, and therefore _ � µ

�1 �O � µ � J_ =

idLatFD

. Define = _ � µ

�1. Let I 2 O(T(N), then the elements C 2 I aregiven by C = µ(U) and thus (I) =

W

U2µ

�1(I) µ

�1�µ(U)

�

=S

C2I C.The mapping is obviously a (0, 1)-lattice embedding into ANbhd(X).

From this Theorem 6.4.6, which is essentially Birkhoff’s Theorem,we deduce the following properties.

6.4.7 Theorem. Let T(N) be a Morse tiling, then X =S

C2T(N) C.Moreover, C \ C0 = ?, for any pair C, C0 2 T(N).

Proof: The first statement is obvious since : O(T(N)) ! N is anisomorphism. As for the second claim we argue as follows. Assumewithout loss of generality that C < C0 or CkC0, then C ⇢ (#C) andC0 ⇢ (#C)c =

�

(#C)c�, since C0 2 (#C)c. This proves that C \ C0 = ?.


6.4.8 A sublattice A 2 subF0,1Att(X) can be regarded as the image of

a (0, 1)-lattice homomorphism f : L ! Att(X), where L is a finite,distributive lattice. In this case we do not assume that f is injective.By the Birkhoff Representation Theorem may assume that L is alattice of lower sets O(P) for some finite poset (P,). Following thecommuting diagram in Theorem 6.3.1 we obtain:

O (P) A

P J_ (O (P)) J_(A) M

?J_

-f

?J_

� sup �J_( f ) � µ

�1

where A = f (O(P)) and f is a mapping onto its image A. Since fis a surjective (0, 1)-lattice homomorphism, the mapping J_( f ) isan order embedding and g = sup �J_( f ) � µ

�1 : M ! P is anorder embedding. Via g�1 the sets M 2 M are labeled by P andfor p 62 image(g), the empty set is assigned. If we start from anembedding g : M! P, then

M P

A(M) O (M) O(P)?

O

-g

?

O

�n �O(g)

Since g is injective, then mapping f = n � O(g) : O(P) ! A(M) isa surjective (0, 1)-lattice homomorphism. A prime example is themapping Inv : N ! A, in which case we can represent N by theassociated Morse tiling T(N).

7The Conley Index

A Morse tiling of X gives an order injection of a Morse decomposi-tion into the tiling and a sublattice of attracting neighborhoods givesa lattice homomorphism onto a sublattice of attractors. These map-pings need not be isomorphisms in general and therefore some tilinghave an empty invariant set, and equivalently two different attractingneighborhoods may correspond to the same attractor. By invokingtopological information about tiles/attracting neighborhoods andthe behavior of j on it, we may sometimes be able to decide whetherthere exists a non-trivial invariant set.

7.1 Blocks

We consider attracting neighborhoods that have special propertieswith respect to a flow j.

7.1.1 Definition. A set N ⇢ X is called an attracting block if N 2Invset+(X) and j(t, cl(N)) ⇢ int (N) for all t > 0.

Repelling blocks are defined similarly by using Invset�(X) andt < 0. In order to show that every attractor/repeller allows an attract-ing/repelling block we have the following fundamental theorem.

7.1.2 Theorem. Let (A, A⇤) be an attractor-repeller pair for j. Thenthere exists a continuous function V : X ! R such that(i) V�1(0) = A and V�1(1) = A⇤;(ii) for x 2 X \ A [ A⇤, V(j(t, x)) < V(x), for all t > 0.Such a function V is called a Lyapunov function for (A, A⇤).

Proof: Define the continuous function

dp(x) :=d(x, A)

d(x, A) + d(x, A⇤), (7.1)

for which dp�1(0) = A and dp�1(1) = A⇤. Now let mdp(x) :=supt2T+ dp(j(t, x)), and note that mdp is a mapping from X to [0, 1].


We claim that mdp has the following three properties:

mdp�1(0) = A, mdp�1(1) = A⇤, and mdp(j(t, x)) mdp(x)(7.2)

for all t 2 T+. Indeed, since A is forward invariant and dp � 0,the first property mdp�1(0) = A holds. In the case that A⇤ = ?,then trivially A⇤ ⇢ mdp�1(1). Otherwise, if x 2 A⇤, then dp(x) =

1 so that mdp(x) = 1 and A⇤ ⇢ mdp�1(1). Now consider theopposite inclusion. If A⇤ = ?, then dp is a continuous function ona compact space into [0, 1) so that dp(x) < 1 for all x 2 X, andhence mdp�1(1) = ? = A⇤. Otherwise if mdp(x) = 1, then thereexists a sequence tn ! • with j(tn, x) ! y so that dp(y) = 1and y 2 w(x) \ A⇤. Proposition 5.3.2 implies that x 2 A⇤ so thatA⇤ ⇢ mdp�1(1).

Since mdp is defined as the supremum over t 2 T+, it is not imme-diately clear that mdp is a continuous function on X. Continuity isproved in a more general setting in Lemma 7.1.7 below.

Now define the function

V(x) =Z

T+mdp(j(s, x))dµT+(s),

where dµT+(s) = 2�s�1 Ân2Z+ d(s � n)ds for T+ = Z+ anddµT+(s) = e�sds for T+ = R+. From the definition, mdp(x) 2 (0, 1)for all x 2 X \ (A [ A⇤), and therefore V : X ! [0, 1] is continuous,V�1(0) = A, and V�1(1) = A⇤.

Now we have

V(j(t, x))�V(x)

=Z

T+mdp(j(t + s, x))dµT+(s)�

Z

T+mdp(j(s, x))dµT+(s)

=Z

T+

⇥

mdp(j(t + s, x))�mdp(j(s, x))⇤

dµT+(s) 0,

and equality holds if and only if the integrand is zero for all s � 0.To show that strict inequality holds for x 2 X \ (A [ A⇤), we claimthat mdp(j(s, x)) ! 0 as s ! • for x 2 X \ A⇤. Suppose not. Thenthere exists sn ! • such that j(sn, x) ! y but mdp(j(sn, x)) 6! 0,which implies that y 2 w(x) ⇢ A but mdp(y) 6= 0, a contradiction.Since mdp�1(0) = A, we have strict inequality for x 2 X \ (A [ A⇤).Therefore V is a Lyapunov function, which is normalized to takevalues in the interval [0, 1].

The above construction can be performed in a more general set-ting.

7.1.3 Definition. Let A, R be compact subsets of X. A function dp :S \ (A \ R)! [0, 1] a called a distance potential for the pair (A, R) if

the conley index 141

(i) dp is locally Lipschitz continuous on X \ (A \ R),(ii) dp|A\(A\R) = 0, and(iii) dp|R\(A\R) = 1.

7.1.4 Exercise. Show that if A \ R = ?, then a distance potential is globallyLipschitz continuous.

7.1.5 Exercise. Show that if A \ R = ?, then the function dp defined inequation (7.1) is locally Lipschitz continuous on X \ (A [ R) with Lipschitzconstant less than 1/d(A, R).

In this context, for any distance potential, the function

mdp(x) := supt2T+

dp(j(t, x)) (7.3)

satisfies the properties in equation (7.2) using the same arguments inthe proof of Theorem ??. Thus any distance potential can be used inthe proof of Theorem ?? provided continuity of mdp is established.

7.1.7 Lemma. Let j : T+ ⇥ X ! X a dynamical system on a compactmetric space X. If (A, A⇤) is an attractor-repeller pair in Y, then mdpis continuous for any distance potential dp for (A, A⇤).

Proof: Recall that mdp satisfies the three properties in equation7.2; in particular mdp(A⇤) = 1. If {xi} is a sequence in X withlimi!• xi = x 2 A⇤, then 1 � mdp(xi) � dp(xi) ! 1 as i ! •, andhence mdp is continuous at all x 2 A⇤.

Similarly mdp(A) = 0. Suppose {xi} is a sequence in X withlimi!• xi = x 2 A. By Theorem ?? for every e > 0 there existsa trapping region N

e

⇢ Be

(A) for A. Thus there exists ne

> 0such that xi 2 N

e

for all i � ne

. Since Ne

is forward invariant,j(T+, xi) ⇢ N

e

⇢ Be

(A) for all i � ne

. Fix i � ne

. For each t 2 T+,since j(t, xi) 2 B

e

(A), there exists zt 2 A such that d(j(t, xi), zt) < e.The Lipschitz continuity of dp implies that |dp(j(t, xi))� dp(zt)| Cd(j(t, xi), zt) < Ce. Since zt 2 A, we have dp(zt) = 0 which impliesthat mdp(xi) < Ce. This proves that mdp(xi) ! 0 as i ! •, andhence mdp is continuous at all x 2 A.

For x0 2 X \ (A [ A⇤) let d1 = d(x0, A) > 0 and d2 = d(x0, A⇤) > 0.Then for every 0 < d < min{d1, d2} there exists m > 0 suchthat dp(x) � m > 0 for all x 2 B

d

(x0). Also, there exists 0 <

e < min{d1, d2} such that dp(x) < m for all x 2 Be

(A). Sincedp(j(t, x0)) ! 0 as t ! •, there exists t(x0) > 0 such thatd(j(t(x0), x0), A) < e/2. Since j(t(x0), ·) is uniformly continu-ous, there exists d < min{d1, d2} such that d(j(t(x0), x), A) < e forall x 2 B

d

(x0). Therefore, dp(Bd

(x0)) > dp(j(t(x0), Bd

(x0))), andhence dp(j(t, x)) attains its maximum in [0, t(x0)] for all x 2 B

d

(x0).


For x 2 X, define z(t, x) := dp(j(t, x)). By continuity, for everye > 0 there exists d > 0 such d(x, x0) < d implies that |z(t, x) �z(t, x0)| < e uniformly for t 2 [0, t(x0)]. Let t0 and t1 be times wheremdp(x0) = z(t0, x0) and mdp(x) = z(t1, x) with t0, t1 2 [0, t(x0)].Using the above inequalities

mdp(x0) = z(t0, x0) < z(t0, x) + e z(t1, x) + e = mdp(x) + e.

Similarly mdp(x) < mdp(x0) + e so that |mdp(x)�mdp(x0)| < e.This proves the continuity of mdp at x0 2 X \ (A [ A⇤).

A direct consequence of the existence of Laypunov functions is theexistence of blocks. Let a 2 (0, 1), then Va := {x 2 X | V(x) a}is an attracting block. Indeed, if x 2 Va, then for t > 0, V(j(t, x)) <V(x) a and thus j(t, x) 2 int (Va). The set Va is obviously forwardinvariant. The same holds for the existence of repelling blocks byconsidering Va := {x 2 X | V(x) � a}. Blocks of this type are calledattracting/repelling Lyapunov blocks and forms sublattices of ANbhd(X)

and RNbhd(X) and are denoted by ALBlock(X) and RLBlock(X)

respectively. Since the attracting blocks we use here are attractingLyapunov blocks we often refer to attracting Lyapunov blocks asattracting blocks.

Since a Morse set M is equivalent to a pair of nested attractorsA0 ( A, i.e. M = A \ A0⇤, one can finds a Morse tile T = Va \ V0a0 .Such an isolating neighborhood is called an isolating Lyapunov block,or isolating block for short. The set of isolating Lyapunov blocks isdenoted by ILBlock(X). Based on the previous considerations suchblocks exists for any Morse set M.

7.2 Wazewski’s Principle

An attractor A with attracting block Va is given by A =T

t�0 j(t, Va).It follows from the definition of Lyapunov function and the groupproperty that V(j(·, x)) is strictly decreasing in t for all x 2 X \ (A [A⇤). Therefore, j(t, Va) ⇢ j(s, Va) for all t > s.

If N ) N0 are attracting Lyapunov blocks for the same attractorA = w(N) = w(N0), then there exists a t > 0 such that j(t, N) ⇢ N0

for all t � t. Conversely, if j(t, N) ⇢ N0 for all t � t, then A =

w(N) = w(N0) = A0. Indeed,

A =\

t�0j(t, N) ⇢

\

t�t

j(t, N) =\

t�0j(t, j(t, N)) ⇢

\

t�0j(t, N0) = A0.

By definition A0 ⇢ A, since N0 ⇢ N, and therefore A = A0. These con-siderations suggest that for nested attracting blocks N0 ⇢ N whichcorrespond to the same attractor it holds that N0 is a deformationretract of N. Recall the definition of deformation retract.


7.2.1 Definition. Let A ⇢ X. A deformation retraction of X onto A is acontinuous map h : X⇥ [0, 1]! X such that

h(x, 0) = x for all x 2 X,

h(x, 1) 2 A for all x 2 X,

h(a, 1) = a for all a 2 A.

If such an h exists, then A is called a deformation retract of X. Themap h is called a strong deformation retraction. If the third identity isreinforced as follows:

h(a, s) = a for all a 2 A and all s 2 [0, 1],

then the set A is a strong deformation retract of X.

Note that the map r : X ! A defined by r(x) = h(x, 1) has theproperty r|A = idA. Any continuous map with this property is calleda retraction and its image A is called a retract. Thus a deformationretract is a special case of a retract.

In order to show that in the above example N0 is a strong deforma-tion retract we formulate the WazewskiPrinciple. Let W ⇢ X be anysubset. Define

W0 = {x 2W | there exists t > 0 such that j(t, x) /2W}

andW� = {x 2W | j([0, t), x) 6⇢Wfor all t > 0}. (7.4)

Then W0 and W� are the sets of all points which eventually leave Wand which immediately leave W in forward time respectively. Notethat W� ⇢W0, and both sets could be empty.

7.2.2 Definition. A set W is a Wazewski set if the following conditionsare satisfied.

1. If x 2W and j([0, t], x) ⇢ cl(W), then j([0, t], x) ⇢W.

2. W� is closed relative to W0.

7.2.3 Theorem. If W is a Wazewski set, then W� is a strong deforma-tion retract of W0 and W0 is open relative to W.

Proof: The first step is to construct the strong deformation retrac-tion

r : W0 ⇥ [0, 1]!W0.

Define t : W0 ! R by

t(x) = sup {t � 0 | j([0, t], x) ⇢W } .


By the definition of W0, t(x) is finite and by the continuity of theflow j([0, t], x) ⇢ cl(W). Since W is a Wazewski set, j(t, x) 2W, andin fact the definition of t implies that j(t, x) 2 W�. Observe thatt(x) = 0 if and only if x 2W�.

Assume for the moment that t is continuous and define r byr(x, s) = j(st(x), x). Now notice that

r(x, 0) = j(0, x) = x

r(x, 1) = j(t(x), x) 2W�

and for y 2W�

r(y, s) = j(st(x), y) = j(0, y) = y.

Therefore r is a strong deformation retraction of W0 to W�.Returning now to the question of continuity we first prove that t

is upper semicontinuous. Let x 2 W0 and e > 0, then j([t(x), t(x) +e], x) 6⇢W. By the first condition there exists t0 2 [t(x), t(x) + e] suchthat j(t0, )x) 62 cl(W). Thus we can choose V be a neighborhood ofx · t0 such that V \ cl(W) = ?. Now let U be a neighborhood of xsuch that j(t, U) ⇢ V. Then for y 2 U \W, j(t0, y) 62 W. Note thatthis proves that W0 is open relative to W and that t(y) < t(x) + e.Hence, t is upper semi-continuous.

To prove lower semi-continuity of t, let x 2 W0/W� and let 0 <

e < t(x). Then j([0, t(x)� e], x) ⇢ W0. Since W� is closed relativeto W0, j([0, t(x)� e], x) \W� = ? and hence for all s 2 [0, t(x)� e]

there exists a neighborhood Us of j(s, x) such that Us \W� = ?. Ofcourse {Us} covers j([0, t(x)� e], x) which is compact and hence afinite number {Usi | i = 1, . . . I } covers j([0, t(x) � e], x). Let U =

[Ii=1Usi , then U is open which implies there exists V a neighborhood

of x such that j([0, t(x)� e], V) ⇢ U. Now U \W� = ? implies thatfor all y 2 V, j([0, t(x)� e], y) \W� = ?. Thus, t(y) � t(x)� e.This implies that t is lower semicontinuous and hence continuous.

7.2.4 Lemma. Let N0 ( N be attracting Lyapunov blocks such thatw(N) = w(N0), then N0 is a strong deformation retract of N. Theconverse is not true in general.

Proof: Define W = cl(N \ N0), then, since j(t, N) ⇢ N0 for t � t, itfollows that W0 = W and W� = {x 2W | V0(x) = a0}. The set W is aWazewski set and by Theorem 7.2.3, then W� is a strong deformationretract of W0 = W. This implies that N0 is a strong deformationretract of N by extending the retraction via the identity.

In order for a nested pair of attracting blocks N0 ( N to give anested pair of attractors it is not necessary that N0 is a strong defor-mation retract of N. See for example Morse decomposition M5 in


Example 6.1.3. The above lemma does gives a criterion for nestedattractors however.

7.2.5 Lemma. Let N0 ( N be a pair of nested attracting Lyapunovblocks and N0 is not a strong deformation retract of N. Then, A0 =w(N0) ( w(N) = A.

Proof: If A0 = w(N0) = w(N) = A, then from Lemma 7.2.4it follows that N0 is a strong deformation retract of N, which is acontradiction.

Wazewski’s Theorem can be used in the same way and provides acriterion for non-trivial invariant sets inside Wazewski sets.

7.2.6 Corollary. [Wazewski Principle] If W is a Wazewski set and W�

is not a strong deformation retract of W, then W \ W0 6= ?; that is,there exists complete forward orbits which stay in W for all positivetime. In particular,

Inv(W) 6= ?.

when W is compact.

Proof: The supposition W = W0 contradicts Theorem 7.2.3. If W iscompact and x 2W \ W0, then w(x) 6= ? and thus w(x) ⇢ Inv(W).

Define the Morse tile is T = N \ N0. It is straightforward to verifythat W = cl(T) is a Wazewski set. If N0 is not a strong deformationretract of N, then Inv(T) 6= ? by Corollary 7.2.6. If we invokeLemma 7.2.5 instead of Corollary 7.2.6, then A0 ( A and thereforeInv(T) = Inv(N \N0c) = Inv(A\ A0⇤) = A\ A0⇤ 6= ?. This motivatesthe search of invariants of (N, N0) that guarantee that N0 is not astrong deformation retract of N. For example if H⇤(N, N0) 6= 0, thenthe latter holds and Inv(T) 6= ?.

7.2.7 Exercise. Show that the isolating block W = cl(T) is a Wazewski set.

7.3 The Homological Conley Index for Morse sets

Het H⇤ be the singular homology functor with coefficients in Z

and let N0 ( N be attracting blocks. The attracting blocks are atopological pair and the relative homology H⇤(N, N0) is well-defined.From the above considerations we derive the following result.

7.3.1 Theorem. Let N0 ( N be a nested pair of attracting blocks andH⇤(N, N0) 6= 0, then Inv(N \ N0) 6= ?. In particular, A0 = w(N0) (w(N) = A and M = Inv(N \ N0) = A \ A0⇤.

Conversely, if M is a Morse set, then there exists nested attractorsA0 ( A and an associated pair N0 ( N of attracting blocks. In theprevious section we observed that N0 may be deformation retract


of N in which case H⇤(N, N0) = 0. If X is infinite dimensionalthis may happen since deformation retracts are easy to construct.The homological information in the topological pair motivates thefollowing index for Morse sets.

7.3.2 Definition. Let M be a Morse set and N0 ( N a nested pair ofattracting blocks such that M = Inv(N \ N0). Define the homologicalConley index of M by

HC⇤(M, j) := H⇤(N, N0). (7.5)

The nested pair (N, N0) of attracting blocks is called an index pair ofattracting blocks for M.

We need to verify that the definition does not depend on theparticular choice of index pair. If T = cl(N \ N0) = N \ N0c, thendefine T� = N \ ∂N0c which is called the exit set.

7.3.3 Lemma. Let (N, N0) be an index pair of attracting blocks, thenH⇤(N, N0) ⇠= H⇤(T, T�).

Proof: Define N0e

= {x 2 N | V0(x) a0 + e}, then N0 is astrong deformation retract of N0

e

and therefore H⇤(N0e

, N0) ⇠= 0. As aconsequence H⇤(N, N0) ⇠= H⇤(N, N0

e

). Indeed, the triple N0 ⇢ N0e

⇢ Nyields the long exact sequence

· · ·! Hk(N0e

, N0)! Hk(N, N0)! Hk(N, N0e

)! Hk�1(N0e

, N0)! · · · ,

which reduces to the short exact sequence

0! Hk(N, N0)! Hk(N, N0e

)! 0.

From this the isomorphism follows. The isolating block N0 satisfiesN0 ⇢ int N N0

e

and therefore by excision

H⇤(N, N0) ⇠= H⇤(N \ int N N0, N0e

\ int N N0).

By definition T = clN(N \ N0) = N \ int N N0 and T�e

= N0e

\int N N0 and therefore H⇤(N, N0) ⇠= H⇤(T, T�

e

). Since N0 is a strongdeformation retract of N0

e

it follows that T� is a strong deformationretract of T�

e

. As before H⇤(T, T�e

) ⇠= H⇤(T, T�), which proves thelemma.

This above characterization proves the following lemma.

7.3.4 Lemma. Let (N1, N01) and (N2, N02) be index pairs of attractingblocks such that T = N1 \ N01

c = N2 \ N02c, then H⇤(N1, N01) ⇠=

H⇤(N2, N02) ⇠= H⇤(T, T�).The last step is to show that for two Morse tiles T, T0, for which

Inv(T) = Inv(T0) = M, the index is the same.


7.3.5 Lemma. Let (N1, N01) and (N2, N02) be index pairs of attract-ing blocks such that Inv(N1, N01) = Inv(N2, N02) = (A, A0), thenH⇤(N1, N01) ⇠= H⇤(N2, N02).

Proof: By Assumption N1 \ N2 is an attracting block for A andN01 \ N02 is an attracting block for A0. Since Inv(N1 \ N2) = Inv(N1) =

A, Lemma 7.2.4 gives that N1 \ N2 is a strong deformation retract ofN1. Consequently, H⇤(N1, N01 \ N02) ⇠= H⇤(N1 \ N2, N01 \ N02). By thesame token N01 \ N02 is a strong deformation retract of N01 and thusH⇤(N01, N01 \ N02) ⇠= 0.

Next consider the triple N01 \ N02 ⇢ N01 ⇢ N1, which gives the shortexact sequence

0! Hk(N1, N01 \ N02)! Hk(N1, N01)! 0.

This shows that H⇤(N1, N01 \ N02) ⇠= H⇤(N1, N01). Combining the towisomorphisms we obtain H⇤(N1 \ N2, N01 \ N02) ⇠= H⇤(N1, N01).

The above arguments also show that H⇤(N1 \ N2, N01 \ N02) ⇠=H⇤(N2, N02), which proves the lemma.

The Lemmas 7.3.3 - 7.3.5 show that HC⇤(M, j) is well-defined. Inaddition, if T is a Morse tile for M, then HC⇤(M, j) ⇠= H⇤(T, T�).The above arguments also show that if M = A \ R, then the tileT = NA \ NR corresponds to an index pair in a canonical way. DefineA0 = A \ R⇤ ⇢ A, then A \ A0⇤ = A \ (A⇤ [ R) = (A \ A⇤) [ (A \R) = M, which gives the index pair N = NA and N0 = NA \ Nc

R. Thisshows that HC⇤(M, j) is well-defined for any Morse tile T.

One can define the notion of isolating block independent of attract-ing and repelling blocks and attracting and repelling blocks can beconstructed.

7.3.6 Definition. A compact set B ⇢ X, which is the closure of anopen set in X, is called an isolating block, if there exists a time t > 0such that for every point x 2 ∂B the following three alternatives hold.Either,

(i) x 2 B� := {x 2 ∂B | j((0, t], x) \ B = ?}, or

(ii) x 2 B+ := {x 2 ∂B | j([�t, 0), x) \ B = ?}, or

(iii) x 2 B� \ B+, the set of ‘tangent’ points, i.e. j([�t, t], x) \ B =

{x}.

In particular, ∂B = B� [ B+.

We show now that the Conley index of a Morse can also be com-puted via an isolating block and therefore the Conley index is inde-pendent of the choice of isolating block.


7.3.7 Lemma. Let M ⇢ X be a Morse set and B ⇢ X is an isolatingblock for M, i.e. Inv(B) = M. Then, HC⇤(M, j) ⇠= H⇤(B, B�).

Proof: Let (N, N0) be an index pair of attracting blocks for M.Define B

t

= j(t, B) and B�t

= j(t, B�) (t is not necessarily the sameas in the definition of isolating block), the (B, B�) and (B

t

, B�t

) arehomeomorphic couples and therefore H⇤(B, B�) ⇠= H⇤(B

t

, B�t

).Choose t > 0 large enough such that B�

t

⇢ N0. Indeed, from theMorse decomposition A0 < M < A⇤ one conclude that points in B�

t

converge to A0. Let W = Bt

\ N0, then W is a Wazewski set withW = W0 and W� = B�

t

is a strong deformation retract of W0 = W.Consequently, H⇤(B

t

, W) ⇠= H⇤(Bt

, B�t

).Define C = N0 [ B

t

and D = N0 \ W, then clC(D) ⇢ int C(N0).The excision property of homology gives: H⇤(C, N0) ⇠= H⇤(C \D, N0 \ D) = H⇤(B

t

, W). Combining the three isomorphisms we haveH⇤(B, B�) ⇠= H⇤(C, N0).

The set C is an attracting block contained in N. The index pairs(N, N0) and (C, N0) define the same pair (A, A0) and thereforeby Lemma 7.3.5, H⇤(N, N0) ⇠= H⇤(C, N0) and thus H⇤(N, N0) ⇠=H⇤(B, B�), which proves the lemma.

7.4 The Morse relations

The order structure of a Morse decomposition M contains informa-tion about the connecting orbits between Morse sets M 2 M. IfMkM0, then no connections exist. On the other hand if two Morsesets M, M0 are ordered, then connections need not exist between theMorse sets. Before addressing the problem of connections we firstconsider relations between the Conley indices of Morse sets in thespirit of Morse Theory.

Morse decompositions for X

If we discard all information about non-existence of connectionsand coarsen a Morse decomposition M to a totally ordered Morsedecomposition M0 > M (still consisting of the same Morse sets). LetA(M) be the lattice of attractors associated with M. For each A 2 A

choose an attracting block BA and define

B = [{BA}A2A] =\

�

L 2 subF0,1Att(X) | {BA}A2A ⇢ L

.

This yields a lattice surjection w : B! A. The lattice B yields a Morsetiling T such that M ! T is an order injection. Now coarsen T ! T0

such that T0 is totally ordered. Via the functoriality of T0 ! B0 thisyields an order injection M0 ! T0, where M0 = M(A0) and A0 = w(B0).The lattice B0 is a totally ordered lattice of attracting neighborhoods.


In the following B0 ! A0 can be either an isomorphism or asurjection. Every pair (B, B0), with B 2 J_(B0) and B0 = pred(B),corresponds to a Morse tile T = B \ B0c. Consider the topological pairB0 ⇢ B, which yields the long exact sequence

· · ·∂k+1��! Hk(B0)

i⇤k�! Hk(B)j⇤k�! Hk�1(B, B0)

∂k�! · · · ,

with ker i⇤k = im ∂k+1, ker j⇤k = im i⇤k and ker ∂k = im j⇤k . For the ranksof the subspaces involved we obtain:

rank Hk(B0) = rank im i⇤k + rank ker i⇤k = rank im i⇤k + rank ∂k+1

rank Hk(B) = rank im j⇤k + rank ker j⇤k = rank im j⇤k + rank im i⇤krank Hk(B, B0) = rank ∂k + rank ker ∂k = rank ∂k + rank im j⇤k ,

which gives the relation

rank Hk(B0)+ rank Hk(B, B0)� rank Hk(B) = rank ∂k + rank ∂k+1, 8k.

Define the Poincare polynomial for a topological pair (A, B) by:Pt(A, B) = Âk�0 rank Hk(A, B)tk, then

Pt(B0) + Pt(B, B0)� Pt(B) = (1 + t)Qt,

where Qt = Âk�0 rank ∂ktk � 0. Enumerate B0 by the P0 ⇢ N [ {0},i.e. P0 ! N0 is an order-isomorphism, then B0 = ?, Bn = X andBp 2 J_(B0) for all p � 1. For the the topological pairs (Bp, Bp�1) wehave

Pt(Bp�1) + Pt(Bp, Bp�1)� Pt(Bp) = (1 + t)Qt(p),

Summing over p 2 P0 gives

Pt(?) + Âp2P0

Pt(Bp, Bp�1)� Pt(X) = Âp2P0

(1 + t)Qt(p) = (1 + t)Qt.

The surjection B0 ! A0 induces the injection M0 ! P0, selectingthe Morse blocks for M 2 M0. It follows from Lemma 7.2.4 thatPt(Bp, Bp�1) = 0 for all p 62 im (M0 ! P0) and therefore

ÂM2M

CPt(M, j) = Pt(X) + (1 + t)Qt,

where CPt(M, j) = Pt(B, B0), with (B, B0) an index pair for M fromthe lattice B0. Summarizing we have the following theorem.

7.4.1 Theorem. Let M be a Morse decomposition for X, then thefollowing identity of formal series in Z[t] holds:

ÂM2M

CPt(M, j) = Pt(X) + (1 + t)Qt, (7.6)

where Qt � 0 is a formal series with non-negative coefficients in Z.

The relation (7.6) are called the Morse relations for M and provideobstructions for CPt(M, j) with respect to the topology of X. Infor-mation about connecting orbits is contained in the term Qt.


Morse decompositions for a Morse set

Let S be a Morse and S = A \ A0⇤, where A0 ( A. If M(S) is a Morsedecomposition of S, then a generalization of Theorem 7.4.1 can beproved. Let M0 be linearly ordered poset that is a coarsening of Mand consists of the same Morse sets. Then M(X), consisting of thesets M 2 M0(S), A0 and A⇤, and partially ordered as A0 < M < A⇤

for all M 2 M0(S), is a Morse decomposition of X. Since M(X) istotally ordered it follows from Section 7.4 that B(X) given by

? = B0 ⇢ B1 ⇢ B2 ⇢ · · · ⇢ Bn�2 ⇢ Bn�1 ⇢ Bn = X,

with B1 = B0 and Bn�1 = B, is a sublattice of attracting blocks isomor-phic to A(M(X)) 2 subF

0,1Att(X). As for attractors the sublattice

B0 = B1 ⇢ B2 ⇢ · · · ⇢ Bn�2 ⇢ Bn�1 = B,

is isomorphic to A(M0(S)) 2 subFAtt(X). Such a nested sequence ofsets is also called a filtration of B and Poincaré polynomials of pairssatisfy the following lemma.

7.4.2 Lemma. Let X0 ⇢ · · · ⇢ Xn be a filtration (of metric spaces),then

n�1

Âi=0

Pt(Xi+1, Xi) = Pt(Xn, X0) + (1 + t)Qt, (7.7)

where Qt has non-negative coefficients.

Proof: For a triple Xp�1 ⇢ Xp ⇢ Xp+1 we have the long exactsequence

· · ·∂k+1��! Hk(Xp, Xp�1)

i⇤k�! Hk(Xp+1, Xp�1)j⇤k�! Hk�1(Xp+1, Xp)

∂k�! · · · .

As in the proof of Theorem 7.4.1 this gives

Pt(Xp, Xp�1) + Pt(Xp+1, Xp) = Pt(Xp+1, Xp�1) + (1 + t)Qt(p).

Equation (7.7) holds for n = 2. Suppose Equation (7.7) is true for n,then consider the topological triple X0 ⇢ Xn ⇢ Xn+1, which yieldsPt(Xn, X0) + Pt(Xn+1, Xn) = Pt(Xn+1, X0) + (1 + t)Qt. Then,

n

Âi=0

Pt(Xi+1, Xi) = Pt(Xn, X0) + Pt(Xn+1, Xn) + (1 + t)Qt,

= Pt(Xn+1, X0) + (1 + t)Qt(n) + (1 + t)Qt

= Pt(Xn+1, X0) + (1 + t)Qt,


We apply Lemma 7.4.2 to establish the following generalization ofTheorem 7.4.1.


7.4.3 Theorem. Let S ⇢ X be a Morse set and M(S) is a Morse decom-position for S. Then,

ÂM2M(S)

CPt(M, j) = CPt(S, j) + (1 + t)Qt, (7.8)

where Qt � 0 is a formal series with non-negative coefficients.

Proof: The filtration B0 = B1 ⇢ B2 ⇢ · · · ⇢ Bn�2 ⇢ Bn�1 = B,corresponds to A(M0(S)) and each pair (Bp, Bp�1) gives a index pairfor a non-empty Morse set in M0(S). By Lemma 7.4.2 we have

ÂM2M(S)

CPt(M, j) =n�2

Âi=1

Pt(Bi+1, Bi),

= Pt(Bn�1, B1) + (1 + t)Qt = Pt(B, B0) + (1 + t)Qt,

= CPt(S, j) + (1 + t)Qt,


7.4.4 Remark. Since CPt(S, j) only depends on (T, T�), where T =

cl(B \ B0), the Morse relations hold for any isolated invariant set S. InSection ?? we show that If S is any isolated invariant set, then S canbe regarded as a Morse set for an appropriate dynamical system j

(j itself if S is already a Morse set) and therefore allows an isolatingblock. We have the following filtration

T� = U1 ⇢ U2 ⇢ · · · ⇢ Un�2 ⇢ Un�1 = T,

where Up = Bp \ int BB0.

7.5 Connecting orbits

So far we defined a local index, the Conley index, to provide infor-mation about Morse sets inside Morse tile. The topology of X, ormore specifically the Conley index of Morse set S, is related to thelocal indices of the Morse sets M 2 M(S) via the Morse relations. Inorder to derive the Morse relations we do not need the precise orbitstructure of connecting orbits between Morse sets.

If M is a Morse decomposition, then M||M0 implies that C(M, M0) =C(M0, M) = ?. The fact that M < M0, or M0 < M does not containinformation about C(M, M0), or C(M0, M) a priori. As for the Conleyindices, the global topological properties of X put constraints onorbit structure in X. Let us start with a simple example. Let (A, A⇤)be a attractor-repeller pair in X, i.e. a two set Morse decompositionA < A⇤, then the Morse relations give:

CPt(A) + CPt(A⇤) = Pt(X) + (1 + t)Qt.


If C(A⇤, A) = ?, then X = A t A⇤ and thus blocks for A and A⇤

are given by N = A and Nc = A⇤. Consequently, CPt(A) = Pt(A),CPt(A⇤) = Pt(A [ A⇤, A) = Pt(A⇤), and Pt(X) = Pt(A [ A⇤) =

Pt(A) + Pt(A⇤), which shows that Qt = 0, or in other words ∂k = 0 forall k. This yields the following lemma.

7.5.1 Lemma. If (A, A⇤) is an attractor-repeller pair in X and CPt(A) +

CPt(A⇤) 6= Pt(X), then C(A⇤, A) 6= ?.

A more general version of this lemma can be proved for attractor-repeller pair decompositions of Morse sets, or, more generally, iso-lated invariant sets.

7.5.2 Lemma. Let S be a Morse set such that S = M t M0, withM, M0 6= ?, then M(S) = {M, M0} is a Morse decomposition andCPt(M, j) + CPt(M0, j) = CPt(S, j).

Proof: It is clear that M(S) is a Morse decomposition for S andM(S) yields in Morse decomposition M(X) for X, as in indicatedin Figure 7.1. Consequently, both M and M0 are Morse sets in

sA0@

@@sM

��

sM0

s@@@

��

A⇤

c?

sA0@@@sA0M

��

sA0M0

c@@@

��

A

sXFigure 7.1: The Morse set S is givenby the attractors A0 ( A and corre-sponds to the Morse decompositionA0 < S < A⇤. This yields the Morsedecomposition M(X). In the lattice ofattractors [right] A0M and A0M0 are theattractors given by A0M = n(A0, M) andA0M0 = n(A0, M0).

Morse(X). One can choose tiles TM and TM0 that are disjoint. Sincethe Conley index only depends on the tiles it follows that CPt(M) +

CPt(M0) = Pt(TM, T�M) + Pt(TM0 , T�M0) = Pt(TM t TM0 , T�M t T�M0) =

CPt(S), which proves the lemma.

7.5.4 Remark. Let NA0M and NA0M0 be attractor blocks for A0M andA0M0 respectively. The Mayer-Vietoris exact sequence yields the shortexact sequence

0! Hk(NA0M, NA0M \ NA0M0)� Hk(NA0M0 , NA0M \ NA0M0)!

! Hk(NA0M [ NA0M0 , NA0M \ NA0M0)! 0,

which implies that CPt(M) + CPt(M0) = CPt(S).


7.5.5 Corollary. Let M < M0 be a two set Morse decomposition ofa Morse set S ⇢ X and CPt(M, j) + CPt(M0, j) 6= CPt(S, j), thenC(M0, M) 6= ?.

Proof: Assume that C(M0, M) = ?, then S = M tM0 and by Lemma7.5.2 CPt(M) + CPt(M0) = CPt(S), which is a contradiction.

7.6 Isolated invariant sets and the Conley index

The theory so far has been about decompositions of compact in-variant sets, with the space X being the largest invariant set. InSection 6.4 we introduced the notion of isolating neighborhoodand isolated invariant set. The space X is isolated and X itself is anisolating neighborhood. The Conley index therefore only containstopological information about X. Via Morse decompositions thisinformation puts obstructions on the dynamics. Morse sets are alsoexamples of isolated invariant sets and if there are isolating neighbor-hoods/blocks that are strictly larger, then the Conley index may alsocontain information about dynamics on the isolating neighborhoodand may therefore yield more detailed information about the dy-namics inside the Morse set. This theory can be applied to arbitraryisolated invariant sets, not just Morse sets.

Existence of blocks

We show that every isolated invariant set contains an isolating Lya-punov block, isolating the same same invariant set.

7.6.1 Theorem. Let N ⇢ X be an isolating neighborhood. Then, thereexists an isolating block B ⇢ cl(N) such that S = Inv(cl(N)) =

Inv(B).


The existence of a block for isolated invariant sets allows one toextend the theory of Morse decompositions of X and Conley index toisolated invariant sets.

7.6.2 Definition. Let S = Inv(N) ⇢ X be an isolated invariant set.Then the Conley index of S is defined by HC⇤(S, j) := H⇤(B, B�),where B ⇢ cl(N) is an isolating block for S.

In previous we introduced the Conley index for Morse sets.It remains to prove that the Conley index is well-defined. Well-definedness is proved in the same way as Lemma 7.3.7.


7.6.3 Lemma. Let B, B0 ⇢ N ⇢ X be isolating blocks for S = Inv(N),then H⇤(B, B�) ⇠= H⇤(B0, B0�) and therefore the Conley index iswell-defined.

Proof: Since isolating blocks form a meet semi-lattice, B00 = B \ B0 isalso an isolating block for S and B00 ⇢ B. Explain the extension of the dynamical system via the product

system extension, see notesDefine a flow j(s, x) = j(t(s, x), x), where t(s, x) solves dt

ds =

a(j(t, x)) and a : B = j([�t, t], B) ! [0, 1] is a continuous functionsuch that a|B = 1, a|

j(�t,B�) = 0 and a|j(t,B�) = 0. For B, j(�t, B�)

is a repeller, j(t, B�) is an attractor and j|B = j. The set S is aMorse set and thus by Lemma 7.3.7, H⇤(B, B�) ⇠= H⇤(B00, B00�). Thesame construction applies to B0, which shows that H⇤(B0, B0�) ⇠=H⇤(B00, B00�) and therefore H⇤(B, B�) ⇠= H⇤(B0, B0�).

7.6.4 Exercise. Prove Lemma 7.6.3 directly without embedding S into adynamical system j.

7.6.5 Theorem. Let N ⇢ X be an isolating neighborhood and S =

Inv(N). If HC⇤(S, j) 6= 0, then S 6= ?.

Proof: By Theorem 7.6.1 there exists an isolating block B ⇢ N, andHC⇤(S, j) ⇠= H⇤(B, B�) 6= 0. The implies that B� is not a strongdeformation retract of B and since B is a Wazewski set Corollary7.2.6 yields that Inv(B) = S 6= ?.

For Morse decompositions of isolated invariant sets the Morserelations holds.

7.6.6 Theorem. Let M(S) be a Morse decomposition of an isolatedinvariant sets S. Then, the Morse sets M ⇢ S are isolated invariantsets and

ÂM2M(S)

CPt(M, j) = CPt(S, j) + (1 + t)Qt, (7.9)

where CPt(M, j) are the Poincaré polynomials of HC⇤(M, j).

Proof: Let B be an isolating block for S and define j as before. TheTheorem follows by applying Theorem 7.4.3.

Non-compact spaces

So far we have assumed X to be a compact metric space. The com-pactness of X guarantees the existence of non-empty attractors/repellersin trapping/repelling regions and allows the existence of Lyapunovfunctions and therefore the existence of blocks. However, when westudy blocks for a dynamical system, then compactness of X is notrequired. Of course, trapping regions may not contain attractors. TheConley Theory for tilings of blocks still satisfies the Morse relationsand can be summarized as follows.


Let X be a topological space and j : R ⇥ X ! X is a flow on X.Let B ⇢ X be an isolating block and T(B) a Morse tiling. For a tileT 2 T be define the Conley index by HC⇤(T, j) := H⇤(T, T�). TheConley index is an invariant for blocks in the sense that if B, B0 areblocks with B \ B0 6= ? and Inv(B) = Inv(B0), then H⇤(B, B�) ⇠=H⇤(B0, B0�). This follows from the above considerations and does notneed compactness. This yields the following generalization of theMorse relations.

7.6.7 Theorem. Let T(B) be a Morse tiling for a isolating block B.Then,

ÂT2M(S)

CPt(T, j) = CPt(B, j) + (1 + t)Qt, (7.10)

where CPt(T, j) are the Poincaré polynomials of HC⇤(T, j).

7.7 Continuation of the Conley index

Some questions. Are block locally stable, i.e. a block for j is a blockfor nearby flows? Answer is no. Find an easy argument to show thatthe Conley index continues.

8Discrete Parabolic Dynamics

This chapter entails an application of Conley Theory to an importantclass of dynamical systems; parabolic recurrence relations. We con-struct a Morse-type theory on certain spaces of braid diagrams. Atopological invariant of closed positive braids is defined and is corre-lated with the existence of invariant sets of parabolic flows defined ondiscretized braid spaces via parabolic recurrence relations. Parabolicflows, a type of one-dimensional lattice dynamics, evolve singularbraid diagrams in such a way as to decrease their topological com-plexity; algebraic lengths decrease monotonically. This topologicalinvariant is derived from the homological Conley index. This culmi-nates in very general forcing theorems for the existence of infinitelymany braid classes.

8.1 Parabolic recurrence relations

We start with a class of discrete dynamical systems defined by recur-rence relations with next-neighbor coupling. These will be referred toas parabolic recurrence relations.

8.1.1 Definition. On the space of sequences RZ, define a sequence ofC•-functions {Ri}i2Z, Ri : R3 ! R, which satisfy

(i) ∂1Ri > 0 and ∂3Ri > 0;

(ii) Ri+d = Ri, for some d 2 N.

A parabolic recurrence relation is given by the equation

Ri(xi�1, xi, xi+1) = 0, (8.1)

with {xi}i2Z 2 RZ and {Ri} satisfying (i) and (ii).

A parabolic recurrence relation with the periodicity condition (ii)define finite iterative system of mappings. Let z = (u, v), then solvewi from Ri(ui, vi, wi) = 0, which gives wi = gi(ui, vi). Define themappings Fi(u, v) =

�

v, gi(u, v)�

. The iterative system zi+1 = Fi(zi)


defines a discrete dynamical system and since Since ∂ui+1∂vi

> 0, theseare twist maps. Parabolic recurrence relations occurs is variousapplications in dynamical systems, as well as in physical models. Weare interest in fixed points and periodic points of {Fi}, i.e. z = bF(z),where bF = Fd�1 � · · · � F0, or z = bFn(z). In terms of the parabolicrecurrence relation this means sequence {xi}i2Z, with xi+d = xi and{xi} satisfies (8.1).

The space of d-periodic sequences in RZ will be definoted by Wdand the smooth mapping

Wd ! Rd, {xi} 7! x = (x0, · · · , xd�1),

yields global coordinates on Wd, which makes Wd a smooth manifold.We may represent d-periodic sequences in Wd by piece wise linearinterpolations

b(x)(s) = xbd·sc + (d · s� bd · sc)(xdd·se � xbd·sc),

for s 2 [0, 1]. We identify Wd with the space of piecewise linear d-functions as defined above. In principle, connecting the anchor pointsxi with line segment enables us to distinguish between differentsequences geometrically, i.e. sequences x and x

0 with different coor-dinates represents different piecewise linear functions b(x) and b(y).We will show now that periodic sequences x that occur as solutionsof parabolic recurrence relations has special geometric propertieswith respect to their piecewise linear interpolations b(x).

8.1.2 Definition. Two sequences x, y 2 Wd, with x 6= y, are said to betransverse if

(xi�1 � yi�1)(xi+1 � yi+1) < 0,

for every index i for which xi = yi. We write x t y. Transversality is alocal property and therefore the same applies to sequences x, y 2 RZ.

Transversality is a natural property of solutions of parabolicrecurrence relations as the following lemma shows.

8.1.3 Lemma. Let x, y 2 Wd, with x 6= y, be solutions of Equation(8.1), then x t y.

Proof: Consider the case where x and y coincide for at least onecoordinate. Since x 6= y, one can choose an index i such that xi = yiand xi+1 > yi+1 (the reverse inequality is the same). Suppose xi+1 �yi+1, then, since ∂3Ri > 0, it follows that 0 = Ri(xi�1, xi, xi+1) >

Ri(yi�1, yi, yi+1) = 0, which is a contradiction. Therefore, (xi�1 �yi�1)(xi+1 � yi+1) < 0. This implies that indices for which xi = yi areisolated.

discrete parabolic dynamics 159

Graphically two sequences x and y are represented by intersectinggraphs b(x) and b(y) and as such transverse sequences correspondto transverse graphs. Intersections can be counted and leads to thenotion of linking, see Fig. ??.

8.1.4 Definition. The linking number of two transverse sequencesx t y the linking number is defined as

link(x, y) = #{intersections of b(x) and b(y)},

which is an even integer.

The linking number link(x, y) is also equal to two times the num-ber of connected components of the set {s | (b(x)� b(y)(s) > 0}. Thelinking number is a local invariant, i.e. the linking number does notchange under small perturbations of the sequences x and y.

Figure 8.1: Linking sequences [left]and non-transverse intersections, ortangencies [right].

For two transverse sequences an occurrence xi = yi is calledan intersection at an anchor point. However, if x and y are non-transverse sequences, then xi = yi and (xi�1 � yi�1)(xi+1 � yi+1) � 0for some i, and such a an occurrence is called a tangency, see Fig. ??.

8.2 Parabolic flows and braids

In order to study solutions of Equation (8.1) we embed Equation (8.1)in a canonical dynamical system. Parabolic recurrence relations maybe regarded as vector field in a suitable way. Since Ri is d-periodic,R = (R0, · · · , Rd�1) 2 T

x

Wd defines a vector field on Wd. We inte-grate the vector field via the equation

x0i = Ri(xi�1, xi, xi+1), (8.2)

which generates a local C•-flow j on Wd. Such a flow j is called aparabolic flow on Wd. Of course a vector field R only provide a localflow in gerenal and a global flow may be obtained by considering forexample R

1+|R|2 . However, since we will be studying blocks it doesnot matter whether j is a local or global flow. The following lemmamotivates why linking of periodic sequences is naturally related toparabolic flows.


8.2.1 Lemma. Let x, y 2 Wd, with x 6= y and x 6t y, then there existsan e > 0 such that j(t, x) t j(t, y) for all t 2 [�e, e] \ {0} and

link�

j(�t, x), j(�t, y)�

> link�

j(t, x), j(t, y)�

,

for all t 2 (0, e].


The behavior of parabolic flows with respect to transversalitymotivates the following definition.

8.2.2 Definition. An unordered set of sequences x = {x

1, · · · , x

n},is called a discrete braid on n strands if the following conditions aresatisfied:

(i) x

a = (xa

0 , · · · , xa

d) 2 Rd+1, for all a = 1, · · · , n;

(ii) there exists a permutation s 2 Sn such that xa

d = xs(a)0 , for all

a = 1, · · · , n;

(iii) x

a t x

a

0 for all pairs a 6= a

0.

The piecewise linear interpolation b(x) = {b(x

a)} defined a braiddiagram with positive crossings.

Fig. ?? gives an example of a positive piecewise linear braid dia-gram and therefore a discrete braid. Sometimes a braid is denoted by(x, s). Two braids (x, s) and (x

0, s

0) are close if there exists a permu-tation p 2 Sn such that x

p(a) and x

0a are close in Rd+1 for all a andp � s

0 = s � p. The space of all discrete braids on n strands, topolo-gized as above, is denoted by Wn

d . The space of braids Wnd is a metric

space. The permutation is often omitted from the notation. Twobraids x and x

0 are homotopic, or equivalent, x ⇠ x

0, iof there existsa continuous path x(t) in Wn

d , such that x(0) = x and x(1) = x

0. Thepath components of Wn

d are called braid classes and are denoted by[x] ⇢ Wn

d . Define Wnd as the space of unordered sets x = {x

1, · · · , x

n},satisfying (i) and (ii) of Definition 8.2.2. The set Sn

d := Wnd \ Wn

d iscalled the set of singular braids. The set Wn

d is a complete metric spaceand Wn

d ⇢ Wnd is open in Wn

d . A special subset of singularities is givenby

Sn,�d =

�

x 2 Snd | xa

i = xa

0i , for some a 6= a

0 .

For a discrete braid x 2 Wnd we can define the word-length, or word-

metric

`(x) = #{intersections in b(x)}.

Obviously the word-length is well-defined and can be related to thelinking number of two strands. Let bx 2 Wn

nd be the n-fold extensionof x, i.e. bxa

i = xa

i for i = 1, · · · , d, bxa

i = xs(a)i�d for i = d, · · · , 2d, bxa

i =


xs

2(a)i�2d for i = 2d, · · · , 3d, etc. Then counting of mutual intersections

gives:

`(x) =1

2n Âa 6=a

0link(bxa, bxa

0). (8.3)

The parabolic equation in (8.2) defines a (local) flow Y on Wnd . Since a

parabolic recurrence relation is d-periodic in i we define the flow Y asfollows: \Y(t, x) = Y(t, bx).

8.2.3 Exercise. Show, using the periodicity of Ri with respect to i, that Y iswell-defined on the space Wn

d .

From Lemma 8.2.1 we derive how the word-length behaves withrespect to parabolic flows on Wn

d , see Fig. ??.

8.2.4 Lemma. Let x 2 Snd \ Sn,�

d , then there exists an e > 0 such thatY(t, x) 2 Wn

d for all t 2 [�e, e] \ {0} and

`�

Y(�t, x)�

> `�

Y(t, x)�

,

for all t 2 (0, e].

Proof: By definition \Y(t, x) = Y(t, bx) and if x 2 Snd \ Sn,�

d , thenbx 2 Sn

nd \ Sn,�nd . If bx 2 Sn

nd \ Sn,�nd , then at least two strands are

not transverse and none are collapsed. Then by Equation (8.3) andLemma 8.2.1 it follows that `

�

Y(�t, x)�

> `�

Y(t, x)�

. Since thisholds for any non-tranverse strands Lemma 8.2.1 also implies thatY(t, x) 2 Wn

d for all t 2 [�e, e] \ {0}.

Figure 8.2: The behavior of parabolicflows with respect to intersections andbraid classes.

Define Wnd rel Wm

d be the space of ordered pairs (x, y) 2 Wnd ⇥Wm

dsuch that {x

1, · · · x

n, y

1, · · · , y

m} 2 Wn+md . The path components of

Wnd rel Wm

d are denoted by [x rel y] and are called relative braid classes.Define the projection p : Wn

d rel Wmd ! Wm

d by (x, y) 7! y and forgiven y 2 Wm

d , called the skeleton, the set p

�1(y) = [x] rel y is called arelative braid class fiber and can be canonically embedded into Wn

d . Wewill always identify [x] rel y with a subset of Wn

d !


Figure 8.3: A improper class [left] and aproper class [right].

8.2.5 Definition. A relative braid class [x rel y] 2 Wnd rel Wm

d isbounded if every fiber [x] rel y is a bounded set in Wn

d . A relativebraid class is proper if

p

�1(y0) \ Sn+m,�d = ?,

for every fiber p

�1(y0) = [x0] rel y

0 ⇢ [x rel y]. If a relative braidclasses is not proper it is called improper.

Intuitively, proper classes have the property that strands in x

cannot be collapsed onto strands in y, nor can strands in x reduce toa coarser braid with fewer than n strands, see Fig. ??.

8.3 Isolating blocks and braid classes

We will now relate to concept of relative braid classes to parabolicflows. Suppose that y 2 Wm

d is braid consisting of solutions ofEquation 8.1. To be more precise, by = {bya}, with by

a 2 Wnd, is a setof nd-periodic solutions of Equation (8.1). Conversely, if we have a k-periodic solutions by 2 Wk, with m = k/d 2 N, then the translate {y

a}over the indices i = 1, · · · , d, represents an element y = {y

a} 2 Wmd .

If the period k is minimal, then y 2 Wmd due to Lemma 8.1.3. If we

have more than one periodic solution, the union yields a braid y byLemma 8.1.3. In terms of parabolic flows we have that Y(t, y) = y,for all t 2 R, which implies that y is stationary for Y and is thereforean invariant set of Y.

Let y 2 Wmd be stationary for a parabolic flow Y and consider a rel-

ative braid class fibers [x] rel y 2 Wnd rel y. As explained above braid

class fibers are considered as subsets of Wnd . Since y is stationary for

Y, Wnd rel y

⇠= Wnd is an invariant set for Y and we can restrict Y to

Wnd after identification. In the following we consider the special case

that n = 1 and Y|Wd= j. The connected components of Wd rel y are

subsets of Wd.


8.3.1 Lemma. Let y 2 Wmd be a stationary braid (skeleton) for Equa-

tion (8.1) and let [x rel y] be a proper and bounded relative braidclass. Then any fiber

B := [x] rel y ⇢ Wd ⇢ Wd,

is an isolating block for the associated parabolic flow j.

Proof: Since B is bounded, cl(B) is compact. Let x 2 ∂B, then, sincethe braid class is proper, it follows from Lemma 8.2.4 that there existsan e > 0 such that j([�e, 0), x) 6⇢ B, j((0, e], x) 6⇢ B, or neitherare contained in B. This implies that boundary points are in eitherB�, B+, or in B� \ B+. By compactness we can choose a uniforme = t > 0, which proves that B is an isolating block.

The Conley index of B is well-defined and will be denoted byHC⇤(B, j) ⇠= H⇤(cl(B), B�). Lemma 8.2.4 characterizes B� as theset of boundary points for which the word-length is decreasing byflowing from B to Bc. To be more precise, for every point x 2 ∂Bwe can choose a neighborhood W ⇢ Wd such that B \ W consists offinitely many connected components Wj and W0 = B \W. Then,

B� = cl�

x 2 ∂B | `(W0) > `(Wj), 8j

.

The set B+ is defined similarly. Since the sets int∂BB� and int

∂BB+

are smooth codimension-1 submanifolds in Wd a coorientation isdefined by the unit normal in the direction of decreasing `. Therefore,B, B� and B+ are completely determined by the topological type ofthe relative braid class. The dynamics of j always complies with thecoorientation of the boundary of a proper braid class.

The next theorem shows that the Conley index of a braid classfiber is topological invariant of a discrete relative braid class.

8.3.2 Theorem. Let y 2 Wmd be a stationary braid (skeleton) for Equa-

tion (8.1) and let [x rel y] be a proper and bounded relative braidclass. Then for any fiber B = [x] rel y the Conley index HC⇤(B, j) iswell-defined, and(i) if Y0 is any parabolic flow for which Y0(t, y) = y, for all t 2 R,

then HC⇤(B, j

0) ⇠= HC⇤(B, j), where j

0 = Y0|Wd;

(ii) if [x0] rel y

0 and [x] rel y are fibers in [x rel y], then HC⇤(B, j) ⇠=HC⇤(B0, j

0), where B = [x] rel y, B0 = [x0] rel y

0 and j and j

0

parabolic flows for which y and y

0 are stationary respectfully.The Conley index HC⇤(B, j) is a braid class invariant for [x rel y] andis denoted by h(x rel y).


Let us now discuss some examples of the proper and boundedbraid classes and compute their braid class invariants.


8.3.3 Example. Consider the proper period-2 braid illustrated inFig. ??[left]. (Note that deleting any strand in the skeleton yields animproper braid.) There is exactly one free strand with two anchorpoints (recall that these are closed braids and the left and right sidesare identified). The anchor point in the middle, x1, is free to movevertically between the fixed points on the skeleton. At the endpoints,one has a singular braid in S which is on the exit set since a slightperturbation sends this singular braid to a different braid classwith fewer crossings. The end anchor point, x2 (= x0) can freelymove vertically in between the two fixed points on the skeleton. Thesingular boundaries are in this case not on the exit set since pushingx2 across the skeleton increases the number of crossings.

Figure 8.4: The braid of Example 1 [left]and the associated configuration spacewith parabolic flow [middle]. On theright is an expanded view of W2 rel y

where the fixed points of the flowcorrespond to the four fixed strandsin the skeleton v. The braid classesadjacent to these fixed points are notproper.

Since the points x1 and x2 can be moved independently, the con-figuration space B in this case is the product of two compact inter-vals. The exit set B� consists of those points on ∂B for which x1 is aboundary point. Thus, the homotopy index of this relative braid is[B/B�] ' S1.

8.3.4 Example. Consider the proper relative braid presented inFig. ??[left]. Since there is one free strand of period three, the con-figuration space B is determined by the vector of positions (x0, x1, x2)

of the anchor points. This example differs greatly from the previousexample. For instance, the point x0 (as represented in the figure)may pass through the nearest strand of the skeleton above and belowwithout changing the braid class. The points x1 and x2 may not passthrough any strands of the skeleton without changing the braid classunless x0 has already passed through. In this case, either x1 or x2 (de-pending on whether the upper or lower strand is crossed) becomesfree.

To simplify the analysis, consider (x0, x1, x2) as all of R3 (allowingfor the moment singular braids and other braid classes as well).The position of the skeleton induces a cubical partition of R3 byplanes, the equations being xi = ya

i for the various strands va ofthe skeleton v. The braid class B is thus some collection of cubes inR3. In Fig. ??[right], we illustrate this cube complex associated to B,


claiming that it is homeomorphic to D2 ⇥ S1. In this case, the exit setB� happens to be the entire boundary ∂B and the quotient space ishomotopic to the wedge-sum S2 _ S3.

Figure 8.5: The braid of Example 8.3.4and the configuration space B.

8.3.5 Example. To introduce the spirit behind the forcing theoremsof the latter half of the paper, we reconsider the period two braidof Example 1. Take an n-fold cover of the skeleton as illustrated inFig. ??. By weaving a single free strand in and out of the strands asshown, it is possible to generate numerous examples with nontrivialindex. A moment’s meditation suffices to show that the configurationspace B for this lifted braid is a product of 2n intervals, the exit setbeing completely determined by the number of times the free strandis “threaded” through the inner loops of the skeletal braid as shown.

For an n-fold cover with one free strand we can select a family of3n possible braid classes describes as follows: the even anchor pointsof the free strand are always in the middle, while for the odd anchorpoints there are three possible choices. Two of these braid classes arenot proper. All of the remaining 3n � 2 braid classes are boundedand have homotopy indices equal to a sphere Sk for some 0 k n.Several of these strands may be superimposed while maintaining anontrivial homotopy index for the net braid: we leave it to the readerto consider this interesting situation.

Figure 8.6: The lifted skeleton ofExample 1 with one free strand.

Stronger results follow from projecting these covers back down tothe period two setting of Example 1. If the free strand in the coveris chosen not to be isotopic to a periodic braid, then it can be shown


via a simple argument that some projection of the free strand downto the period two case has nontrivial homotopy index. Thus, thesimple period two skeleton of Example 1 is the seed for an infinitenumber of braid classes with nontrivial homotopy indices. Usingthe techniques of 1, one can use this fact to show that any parabolic 1

recurrence relation (R = 0) admitting this skeleton is forced to havepositive topological entropy: cf. the related results from the Nielsen-Thurston theory of disc homeomorphisms 2. 2

8.4 Morse decompositions, Morse relations and connecting orbits

8.5 Stabilization and global invariants

8.6 Scalar parabolic PDE’s

9Morse theory

Variational methods are used for finding critical points of differen-tiable functions. In general the functions are defined on an infinitedimensional space. Such problems typically occur for large classesof ordinary and partial differential equations and are called varia-tional problems. In classical mechanics the functions, or functionalsare called Lagrangians and the critical point equations are referred toas the Euler-Lagrange equations. In this Chapter we discuss a classof methods, also referred Morse Theory, that is used to find criticalpoints of functions on finite and infinite dimensional spaces. Themain characteristic is to link topological properties of the space to theset of critical points of the function in question. Certain aspects ofMorse Theory can regarded as a special case of Conley Theory in thecase of gradient and gradient-like flows.

9.1 Gradient-like flows on compact spaces

An important class of flows that mimic gradient flows in the contin-uous case are called gradient-like flows and are defined as follows.Assume that X is a compact metric space.

9.1.1 Definition. A dynamical system j : R ⇥ X ! X is calledgradient-like if there exists a continuous function V : X ! R such thatV(j(t, x)) is strictly decreasing in t for all x 62 E = {x 2 X | j(t, x) =x, 8t 2 R}, where E is the set of equilibria of j.

Every gradient flow is trivially gradient-like. Compact invariantsets S have a special structure.

9.1.2 Lemma. If S is a compact invariant set of a gradient-like dynam-ical system, then S consists of equilibrium points and heteroclinicconnecting orbits, i.e. bounded orbits gx with a(x), w(x) ⇢ E.

Proof: Since V(j(t, x)) is non-increasing, if gx = j(R, x) is abounded orbit, then V(j(t, x)) is a decreasing, bounded function


which has limits V(j(t, x)) ! c± as t ! ±•. Suppose y 2 w(x).Then by the definition of omega limit set there exist times tn ! •such that j(tn, x) = y, and hence V(y) = c+. Suppose y /2 E. Thenfor any t > 0, we have V(j(t, y)) < c+. However, j(tn + t, x) !j(t, y) so that V(j(tn + t, x)) ! c+ implying V(j(t, y)) = c+, acontradiction.

In particular, when E \ S is a finite set, then every orbit gx ⇢ Sconverges to equilibrium points in E. Indeed, since a(x) and w(x) areconnected sets, it follows that if E is finite, then a(x) and w(x) consistof single points.

Let us continue with the case where E is a finite set. Then Lemma9.1.2 implies that E is a Morse decomposition for X. Note that thesame holds in general (E not necessarily finite) if we take finitelymany connected components of E. Let S ⇢ X be an isolated invariantset (for example when S = X), then the Morse relations in Theorem7.6.6 yield:

Âx2E\S

CPt({x}, j) = CPt(S, j) + (1 + t)Qt, (9.1)

where CPt({x}, j) are the Poincaré polynomials of HC⇤({x}, j). Thelatter is defined via an isolating block for x 2 E\ S.

There is a way to compute the Conley index of an equilibriumpoint by using the function V. In the case that V(x) = c is a differentfor every x 2 E\ S, then Vc+e

c�e

= {x 2 X | c� e V(x) c + e}, withe > 0 sufficiently small, is an isolating block. Then, (Vc+e, Vc�e) is anindex pair of attracting blocks, where Vc±e = {x | V(x) c ± e} andHC⇤({x}, j) ⇠= H⇤(Vc+e, Vc�e). Since the Conley index of x 2 E doesnot depend on e, the limit e! 0 formally implies that HC⇤({x}, j) ⇠=H⇤(Vc+e, Vc�e) ⇠= H⇤(Vc, Vc \ {x}). We will give a complete proofof this fact in Section 9.4 via the deformation lemma. The latterisomorphism also gives an idea on how to compute CPt({x}, j)

when the values of V are not necessarily different on E\ S. In that casethe deformation lemma gives HC⇤({x}, j) ⇠= H⇤

�

Vc \ N, (Vc \ {x}) \N�

, where N is an isolating neighborhood for x.The Morse relations for gradient-like systems described above can

be applied to the special case of gradient flows. Let X be a smooth,closed (compact, no boundary) manifold of dimension n < •, andf : X ! R is a smooth function. Then the differential equation

x0 = �rg f (x),

defines a smooth flow j : R⇥ X ! X, where rg f (x) is the gradientof f with respect to a chosen Riemannian metric g on X. The equilib-rium points of j are exactly the critical points of f . Equation 9.1 givesthe classical Morse relations for functions f with finitely many critical

morse theory 169

points. In case the critical points are all non-degenerate, i.e. f 00(x) isinvertible for all critical points x, then the Conley index is given byCPt({x}, j) = tµ(x), where µ(x) = #{negative eigenvalues of f 00(x)}.The latter is called the Morse index and the function f is a Morsefunction. These consideration remain valid for open manifolds (finitedimensional) if we consider compact isolated invariant sets S. If j

is a gradient flow on a compact, finite dimensional manifold withboundary, then the same Morse relations apply.

When X is not compact, or locally compact, then finite sets ofequilibrium points of a gradient, or gradient-like flow do not neces-sarily define Morse decompositions. However, one does find Morsetiling for which the Morse relations hold. For gradient systems weuse a compactness condition, called the Palais-Smale condition, thatreplaces compactness of X and makes Conley Theory applicable togradient flows on non-compact spaces.

9.2 Palais-Smale functions and compactness

Let X be a real Hilbert space with inner product (·, ·) and f : X ! R

is a continuously differentiable function. We are interested in theequation

f 0(x) = 0, x 2 X. (9.2)

Equation (9.2) is a special case of the equations studied in Part ??.Indeed, the gradient of f at x is the unique vector r f (x) given bythe Riezs Representation Theorem: (r f (x), y) = f 0(x)y for ally 2 X. The gradient define a continuous mapping r f : X ! Xand Equation (9.2) is equivalent to r f (x) = 0. Since the mappingin question is a gradient the problem has additional structure and iscalled a variational problem. The additional structure allow differenttechniques for finding zeroes of r f (x) = 0.

Variational problems, especially those coming from differentialequations are stated on infinite dimensional spaces and thereforecompactness issues may occur when searching for critical points. Forfunction f on a smooth, closed manifold (compact, no boundary) itholds that if f 0(xn)! 0, then there exists a subsequence xnk ! x, andf 0(x) = 0. Such a sequence is called a Palais-Smale sequence and theyplay an important role in variational methods. Consider the functionf (x) = arctan(x). Clearly, f has no critical values, and thus no criticalpoints on R. The values c = ±p/2 are special however. One can forexample take sequences {xn}, xn = ±n, such that f (xn) ! c andf 0(xn) ! 0. Regardless of the fact that f 0 goes to zero along suchsequences there are no critical points. One could argue that thereexist critical points at ‘infinity’. This would require compactifying oursetting.


This simple example already goes to show that due to the non-compactness of R, the domain of definition of f , the notion of criticalvalue and critical point lacks unform estimates. For this very reasonPalais and Smale, in their work on Morse theory in infinite dimen-sions, introduced the following compactness condition.

9.2.1 Definition. A function f 2 C1(X; R) is saidf to satisfy the Palais-Smale condition at c — (PS) for short —, if any sequence {xn} ⇢ Xfor which

f (xn)! c, f 0(xn)! 0,

has a convergent subsequence.

This condition was referred to as ‘Condition (C)’ in the original workof Palais and Smale. Functions may also satisfy (PS) for an interval ofvalues c, i.e. a function satisfies (PS) on an interval I if it satisfies (PS)for every c 2 I. The same holds for the Palais-Smale condtion on X.In that case we do not specify c beforehand.

The (PS) condition has various consequences for Palais-Smalefunctions. Denote by Crit( f , I) the set of critical points of f withcritical values restricted to the interval I.

9.2.2 Lemma. Let f 2 C1(X) satsify (PS) for some c, then the setCrit( f , c) is compact. Moreover, Crit( f , I) is compact whenever I iscompact. Proof: Compactness

is established by pointing out that compactness for a metric space isequivalent to sequential compactness. The space (Crit( f , c), d) withthe induced metric is a metric space itself. For any sequence {xn} wehave that f (xn) = c, and f 0(xn) = 0. The (PS)-condition then impliesthat xnk ! x 2 X. Consequently, f (x) = c and f 0(x) = 0, and thusx 2 Cf (c), which establishes sequential compactness. The same holdsfor Crit( f , I).

Another important consequence of the (PS)-condition is uniformityon lower bounds for f 0 at regular values.

9.2.3 Lemma. Let c be a regular value for f . Then there exists ane > 0, such that k f 0(x)kX⇤ � d > 0 for all x 2 f�1[c� e, c + e]. Proof:

The fact that c is regular implies that a neighborhood [c � e, c + e],for some e > 0, consists of regular values. If not, one can choosecn ! c, and xn, with f (xn) = cn, and f 0(xn) = 0. By (PS) we have thatxnk ! x, with f (x) = c and f 0(x) = 0, a contradiction.

For any c⇤ 2 [c � e, c + e] one can find a dc⇤ > 0 such thatk f 0(x)kX⇤ � dc⇤ > 0 for all x 2 f�1(c⇤). Indeed, otherwise onecan find sequences {xn} such that f 0(xn) ! 0, which, by (PS), haveconvergent subsequences converging to a critical point at level c⇤, acontradiction.

morse theory 171

Finally, dc⇤ � d > 0 for all c⇤ 2 [c � e, c + e], from the abovearguments.

9.3 The Morse relations for critical points

In order to study zeroes of f 0 we use the canonical dynamical systemgenerated by the gradient flow equation

x0 = �r f (x). (9.3)

From this point on we will assume that f is twice continuouslydifferentiable. In that case Equation (9.3) generates a local C1-flow j

on the Hilbert space X. An important property, intrinsic to gradientflows, involves the following identity:

�

f � j

�0= f 0(j)j

0 = � f 0(j)r f (j) = �kr f (j)k2 0,

which implies that f is a Lyapunov function on X. Indeed, f is con-stant at Crit( f , R) and f � j is strictly decreasing outside Crit( f , R).

Gradient flows and Morse decompositions

Let a < b be regular values of f . Note that for smooth functionsthe regular values form a dense subset of R. Consider the set f b

a =

{x 2 X | a f (x) b}. We also use the convention that f •�• = X,

f b�• = f b = {x 2 X | �• < f (x) b} and f •

a = fa = {x 2 X | a f (x) < •}.

9.3.1 Lemma. Let f 2 C2(X) satisfy (PS) and let a 0 such that [a � e, a + e]

and b� e, b + e] are intervals of regular values and kr f (x)k � d > 0for all x 2 f a+e

a�e

[ f b+e

b�e

. In order to show that B is a block we startwith identifying sets B� and B+ as subsets of ∂B = {x 2 X | f (x) =a} [ {x 2 X | f (x) = b}. Define B� = {x 2 X | f (x) = a} andB+ = {x 2 X | f (x) = b}. We note that B� \ B+ = ?. If x 2 B+, i.e.f (x) = b, then for t < 0 it holds that

f (j(0, x))� f (j(t, x)) = �Z 0

tkr f (j)k2 td2,

which implies that f (j(t, x)) � b � td2 > b for all t < 0. In samewhy if follows that f (j(t, x)) a� td2 < a for all t > 0. Summaring,j((�•, 0), x) \ B = ? for all x 2 B+ and j((0, •), x) \ B = ? for allx 2 B�, which proves that B = f b

a is an isolating block.


In the same way it follows that sets f a are attracting blocks and setfa are repelling blocks (a a regular value). By the theory in Section 7.6we have that the Conley index of f b

a is well-defined and is given byHC⇤( f b

a , j) = H⇤(B, B�). Since X is a non-compact space (not evenlocally compact in general), the interpretation of the Conley indexneeds to be discussed in this case. The block B = f b

a is even a Morseblock via the observation that

? ⇢ f a ⇢ f b ⇢ X,

gives a lattice of attracting neighborhoods and a Morse tiling f a <

f ba < fb. From the Lemmas 7.2.4 and 7.3.3 and the integral estimates

in Lemma 9.3.1 it follows that HC⇤( f ba , j) = H⇤(B, B�) ⇠= H⇤( f b, f a).

We can find finer Morse tilings by considering sequences of regu-lar values. Let

a = a0 < a1 < · · · < an�1 < an = b,

be regular values of f . Then,

? ⇢ f a0 ⇢ f a1 ⇢ · · · ⇢ f an�1 ⇢ f an ⇢ X,

is a lattice of attracting neighborhoods (filtration), denoted by B(a)with a = (a0, · · · , an). The associated Morse tiling T(B) of X is givenby f a0 < f a1

a0 < · · · < f anan�1 < fan . A Morse tiling of f b

a is given by

f a1a0 < · · · < f an

an�1.

The Morse relations of Theorem 7.6.7 for the Morse tiling T(B) are

n

Âi=0

CPt( f aiai�1

, j) = CPt( f ba , j) + (1 + t)Qt, (9.4)

where CPt( f aiai�1

, j) = Pt( f ai , f ai�1). Whether a Morse tiling yieldsa real Morse decomposition depends on compactness propertiesof Inv( f b

a ). Morse functions on X and the chosen gradient flow inEquation (9.3) this need not be true. However, for interpreting theMorse relations for critical points this does not play a role.

Morse relations and critical points

The interpretation of the Conley index HC⇤( f ba , j) as an invariant to

detect critical points in B = f ba only uses the (PS) condition.

9.3.2 Theorem. Let f 2 C2(X) satisfy (PS) and let a < b be regularvalues of f . Then, HC⇤( f b

a , j) 6= 0, implies that Crit( f , [a, b]) 6= ?.

Proof: If Crit( f , [a, b]) = ?, then [a, b] consists of regular values andby Lemma 9.2.3 we have that kr f (x)k � d for all x 2 f b

a . Let x 2 f b,

morse theory 173

then, as before, for t > 0,

f (j(t, x))� f (j(0, x)) = �Z t

0kr f (j)k2 �td2,

and therefore f (j(t, x)) b� td2. This implies that for t � t > 0,j(t, f b) ⇢ f a, and thus by the Wazewski Principle (Theorem 7.2.3),f a is a strong deformation retract of f b, see also the proof of Lemma7.2.4. We conclude that HC⇤( f b

a , j) ⇠= H⇤( f b, f a) ⇠= 0, which is acontradiction.

The above theorem also also for a = �•, i.e. f b�• = f b and for

b = •, i.e. f •a = fa, or both, i.e. f •

�• = X.

9.3.3 Theorem. Let f 2 C2(X) satisfy (PS) and let a be a regu-lar values of f . Then, HC⇤( fa, j) ⇠= H⇤(X, f a) 6= 0, implies thatCrit( f , [a, •)) 6= ?.

Proof: Suppose [a, •) consists of regular values, i.e. Crit( f , [a•)) =

?. Define W = fa, W� = f aa and W0 = W. Then by the Wazewski Prin-

ciple, f a is a strong deformation retract of X and therefore H⇤(X, f a) ⇠=0, which is a contradiction.

Theorem 9.3.2 provides very basic information about the criticalpoint of f in the set B = f b

a . Consider the case that Crit( f , [a, b])is a finite set. If Inv( f b

a ) is compact, then Crit( f , [a, b]) provides aMorse decomposition. In general, this need not be the case for j. Leta < c0 < · · · cn 0 that

n

Âi=0

CPt( f ci+e

ci�e

, j) = CPt( f ba , j) + (1 + t)Qt. (9.5)

The fact that we can take f c+e

c�e

, for any sufficiently small e > 0, asa block follows from the fact that the index doesn’t depend on theparticular choice of the block. If we further analyze j at critical levelswe will be able to further compute CPt( f ci+e

ci�e

, j). This will lead tothe more traditional Morse relations as we will explain in the nextsection.

9.4 The deformation lemma

In this section we study the flow j near an isolated critical level.

9.4.1 Lemma. Suppose c 2 R is an isolated critical value, thenHC⇤( f c+e

c�e

, j) ⇠= H⇤�

f c \ N, ( f c \ Crit( f , c)) \ N�

, where N is asufficiently small neighborhood of Crit( f , c). In particular, if Crit( f , c)is a finite set, then

HC⇤( f c+e

c�e

, j) ⇠=kM

j=1H⇤

�

f c \ Nj, ( f c \ xj) \ Nj�,


where {xj} = Crit( f , c) and Nj are sufficiently small disjoint neigh-borhoods of the critical points xj.

Proof: We start with proving that f c is a strong deformation retractof f c+e. Define W = f c+e

c , W� = f cc \ Crit( f , c) and W0 = f c+e

c \Ws(Crit( f , c)). It follows that W� is relatively closed in W0 (givemore details) and W is closed and therefore a Wazewski set. Fromthe Wazewski Principle (Theorem 7.2.3) it follows that W� = f c

c \Crit( f , c) is a strong deformation retract of W0 = f c+e

c \ Ws(Crit( f , c)).This also implies that f c is a strong deformation retract of f c+e. This If A ⇢ N is an attractor, then A is a deformation retract of N.

implies thatH⇤( f c+e, f c�e) ⇠= H⇤( f c, f c�e).

Secondly, we show that f c�e is a strong deformation retract off c \ Crit( f , c). As before let W = f c

c�e

\ Crit( f , c), W� = f c�e

c�e

andW0 = W. The set W is a Wazewski set and the Wazewski Principlegives that W� = f c�e

c�e

is a strong deformation retract of W0 =

f cc�e

\ Crit( f , c), which proves that

H⇤( f c, f c�e) ⇠= H⇤( f c, f c \ Crit( f , c)).

Combining the isomorphisms gives H⇤( f c+e, f c�e) ⇠= H⇤( f c, f c \Crit( f , c)).

Let N be a neighborhoods of Crit( f , c) such that N ⇢ f c+e

c�e

and byU =

Smi=1 Ui ⇢ f c+e

c�e

. excision

H⇤( f c, f c \ Crit( f , c)) ⇠= H⇤( f c \ N, ( f c \ Crit( f , c)) \ N).

If Crit( f , c) is a finite set we can choose N as a disjoint union ofisolating neighborhoods of xj 2 Crit( f , c).

9.4.2 Exercise. Give an alternative proof using the initial value problemx0 = �( f (x0)� a)r f (x)/kr f (x)k2, with x(0) = x0 2 f b

a .

9.4.3 Remark. If f c+e

c�e

is locally compact, then follows from the theoryof isolated invariant sets and Morse decompositions. Clearly, everyxj 2 Crit( f , c) is an isolated invariant set and Bj and isolating neigh-borhood and Crit( f , c) is a Morse decomposition with isolating blockf c+e

c�e

. By Theorem 7.6.1 there exist isolating blocks for xj. See Section9.1 for more details.

For a critical value c 2 R we can define the critical groups as

C⇤(Crit( f , c)) = H⇤( f c \ N, ( f c \ Crit( f , c)) \ N),

where N is a neighborhood of Crit( f , c). The same can be definedfor any isolated connected component of Crit( f , c). The definitionof critical group is independent of the choice of neighborhood N.

morse theory 175

For an isolated critical point xj 2 Crit( f , R) we define C⇤(xj) =

H⇤( f c \ Nj, ( f c \ xj) \ Nj), where Nj is a neighborhood of xj.From the deformation lemma we derive the following version of

the Morse relations for critical points.

9.4.4 Theorem. Assume f 2 C2(X) satisfies the (PS) condition and leta < b be regular values such that Crit( f , [a, b]) is a finite set. Then,

Âxj2Crit( f ,[a,b])

it(xj) = Pt( f b, f a) + (1 + t)Qt,

where it(xj) = Pt( f c \ Nj, f c \ xj \ Nj), the Poincaré polynomial ofC⇤(xj).

In the case that 0 is an isolated element of the spectrum of f 00(xj)

the Gromoll-Meyer Theorem implies that it(xj) is a polynomial andcontains index information about the spectrum. This property issatisfied by Fredholm functionals on X.

9.5 Homotopy types and the Morse index

A Morse function satisfies the property that all elements in Crit( f , R)

are non-degenerate, i.e. the operator f 00(x) : X ! X⇤ is invertiblefor all x 2 Crit( f , R). For any critical point x 2 Crit( f , R) we maylinearize f , i.e. compute its Taylor expansion up to order 2:

f (x) = f (x0) +12�

f 00(x0)(x� x0), x� x0�

+ o(kx� x0k2).

The Morse Lemma provides a local change of coordinates to thequadratic form given by f 00(x).

9.5.1 Lemma. Let x0 be a non-degenerate critical point. Then thereexists an neighborhood N of x0, and a diffeomorphism h : N !h(N) ⇢ X, with h(x0) = 0, such that

f (h�1(y)) = f (x0) +12( f 00(x0)y, y),

for all y 2 h(N). A function f is a Morse function on f ba if

all critical points in Crit( f , [a, b]) are non-degenerate. Morse functionsare prevalent.

9.5.2 Proposition. Let f 2 C2(X) satisfying (PS) on the strip f ba for

regular values • < a 0 there exists aMorse function bf such that k f � bf kC2 < e, and which satisfies (PS) onthe strip bf b

a .

The next step is to investigate the homology H⇤( f c, f c\Cf (c)). Forthat we invoke the Morse lemma. Before we state the main result ofthis section we first introduce a characterization of the critical points.


9.5.3 Definition. For a critical point x 2 Crit( f , R) of a C2 Morsefunction the Morse index µ(x) is defined as dim E�, where E� is thenegative eigenspace in the decomposition X = E+ � E�.

The Morse co-index µ

c(x) is defined as the dimension of E+. Inthe infinite dimensional case both indices can be infinite, and Morsetheory, as we present it here, will be empty. Extensions such as Floerhomology can be a way to obtain a working Morse theory in thatcase and will be discussed in Section 9.6. In the infinite dimensionalexamples that we will see in the next chapter the Morse index of acritical point is always finite. Such problems can be characterizedas semi-definite. Of course in the finite dimensional both the Morseindex and co-index are always finite.

Before we use the Morse lemma we recall that for a Morse functionit holds that

H⇤( f c, f c \ Crit( f , c)) ⇠=M

jH⇤( f c \ Nj, ( f c\{xj}) \ Nj).

9.5.4 Lemma. Let x 2 Crit( f , R) be a non-degenerate critical pointand N a sufficiently small neighborhood of x. Then

Hk( f c \ N, f c\{x} \ N) =

8

<

:

F for k = µ(x),

0 otherwise,

where F is the coefficient field of H⇤.

In the next section we will use this information to count criticalpoints.

9.6 Other homology invariants and the Morse inequalities

Here also discuss other (co)-homology theories in order to dostrongly indefinite problems.

In the previous section we have established a link between thetopology of the pair ( f b, f a) and the critical points at the (only)critical level c. We recall

Hk( f b, f a) 'M

i : µ(xi)=kZ = Zmk(a,b), (9.6)

where mk(a, b) = #{i : µ(xi) = k, xi 2 Cf ( f ba )}. The next question is,

what happens the Morse index and co-index are infinite.

10Morse Theory for elliptic equations

In this chapter we apply Morse Theory to elliptic equations.

10.1 Variational principles and critical points

In this chapter Morse theory will be applied to a model class ofnonlinear elliptic differential equations. For our purposes here we areconcerned with elliptic problems of the form

�Du = g(x, u), x 2 W,

u = 0, x 2 ∂W,

where W ⇢ Rn is a bounded domain with smooth boundary ∂W, andg(x, u) is a C•-nonlinearity that satisfies the growth estimate

|gu(x, u)| C + C|u|p�1, p > 1,

uniformly in x 2 W, for all u 2 R. We will study solutions of thisproblem using minimax and Morse theory. Standard regularitytheory for this equation reveals that solutions are C•(W). Regularityissues will be postponed till later.

As opposed to applying degree theory and fixed point argumentsthe equation above possesses a alternative formulation for findingsolutions; variational principle. Consider the integral

Z

W

h12|ru(x)|2 � G(x, u(x))

i

dx,

where G(x, u) =R u

0 g(x, s)ds. Clearly, the integral is well-definedfor all u 2 C•(W) \ C1

0(W). Denote the integral as functional onfunctions u(x) by f . Let us consider the first variation of the integralwith respect to test functions j 2 C•

0 (W). This yields

f (u + j)� f (u) =Z

W

h

ru ·rj� g(x, u)j

i

dx

+Z

W

h12|rj|2 � gu(x, u + qj)j

2i

dx.


As explained before, under the assumption that p < n+2n�2 , when n � 3,

the function f extends to the Sobolev space H10(W) = closH1(C•

0 (W)),with equivalent norm

kukH10

:=r

Z

W|ru(x)|2dx.

This uses the compact embeddings

H10(W) ,!

8

>

>

<

>

>

:

C0(W), n = 1,

Lp+1(W), n = 2, 0 p < •,

Lp+1(W), n � 3, 0 p < n+2n�2 .

If we use the above variation formula we obtain that for fixed u 2H1

0(W) it holds that�

�

�

f (u + j)� f (u)�Z

W

h

ru ·rj� g(x, u)j

i

dx�

�

�

= o(kjkH10),

which proves that f is differentiable on H10(W). Notation:

f 0(u)j =Z

W

h

ru ·rj� g(x, u)j

i

dx.

10.1.1 Exercise. Prove the above identity for the Fréchet derivative in the caseg(x, u) = lu + |u|p�1u, using the first variation and the Sobolev emebeddings.

10.1.2 Exercise. †† In Section 9.4 we introduced the notion of gradient. Compute thegradient r f (u) in H1

0(W).

Similarly, the second variation yields�

�

�

f 0(u+y)j� f 0(u)j�Z

W

h

ry ·rj� gu(x, u)yj

i

dx�

�

�

= o(kykH10)kjkH1

0,

which proves that f twice continuously differentiable on H10(W).

Notation: Notation:

f 00(u)yj =Z

W

h

ry ·rj� gu(x, u)yj

i

dx.

10.1.3 Exercise. Establish the expression for the second derivative in the caseg(x, u) = lu + |u|p�1u, by proving the identity for the second Fréchet deriva-tive. The expression for

the first derivative explains that the elliptic equation is satisfied in a‘weak’ sense, i.e. weak solution u 2 H1

0(W). If additional regularity isknown then a simple integration by part provides the identity

Z

W

h⇣

�Du� g(x, u)⌘

j

i

dx = 0,

for all j 2 H10(W, which reveals the equation again and u is a ‘strong’

solution. This identity, without a priori regularity, can also be in-terpreted in distributional sense, i.e. �D is regarded as a map fromH1

0(W) to its dual Sobolev space H�1(W).

morse theory for elliptic equations 179

10.1.4 Exercise. Interpret the above identity in the dual space H�1(W).

Having established all these preliminary differentiability prop-erties we conclude that solutions of the elliptic equation can beregarded as critical points of the function f . This variational principleallows us to attack the elliptic problem via critical point theory.

Before going to the actual application in the next section we firstprove a result concerning the Palais-Smale condition.

10.1.5 Lemma. Let g and W be as above and let 1 2,

0 < gG(x, u) ug(x, u), for |u| � r > 0.

Then, the function f satisfies the Palais-Smale condition on H10(W).

Proof: The requirements on p are needed in order for f to be well-defined and differentiable. Let {un} be a sequence satisfying

f (un)! c 2 R, and f 0(un)! 0,

as n! • — a Palais-Smale sequence. In terms of the above integralsthis reads:

Z

W

h12|run|2 � G(x, un))

i

dx ! c, and�

�

�

Z

W

h

run ·rj� g(x, un)j

i

dx�

�

�

enkjkH10, en ! 0, 8j 2 H1

0(W).

10.1.6 Exercise. Derive the above inequalities from the definitions of f and f 0.The first step is to show that a Palais-Smale sequence {un} is

uniformly bounded in H10(W), with the bound only depending on c.

In the expression for the derivative we choose j = g

�1un. This gives,upon substitution, that

�enkunkH10

Z

W

h12|run|2 � g(x, un)un

i

dx enkunkH10.

Combining this inequality with the expression for f (un) we obtain:⇣1

2� 1

g

⌘

Z

W|run|2dx =

Z

W

h

G(x, un)� g

�1ung(x, un)i

dx

+ c + en + eng

�1kunkH10 C + g

�1enkunkH1

0.

This inequality yields the estimate kunkH10 C.

Since H10(W) is a Hilbert space the boundedness of {un} implies

that unk * u in H10(W). Since the embeddings of H1

0(W) into Lp+1(W)

are all compact, provided p < n+2n�2 , n � 3, it holds that unk ! u in

Lp+1. Consequently,R

W G(x, unk )dx !R

W G(x, u)dx. If we combinethis with the convergence of f we obtain:

12

Z

W|run|2dx = f (un) +

Z

WG(x, unk )dx ! c +

Z

WG(x, u)dx,


which proves that kunkH10! kukH1

0, and convergence of {unk} in

H10(W), completing the proof.

Having establish the Palais-Smale condition for f allows us now toapply critical point methods.

10.2 Solutions via Morse Theory

Knowing now that our function f is a proper C2-function on H10 ,

which satisfies also the other conditions of (H0f ), we can use theMorse relations of Section 6 (Chapter II). We recall

it(Cf \ f ba ) = Pt( f b, f a) + (1 + t)Qt,

where a 1.

By studying the geometry of the sets f a, for different values of a,we shall try to compute the dimension of certain homology groupsHn( f b, f a), for certain values of a < b. Using the Morse relations asmentioned above we can find parts of the Morse series it(Cf \ f b

a ). Bymeans of Theorem 7.3 (Chapter II) we can obtain information aboutthe existence of critical points of certain index.

Theorem 10.2.1 Let D ⇢n be a smooth bounded domain. Assume thatl < l1, 1 < p < • if n 2 and 1 < p < n+2

n�2 if n � 3. Then f (u) has atleast one non-trivial critical point u 2 H1

0(D), with

µ(u) 1 µ

⇤(u).

Proof: From Section 2 we already that f (u) satisfies Hypotheses(H0f ). Because l < l1 we can define

kuk2⇤ =

Z

D|ru|2 � l

Z

Du2,

as an equivalent norm on H10(D). We then have

f (u) =12kuk2

⇤ �1

p + 1kukp+1

p+1.

Using the sobolev-inequality (Lemma 1.2) we obtain the estimate

f (u) =12kuk2

⇤ �1

p + 1kukp+1

p+1 �12kuk2

⇤ �C

p + 1kukp+1

⇤ � a > 0,

provided u 2 ∂Br(0) 2 H10(D) and r > 0 sufficiently small. Com-

puting f 00(0) one can easily see that 0 is a non-degenerate minimum


(thus isolated), with it({0}) = 1. For any other critical point u of f (u)one can compute the critical values. We have

c =12

p� 1p + 1

kukp+1p+1, 8u 2 Cf .

From (3.3) one easily sees that 0 is the only critical point at ’energy-level’ 0. The levels below 0 (c < 0) are regular values of f , which isclear by (3.3) and the (PS)-condition. The level c = 0 is also an iso-lated energy-level, because suppose not, then there exists a sequenceof positive critical values cn ! 0, critical points {un} with f (un) = cn

and f 0(un) = 0. By the (PS)-condition one deduces that unk ! 0 inH1

0(D), which contradicts the isolatedness of 0.One can choose e > 0 sufficiently small, such that c = e is a

regular value of f (u). From (3.2) we have that an annalus {u; r1 kuk⇤ r2} is not cantained in f e, provided e > 0 is small enough.Therefore f e is not path-connected and f e has at least two path-connected components, i.e. a small neighbourhood of 0 (use theMorse Lemma) and the set {u; kuk⇤ � R}, R large enough. Thisyields

dim H0( f e) � 2.

In order to find critical points of f (u) now we consider the pair( f •, f e) = (H1

0 , f e). Let us consider the exact sequence

�! H1(H10 , f e)

∂1�! H0( f e, ∆)i0�! H0(H1

0 , ∆) �! .

Clearly dim H0(H10 , ∆) = 1 and dim H0( f e, ∆) = dim H0( f e) � 2.

Using the exactness of the above sequence we deduce that

dim H1(H10 , f e) � 1.

From Theorem 9.3.3 we conclude that Crit( f , [a, •)) 6= ?.

10.2.2 Remark. In Theorem 3.1 the existence of a second critical pointis given, with additional information on the Morse-index of the criti-cal point. If one is only interested in the existence of a second criticalpoint one can also do this using the Morse relations of Theorem 4.6.One argues as follows. Suppose 0 is the only critical point of f (u).Clearly f satisfies (Hf ) in that case. The Morse relations then give

1 = t + p(t) + (1 + t)Qt,

which cannot be true for any t, therefore yielding a contradiction. Sothe assertion that Cf = {0} was false and so f must have at least oneadditional critical point. The details of this reasoning are left to thereader as an exercise.


Theorem 10.2.3 Let D ⇢N be a smooth bounded domain. Assume thatl 62 s(�D), 1 < p < • if N 2 and 1 < p < N+2

N�2 if N � 3. Then f (u)has at least one non-trivial critical point u 2 H1

0(D), with

µ(u) n + 1, n� 1 µ

⇤(u),

where n = m(0).

Proof: From the previous we know that if u 2 Cf its critical valuec is non-negative. Thus every negative level is therefore regular. Inorder to prove this theorem we shall therefore compute the relativehomology groups of the pair (H1

0(D), f�a), with a > 0. We prove thatHk(H1

0(D), f�a) = 0 for all k � 0. To do so we proceed as follows; Letu 2 ∂B1(0) = S•, then

f (tu) =t2

2� l

2t2kuk2

L2 �tp+1

p + 1kukp+1

p+1.

It is clear that ddt f (tu) < 0, whenever f (tu) �a, a > 0. Using the

Implicite Function Theorem (see e.g. ??HV]) one concludes that thereis an unique function T(u), i.e. T 2 C(S•,+ ), such that

f (T(u)u) = �a, 8u 2 S•.

Using the Sobolev embeddings and the fact l 62 s(�D) we obtain;

f (tu) � t2

2C(l)� tp+1

p + 1S�

p+12 ,

where C(l) is nonzero, which implies

|T(u)| � d(l) > 0.

We have that Bd

(0) 6⇢ f�a, so we define

h(s, u) =

8

<

:

(1� s)u + sT( ukukH1

0

) ukukH1

0

, kukH10� d, f (u) � �a,

u, f (u) �a.

It is clear that h 2 C([0, 1] ⇥ H10(D)\B

d

(0), H10(D)\B

d

(0)). Oneobserves now that f�a is a strong deformation retract of H1

0(D)\Bd

(0)and thus

Hk(H10(D), f�a) = Hk(H1

0(D), H10(D)\B

d

(0)), 8k.

Furthermore we define x(s, u) = ukukH1

0

, u 2 H10(D)\B

d

(0). Using the

map x(s, u) one easily proves that

Hk(H10(D), H1

0(D)\Bd

(0)) = Hk(B•, S•), 8k,


which, using Remark 3.2 (Chapter II), proves our assertion. We have

Pt(H10(D), f�a) = 0.

Let us continue now with the proof. Whenever l 62 s(�D), themap f 00(0) = �D� l, seen as bounded map from H1

0(D) to H�1(D),is invertible. In that case 0 is a non-degenerate critical point withMorse-index m(0) = # negative eigenvalues of f 00(0) (seen now aunbounded map in L2(D)). From Section 3 (Chapter II) we repeat

Cn( f , 0) = , n = m(0), 0, n 6= m(0),

and thus it({0}) = tm(0). For the Morse-index of the set Cf \ f •�a this

yieldsit(Cf \ f •

�a) = tm(0) + p(t),

where p(t) is some positive formal series. From the Morse relationsof Theorem 6.7 we have

tm(0) + p(t) = (1 + t)Qt,

which indicates that Qt either contains the monomials tm(0) or tm(0)�1.For that reason p(t) must contain either tm(0)+1 or tm(0)�1. Thisyields;

it(Cf \ f •�a) =

8

<

:

tµ(0) + tµ(0)+1 + z1(t),

tµ(0) + tµ(0)�1 + z2(t).

By Theorem 7.3 we then deduce the existence of a non-trivial criticalpoint u 2 Cf \ f •

�a with the additional property as stated in thistheorem.

10.3 Multiplicity results for critical points

If one take a closer look at the function f of our example, one ob-serves that the function is even, i.e.

f (u) = f (�u)

Exploring this symmetry property together with the Morse relationswe can find infinitely many solutions of Problem (I).

Theorem 10.3.1 Let D ⇢ Rn be a smooth bounded domain. Assume thatl 62 s(�D), 1 < p < • if n 2 and 1 < p < n+2

n�2 if n � 3. Then f (u)has infinitely many critical points u 2 H1

0(D).

Proof: The conditions in order to apply Theorems 6.6 and 6.7are satisfied by our previous considerations. For the proof of thistheorem we need some refinement of the Morse relations in that case


that f possesses certain symmetry properties. Denote by G the classof functions satisfying (H0f ) and in addition possess an symmetryproperty. The approximation procedure we carried out in section6 can now be performed using approximations having the samesymmetry properties. We obtain

it(Cf \ f ba ;G) = Pt( f b, f a;G) + (1 + t)Qt(G).

We argue now by contradiction (see Remark 3.2). Assume nowthat the number of critical points of f is finite. Again 0 is a non-degenerate critical point and all other critical points are of courseisolated. By the symmetry in the function f we see that if u 2 Cf also�u 2 Cf . For it(Cf � {0}) this yields, using Theorem 7.1;

it(Cf � {0}) = 2 Âk�0

aktk,

where the sum is finite. The latter observation can be justified asfollows. The Morse-index of critical points of f is finite. This canbe seen by analysing f 00(u) (see e.g. [HV]). Nevertheless if criticalpoints would have infinite index, they would not appear in the Morseseries it and therefore we can restrict ourselves to critical points withfinite index µ(u). By Theorem 5.2 (Chapter II) it follows then that thedimensions of Cn( f , u) are finite whenever n 2 [µ(u), µ

⇤(u)] and thedimensions are zero if n 62 [µ(u), µ

⇤(u)]. This finally yields that thesum in (4.3) consists of only finitely many terms.

Take a > 0, then

it(Cf ) = it(Cf \ f •�a) = tm(0) + 2 Â

k�0aktk.

Now we use the Morse relations (4.2) together with (4.4);

tµ(0) + 2 Âk�0

aktk = Pt(H10(D), f�a) + (1 + t)Qt = (1 + t)Qt,

where Qt is a proper polynomial. Therefore we can evaluate (4.5) att = 1. We obtain

1 + 2M = 2N, N, M < •,

which is a conttradiction unless the number of critical points isinfinite.

10.4 Functions lacking compactness

So far we have seen that if p < N+2N�2 , the (PS)-condition is satisfied.

In this section we shall take a closer look at the case p = N+2N�2 . For

example, if l = 0 and D is starshaped, Problem (I) does not have a


solution except from the trivial one (u ⌘ 0) and for this reason f hasno critical points different from u = 0 (see Theorem 1.4, Chapter III).This yields that Cf = {0} and it(Cf ) = 1. If the Morse relations ofChapter II were to hold we would have

1 = Pt( f b, f a), a < b • (regular), (5.1)

where Qt = 0. Choosing b = • and a < 0, we immediately obtaincontradiction because Pt( f •, f a) = 0 in that case (choosing differentvalues of a and b will also lead to a contradiction). What is exactlythe reason the Morse relations do not hold? This question can beanswered as follows. In the proof of Lemma 4.2 and 4.3 one uses the(PS)-condition in an essential way. If the condition is not satisfiedstrange things can happen and the Lemmas 4.2 and 4.3 are no longervalid. This can be seen by going through the motions of the proofsof these lemmas. To make this more clear we shall illustrate this bymeans of the following example.

Consider the function

g(u) =u

1 + u2 2 C•(R, R). (5.2)

Clearly c = {1/2,�1/2} are the only critical values of f . The valuec = 0 is exceptional, because there are sequences of points {un} suchthat g(un) ! 0, g0(un) ! 0 and un ! •. This implies that g cannever satisfy (PS) in the strip gb

a, with a < b < 0 or 0 < a < b. Onecan easily picture what happens if one tries to deform gb onto ga;

Let us consider the following function

f (u) =12

Z

D|ru|2 � N � 2

2N

Z

D|u|

2NN�2 . (5.3)

This function does not satisfy the (PS)-condition in the strip f •a , a < 0.

We have in fact the following lemma;

Lemma 5.1. (Brezis & Nirenberg [BrN]) Let D ⇢ RN be a smoothbounded domain and N � 3. Then function f , defined in (5.3),satisfies the (PS)-condition in f b

a if and only if

b <1N

SN2 , (5.4)

where S is the largest constant such that

Skuk2

L2N

N�2 kruk2

L2 . (5.5)

The constant is given by

S = N(N � 2)p⇣G(N

2 )

G(N)

⌘

2N . (5.6)


Lemma 5.1 gives that (PS) is not satisfied in f •a . One can get more

precise statements about the behaviour of f as its ’energy’ increases.For this we refer to Lions [L1,2] concerning the so-called concertra-tion compactness method. Using Lemma 5.1 one can compute thePoincare-series of the pair ( f b, f a) for the function f given in (5.3).

Corollary 5.2. Let D ⇢ RN be a smooth bounded domain andN � 3. Then for the function f , defined in (5.3), we have

Pt( f b, f a) = 0, a > 0, 1, a < 0,

provided b < 1N S

N2 .

Proof of Corollary 5.2. From Lemma 5.1 we have that f satisfies(PS) in f b

a , there b < 1N S

2N . The point 0 is a non-degenerate local

minimum of f . Let us assume there are more critical points in f ba . Let

u be a non-trivial critical point. Then u satisfies the following Eulerequation

�Du = uN+2N�2 , in H�1(D) (5.7)

andZ

D|ru|2 =

Z

D|u|

2NN�2 , (5.8)

Z

D|ru|2 � N � 2

2N

Z

D|u|

2NN�2 = c <

1N

SN2 .(5.9)

combining (5.5) and (5.7) we obtainZ

D|ru|2 � S

N2 . (5.10)

Substituting (5.8) into (5.9) we get

c =Z

D|ru|2 � N � 2

2N

Z

D|u|

2NN�2 =

1N

Z

D|ru|2 � S

N2 ,

using (5.10), which leads to a contradiction. Therefore 0 is the onlycritical point in f b

a . Thus it(Cf \ f ba ) = 1, when a < 0 and it(Cf \ f b

a ) =

0, when a > 0. Using the Morse relations we conclude the proof.Now we perturb the above problem slightly, i.e. we consider the

function;

f (u) =12

Z

D|ru|2 � l

2

Z

Du2 +

N � 22N

Z

D|u|

2NN�2 . (5.11)

As before f satisfies the (PS)-condition beneath some fixed energy-level. We have the following lemma;

Lemma 5.3. (Brezis & Nirenberg [BrN]) Let D ⇢ RN be a smoothbounded domain and N � 3. Then function f , defined in (5.11),satisfies the (PS)-condition in f b

a if and only if

b <1N

SN2 . (5.12)


Proof of Lemma 5.3. The prove of this lemma is similar to the proofof Lemma 5.1. Because the pertubation l

R

D u2 has compact deriva-tive (compactness of the embedding of H1

0(D) into L2(D)), the criticalenergy-level is the same as in Lemma 5.1.

Using the Morse relations again we shall prove existence of criticalpoints of f .

Theorem 5.4. Let D ⇢ RN be a smooth bounded domain andlet N � 4. If furthermore 0 < l < l1 (first eigenvalue of �D), fhas at least one non-trivial (positive) critical point and consequentlya solution (weak) to equation (0.1) with p = N+2

N�2 . If the abovecondition l is not satisfied there are no (positive) critical points andno solutions to (0.1).

Proof of Theorem 5.4. From Lemma 5.3 we have that f satisfies (PS)in the strip f b

a when b < 1N S

N2 . In order to prove the existence of

critical points we argue by contradiction. Suppose f has no criticalpoints besides 0 in f c, c = 1

N SN2 . Clearly then c� d, d > 0 small, is

a regular value of f . We shall apply the Morse relations now in thestrip f c�d

a , for suitably choosen a, d > 0. From the proof of Theorem3.1 (Chapter III) we already know that dim H(

0 f a) � 2, provided a > 0is sufficiently small. Next we want to be able to choose a d > 0 suchthat f c�d contains a path connecting 0 and an arbitrary point u0 2 f a,with the additional property that f (u0) < 0. In order to realize thelatter one needs to make some technical estimates.

Consider the following function

v(x) =f(x)

(2+|x|2) N�22

, (5.12)

where f(x) 2 C•0 (D) is a smooth cutt-off function. Let us now study

f on the half-line {tv(x)}t�0. We then have

f (tv) =t2

2

Z

D|rv|2 � l

2t2Z

Dv2 � N � 2

2Nt

2NN�2

Z

D|v|

2NN�2 . (5.13)

The strategy now is to choose such that max f (tv) < 1N S

2N . It is

clear that one can restrict t to the interval [0, t⇤] if t⇤ is sufficientlylarge, i.e. one chooses t⇤ so that f (t⇤v) < 0 (see proof Theorem 3.1).Striaghtforward computation shows that

maxt2[0,t⇤ ]

f (tv) =1N

Z

D|rv|2 � l

Z

Dv2

�

Z

D|v|

2NN�2

�

N�2N

!

N2

. (5.14)

Estimating (5.14) one obtains (see Brezis [Br], Brezis & Nirenberg[BrN] for details);

maxt2[0,t⇤ ]

f (tv) =1N�

S� lC2 + O(N�2)�

N2 , N � 5,

1N�

S� lC2 log(||) + O(2)�

N2 , N = 4.


Clearly if l > 0 and e > 0 (sufficiently small) we see that

maxt2[0,t⇤ ]

f (tv) <1N

SN2 . (5.15)

From (5.15) one deduces that if > 0 is small one can pick a d > 0small so that the line-segment {tv}t2[0,t⇤ ], which connects 0 and t⇤v,is contained in f c�d

a .We conclude the proof by showing that H1( f c�d, f a) is nontrivial.

Because dim H0( f a) � 2, H0( f a) has at least two generators, say [0]and [tv] (see e.g. Spanier [Sp]). Consider the map

i0 : H0( f a) �! H0( f c�d),

induceded by the natural embedding f a ,! f c�d. From the previousit follows that in the set f c�d the elements 0 and tv can be connectedby a path, thus i0([0]� [tv]) = 0. Using the exactness of the sequence

�! H1( f c�d, f a)∂1�! H0( f a)

i0�! H0( f c�d) �!,

it follows that [0]� [tv] 2 Im(d1). This yields H1( f c�d, f a) 6= 0. Forthe Poincare-polynomial (because c� d < 1

N SN2 in the strip f c�d

a , (PS)is satisfied together with the other hypotheses of (H0f ) and thereforewe know from the previous that Pt( f c�d, f a) is finite) this gives

Pt( f c�d, f a) = t + p(t).

By assumption the Morse-polynomial is

it(Cf \ f c�d

a ) = 1.

From the Morse relations we then obtain

1 = t + p(t) + (1 + t)Qt, Qt � 0.

This is clearly a contradiction there t does not appear in the lefthand side and the t in right hand side is inadmissable. We concludetherefore that there is at least one non-trivial critical point u of f inthe strip f c�d

a . This completes the proof.

11Strongly indefinite systems

11.1 Strongly indefinite elliptic systems

11.2 The Weinstein Conjecture for compact contact type hyper-surfaces

AAppendix

A.1 Differentiable mappings

A.1.1 Theorem. (The Inverse Function Theorem). Let W ⇢ Rn be aneighborhood of x0, and let f 2 C1(W; Rn). Assume that f 0(x0) is aninvertible mapping on Rn. Then there exist neighborhoods U 3 x0

and V 3 f (x0) such that f is a local diffeomorphism on U. Moreover,

( f�1(x0))0 = ( f 0(x0))

�1,

and x0 is the unique solution of f (x0) = p in U.

A.1.2 Theorem. (C1-version of the Implicit Function Theorem). LetW⇥L ⇢ Rn ⇥Rk be a neighborhood of (x0, l0), and let f 2 C1(W⇥L; Rn). Assume that f (x0, l0) = 0, and dx f (x0, l0) is an invertiblematrix. Then there exists a neighborhood L0 ⇢ L of l0 and a C1-function g : L0 ! Rn, such that

g(l0) = x0, and f (g(l), l) = 0, 8 l 2 L0.

In addition g0(l) = �[dx f (g(l), l)]�1dl

f (g(l), l).

Approximation

A.1.3 Theorem. Let W ⇢ Rn be a bounded domain and let f 2C`(W; Rm). Then there exists a sequence of maps f k 2 C•(W; Rm)

such thatk f � f kkC` ! 0,

as k! •.

Proof: Construct a sequence f k via mollification. Let z(x) be aC•-function on Rn, satisfying

(i) z(x) ⌘ 0, for |x| � 1, and

(ii)Z

Rnz(x)dx = 1.


Clearly, z 2 C•0 (B1(0)), and the radius of the ball can be altered by

rescaling z as follows: z

e(x) = e

�nz(x/e). Then, z

e 2 C•0 (B

e

(0)), andR

Rn z

e(x)dx = 1, by the choice of the rescaling. The cut-off functionz

e is called a mollifier.Regard f as a function on Rn by extending it by zero outside W,

and define

f k(x) := (z1/k ⇤ f )(x) :=Z

Rnz

1/k(x� y) f (y)dy.

Since z

1/k is smooth and compactly supported, the convolution is aC•-function on Rn for all k � 1. In particular f k 2 C1(W).

A.1.4 Exercise. Show, all partial derivatives of z

1/k ⇤ f exist, and therefore thatf k 2 C•(Rn).

It remains to show that f k converges to f in the C0-topology.

k f � f kkC0 = maxx2W

�

� f (x)� (z1/k ⇤ f )(x)�

�

= maxx2W

�

�

�

Z

Rnz

1/k(x� y)⇥

f (x)� f (y)⇤

dy�

�

�

maxx2W

max|x�y|1/k

k f (x)� f (y)k!

.

Since f is uniformly continuous on W, the right hand side goes tozero as k! •, which proves the lemma.

The theorem’s of Tietze, Sard and Smale

Sard’s Theorem and its inifinite dimensional version due to Smale —the Sard-Smale Theorem — form fundamental steps in establishingtopological tools as we have seen in the previous section. We startwith giving the ‘uncensured’ version of Sard’s Theorem, and proof isgiven in the simplist case (the version of Sard’s Theorem used in theprevious section).

A.1.5 Theorem. Let f : Rn ! Rm be a map of class Ck. Then theset of critical values f (Cf ) has (Lebesgue) measure zero, providedk � max(1, n�m + 1).

Proof: We restrict here to the case that n = m. Let D ⇢ Rn be acube with sides of size `. For points x, x0 2 D, using the fact thatf 2 C1(Rn), Taylor’s Theorem implies that

f (x) = f (x0) + f 0(x0)(x� x0) + R(x, x� x0),

where kR(x, x � x0)k = o(kx � x0k), uniformly in x 2 D, i.e. forany e > 0 there exists a d > 0 such that kR(x, x� x0)k ekx� x0k,for all kx � x0k < d. Suppose x0 2 Cf \ D, then the points L =

appendix 193

{ f (x0) + f 0(x0)(x � x0)} represent an affine subspace of dimensionless or equal to n� 1.

The estimates on the remainder term R imply that if kx� x0k < d,then dist( f (x), L) ed. Since, f is Lipschitz continuous on D withLipschitz constant N, it also holds that k f (x) � f (x0)k Nd. Theimage of the ball B

d

(x0 under f is therefore contained in a ‘cuboid’centered at f (x0) with sides of size less that 2Nd in L, and of sizeless that 2ed orthogonal to L. The volume of the cuboid is less than2n

d

nNn�1e.

Next we divide up the cube D into sub-cubes Dd

with sides oflength d. Thus the number of such cubes is Dn, with D > `/d. Eachsub-cube that contains a critical point x0 has the property that

µ( f (Dd

)) 2n(`/D)nNn�1e.

Consequently, µ( f (Cf \ D)) Dnµ( f (D

d

)) = 2n`nNn�1e, which

proves the theorem since e can be chosen arbitrarily small.

The version of Sard’s Theorem presented above holds for anycombination of n, m � 0. Only when n � m the proof is non-trivial,and the proof here is restricted to the case n = m. In the case n <

m we argue as follows: Let µ denote the Lebesgue measure, thenµ( f (Rn)) = 0, since f (Rn) is an n-dimensional subset of Rm, withn < m ( f is a C1-map). This implies that Sard’s Theorem is triviallysatisfied when n < m.

A.1.6 Theorem. (Tietze’s extension Theorem) Let (X, d) be a metricspace and A ⇢ X a closed subset. Suppose f : A ⇢ X ! R iscontinuous and bounded. Then, there exists a continuous extensionf : X ! R, such that f |A = f .

A.1.7 Theorem. (Dugundji) Let (X, d) be a metric space and A ⇢ Xa closed subset. Let Y be a normed linear space and f : A ⇢ X ! Yis continuous. Then, there exists a continuous extension f : X ! R,such that f |A = f and f (X) is contained in the convex hull of f (A).

A.2 Basic Nemytskii maps

We start off this chapter with a basic result about composition, orsubstitution mappings — also called superposition mappings. Suchmaps are also referred to in the literature as Nemytskii maps, or oper-ators. At a later stage when investigating differentiability propertiesof maps between Banach spaces this result plays a central role.


A.2.1 Definition. Let D ⇢ Rn be a bounded domain, and let g(x, s)that satisfy the following conditions:(i) for each s 2 R the function x 7! g(x, s) is Lebesgue measurable in

D,(ii) the function s 7! g(x, s) is continuous on R for x 2 D a.e.The function g(x, s) is called a Carathéodory function. We will

use Carathéodory functions now to define substitution mappings;Nemytskii maps. Let u(x) be a measurable function on D, then themap u 7! g(·, u(·)), assigns a measurable function to each u, and isdenoted by f . Notation;

f (u)(x) := g(x, u(x)).

Indeed, let un be simple functions converging to u. Then by (i) in thedefinition of Carathéodory functions, the substitution g(x, un(x)) ismeasurable for x 2 D a.e. By (ii) g(x, un(x)) converges to g(x, u(x)),establishing the measurability of g(x, u(x)). The map f is called aNemytskii map.

For our purposes we want a Nemytskii map to act between Lp-spaces, or more general Sobolev space. For this additional growthconditions are needed. We will see that these conditions are in factnecessay and sufficient.

Theorem A.2.2 Let D ⇢ Rn be a bounded domain, and let g(x, s) aCarathéodory function on D ⇥R. If there exist 1 p, q •, a functionh 2 Lq(D), and a constant C > 0 such that

|g(x, s)| h(x) + C|s|pq , (A.1)

then the Nemytskii mapping u(x) 7! f (u)(x), introduced above, is awell-defined mapping from Lp(D) to Lq(D). Moreover, f is bounded andcontinuous.

Proof: We start with the well-definedness of f . Let u 2 Lp(D), then

k f (u)kqq =

Z

D|g(x, u(x))|qdx

Z

D

�

�

�

h(x) + C|u(x)|pq�

�

�

qdx C0

Z

D

�

|h(x)|q + |u(x)|p�

dx

khkqq + Ckukp

p,

which proves that f is a well-defined map from Lp to Lq, and mapsbounded sets in Lp to bounded sets in Lq.

As for the continuity we argue as follows. Let un ! u in Lp(D).Then, we may assume without loss of generality, that |un(x)| k(x)(possibly along a subsequence), with k 2 Lp(D). For the Ne-

mytskii map this implies that | f (un)(x)| h(x) + |k(x)|pq , with

appendix 195

h + |k|pq 2 Lq(D). In order to apply Lebesgue’s Dominated Conver-

gence Theorem now we need to show that f (un)(x) converges a.e. tof (u)(x). Assuming the latter we then obtain that f (un) ! f (u) inLq(D).

We establish the pointwise convergence following the DeFigueiredo1. Define a(x, s) = g(x, s + u(x)) � g(s, u(x)), then a(x, 0) = 0, 1

and by the previous a(x, s) is measurable in x for each s 2 R, anda(x, s) is continuous in s for all x 2 D a.e. Set vn = un � u, thena(x, vn) = g(x, un)� g(x, u). Therefore, if vn ! 0 in measure we needto prove that a(x, vn) ! 0 in measure. For an e > 0, end define thesets

Dke

=n

x 2 D : |s| < 1k

=) |a(x, s)| < e

o

.

Clearly, these sets are nested with [kDke

= D, and�

�[k0k=1Dk

e

�

� = |Dk0e

|.Therefore, we can choose k0 � 1 such that |D| � |Dk0

e

| < e

2 . LetWn = {x 2 D : |vn(x)| < 1

k0}, then then, by assumption, there exists

an n0 � 1 such that |D|� |Wn| < e

2 for all n � n0. Now let Dn = {x 2D : |a(x, vn(x))| < e}, then construction Wn \ Dk0

e

⇢ Dn. Clearly,|D|� |Dn| |D|� |Wn \ Dk0

e

| ⇥

|D|� |Wn|⇤

+⇥

|D|� |Dk0e

|⇤

< e,which proves that

|D\Dn| =�

�{x 2 D : |a(x, vn(x)| � e}�

� < e,

and therefore a(x, vn(x))! 0 in measure.

There are numerous generalizations of this result. For exampleone can consider functions taking values in Rm, or use Orlicz spaces,etc. Another extension is to allow D to be any domain, i.e. the aboveresult remains valid for arbitrary domains. As mentioned before theabove result characterizes Nemytskii mappings. We will mention thefollowing theorem without proof (see 2). 2

Theorem A.2.3 Let D ⇢ Rn be a bounded domain, and let g(x, s) aCarathéodory function on D ⇥ R. If the associated Nemytskii mapping fmaps for Lp(D) to Lq(D), for 1 p, q < •, then there exists a C > 0 anda function h 2 Lq(D) such that

|g(x, s)| h(x) + C|s|pq ,

Moreover, f is bounded and continuous.

This theorem remains valid for arbitrary domains D, and also forq = •.

The next question concernes differentiabilty of Nemytskii maps.The Theorems A.2.2 and A.2.3 reveal that differentiability of g isnot enough to guarantee the well-defined unless we have growthconditions on gs as well. Given an Carathéodory function g whose


partial derivative gs is also a Carathéodory function. Theorem A.2.2gives that if

|gs(x, s)| k(x) + C|s| ab , 8s 2 R, 8x 2 D,

with k 2 Lb(D), and 1 a, b •, then the mappings u(x) 7!gs(x, u(x)) is well-defined from La(D) to Lb(D). The growth condi-tion on gs also yields a growth condition on g itself. We consider afamily of functions g(x, s) as follows: In the next theorem we assumethat

g(x, s) =Z s

0gt(x, t)dt + c(x), c 2 Lq(D).

Using this description we obtain

|g(x, s)| C|s| ab +1 + k(x)|s|+ d(x) C0|s| a

b +1 + C0|k(x)| ba +1 + c(x),

Now let� b

a + 1�

q = b,� a

b + 1�

q = a. Under the assumption a = p,and p > q, we obtain

a = p, b =pq

p� q.

We conclude that u(x) 7! g(x, u(x)) is a Nemytskii map from Lp(D)

to Lq(D), and u(x) 7! gs(x, u(x)) is well-defined from Lp(D) toL

pqp�q (D). Next we investigate the relation between differentiability of

the Nemytskii map f and the well-definedness of u(x) 7! gs(x, u(x)).

Theorem A.2.4 Let D ⇢ Rn be a bounded domain, and let g(x, s) andgs(x, s) be Carathéodory functions on D⇥R. If there exist 1 q 0 such that

|gs(x, s)| k(x) + C|s|1q�

1p ,

then for the functions g defined above the Nemytskii mapping u(x) 7!f (u)(x) is continuously differentiable. Moreover,

⇥

f 0(u)j

⇤

(x) = gs(x, u(x))j(x), 8u, j 2 Lp(D).

Proof: From the previous we know that u(x) 7! gs(x, u(x)) isbounded and continuous from Lp(D) to L

pqp�q (D). Moreover, gs(x, u(x))j(x) 2

Lq(D):

Z

D

�

�gs(x, u(x))j(x)�

�

qdx ⇣

Z

D|gs(x, u(x))|

pqp�q

⌘

p�qp⇣

Z

D|j(x)|p

⌘

qp .

This shows that j 7! gs(x, u(x))j defines a bounded linear map fromLp(D) to Lq(D) with norm bounded by kgs(·, u(·))k pq

p�q, and the map

appendix 197

varies continuously with u 2 Lp(D). It remains to show that this isindeed the Fréchet derivative of f . We have

q(x) = g(x, u(x) + j(x))� g(x, u(x))� gs(x, u(x))j(x)Z 1

0

⇥

gs(x, u(x) + tj(x))� gs(x, u(x))⇤

j(x)dt.

For q we have

kqkq ⇣

Z 1

0kgs(·, u(·) + tj(·))� gs(·, u(·))k pq

p�qdt⌘

p�qp kjkp ekjkp,

for kjkp < d

e

, which follows from the continuity of the Nemytskiimap defined by gs. Using these estimates it follows that

k f (u + j)� f (u)� gs(·, u)jkq = o(kjkp),

for kjkp sufficiently small.

There is a lot more that can be said about differentiability ofNemytskii maps, in particular the case p = q.

A.2.5 Exercise. Investigate the case p = q.

Let us now consider the case of integrating Nemytskii mappings.Given a Carathéodory function g(x, s) we define the integral

G(x, s) =Z s

0g(x, t)dt,

which is also a Carathéodory function. As before we assume that

|g(x, s)| h(x) + C|s|pq , h 2 Lq(D). Integration yields |G(x, s)|

C0|s|pq +1 + |h(x)|

qp +1, and the composition map u(x) 7! G(x, u(x)) is

bounded and continuous from Lp(D) to Lpq

p+q (D). In the special caseq = 1 + 1

p�1 we have a map from Lp(D) to L1(D). Finally we mentionthe following result.

Theorem A.2.6 Let D ⇢ Rn be a bounded domain, and let g(x, s) be aCarathéodory functions on D ⇥R. If there exist 1 p •, a functionk 2 L

pp�1 (D), and a constant C > 0 such that |g(x, s)| h(x) + C|s|p�1,

then for the functional defined by

F(u) =Z

DG(x, u(x))dx,

is a continuously differentiable function on Lp(D). Moreover, its derivativeis continuously differentiable from Lp(D) to L

pp�1 (D), and F0 = f .

Proof: The proof follows along the same lines as the proof of Theo-rem A.3.18 and is therefore left to the reader.


A.3 Sobolev Spaces

Weak derivatives and Sobolov spaces

Assume that D ⇢ Rn is an arbitrary domain, i.e. an open set in Rn.A function u : D ! R is call locally integrable, or u 2 L1

loc(D), ifu 2 L1(D0), for every measurable strict subset D0 ⇢ D. For everyu 2 L1

loc(D) the integral

Tu =Z

Du(x)f(x)dx, f 2 C•

0 (D),

is well-defined and is call a regular distribution.

A.3.1 Definition. A locally integrable function u 2 L1loc(D) has

a weakly partially differentiable with respect to xi there exists afunction vi 2 L1

loc(D) such that

�Z

Du(x)

∂f

∂xi(x)dx =

Z

Dvi(x)f(x)dx,

for all f 2 C•0 (D). The fucntion vi is called the weak partial deriva-

tive with respect to xi, and is again denoted by vi =∂u∂xi

. The weakderivative, if it exists as defined above, has to be regarded as a regu-lar distribution ∂Tu

∂xi= Tvi .

Before we go to the definition of the basic Sobolev spaces let usintroduce some notation. With the partial-symbol ∂

a, with multi-index a = (a1, · · · , an), we denote

∂

au =∂

a1

∂xa11

· · · ∂

an

∂xann

u.

We can now introduce the standard norms. Let m 2 N, then

kukm,p =⇣

Â0|a|m

k∂aukpp

⌘

1p , 1 p < •,

where k · kp is the standard Lp-norm, and |a| = Âi ai. For p = • wedefine

kukm,• = max0|a|m

k∂auk•,

where k · k• is the standard L•-norm. For m = 0 these norm reduceto the Lp-norms.

A.3.2 Definition. The Sobolev space Wm,p(D) is defined as the subsetof functions u 2 Lp(D) for which all weak partial derivatives ∂

auexist for 0 |a| m, and all lie in Lp(D).

Another definition is the space Hm,p(D) which is defined as thecompletion of the set

{u 2 Cm(D) : kukm,p < •}

appendix 199

with respect to the norm k · km,p, and is a Banach space by definition.At first sight these two definition seem to define different classes of

spaces.

A.3.3 Exercise. Prove the inclusion Hm,p(D) ⇢Wm,p(D). One can prove

that also the Sobolev spaces Wm,p are Banach spaces. These spaceshave been used in analysis for quite some time, but it wasn’t until theearly 60’s that Meyers and Serrin proved the following result.

Theorem A.3.4 For 1 p < • it holds that Hm,p(D) = Wm,p(D).

This result immediately proves that the Sobolev spaces Wm,p are Ba-nach spaces, and k · km,p are equivalent norms on Wm,p. We sometimesuse the notation Hm, p, usually for the Hilbert case p = 2, and some-times Wm,p. Sobolev spaces are ‘nice’ Banach space as the followingtheorem reveals:

Theorem A.3.5 For 1 p < • the Sobolev spaces Wm,p(D) are sepa-rable. If in addition 1 < p < •, then these spaces are also reflexive anduniformly convex.

From the proof of the Meyers and Serrin result it follows thatin fact the set C•(D) is a dense subset of Wm,p(D). Under certainconditions on the domain D is also holds that Ck(D), k � m, is densein Wm,p.

Finally we define the space Wm,p0 (D) = Hm,p

0 (D) as the closure ofC•

0 (D) in Wm,p(D).Sobolev spaces for m < 0 can be defined as an interpretation of

dual Sobolev spaces. We mention without proof that for 1 p < •,the dual space (Wm,p(D))0 consists functionals T of the form

Tu = Â0|a|m

Z

Dv

a

(x)∂au(x)dx,

with va

2 Lp0(D) for all a. If u 2 Wm,p0 (D), then the bounded linear

functionals can be regarded as distributions of the form

T = Â0|a|m

(�1)|a|∂ava

.

In this case we define W�m,p0(D) = (Wm,p0 (D))0.

In the literature there exists many variations and generalizations ofSobolev spaces, but for the purposes of this course the basic Sobolevspaces are sufficient. One generalization that is worth mentioning isthat fractional order Sobolev spaces. We will only give an intuitivedefinition on this account. In the case D = Rn, and p = 2 the Sobolevspaces Wm,2(Rn) can also be characterized via the Fourier transformDefine the Fourier transform:

bu(x) =1

(2p)n/2

Z

Rne�ix·xu(x)dx.


Then Wm,2(Rn) =�

u 2 L2(Rn) : (1 + kxkm)bu 2 L2(Rn)

, imme-diately yields a definition of fractional Sobolev spaces Wm,2(Rn) form � 0.

Sobolev inequalities

As spaces of functions the Sobolev spaces Wm,p are very useful forstudying partial and ordinary differential equations as we have seenin the previous chapter. The Sobolev inequalities provide insight intothe nature of Sobolev functions. We start with the case when D = Rn.

A.3.6 Sobolev inequality Let u 2 C•0 (Rn), and let 1 p < n, then there

exists a (universal) constant C(p, n) > 0 such that

kukpn C(n, p)krukp, 3

where 1pn

= 1p �

1n . Proof: We start with the case p = 1. We have that

u(x) =R xi�•

∂u∂xi

(bx)dt, where bx is x with xi replaced by t. Therefore

|u(x)|n n

’i=1

Z

R

�

�

�

∂u∂xi

�

�

�

dxi = f1(x) · · · fn(x),

where fi(x) is independent of xi. If we now apply Hölder’s inequal-ity to |u|

nn�1 :

Z

R|u|

nn�1 dx1 =

Z

R

�

f1 · · · fn�

1n�1 dx1 = f

1n�1

1

Z

R

�

f2 · · · fn�

1n�1 dx1

f1

n�11 ’

i�2

⇣

Z

Rfidx1

⌘

1n�1

Repeat this step now by integrating over x2:Z

R2|u|

nn�1 dx1dx2 =

⇣

Z

Rf2dx1

⌘

1n�1

Z

R

h

f1

n�11 ’

i�3

⇣

Z

Rfidx1

⌘

1n�1

i

dx2

⇣

Z

Rf2dx1

⌘

1n�1

⇣

Z

Rf1dx2

⌘

1n�1 ’

i�3

⇣

Z

R2fidx1dx2

⌘

1n�1

Repeating these steps by integrating over all xi we obtain:Z

Rn|u|

nn�1 dx =

n

’i=1

⇣

Z

Rn

�

�

�

∂u∂xi

�

�

�

dx⌘

1n�1 .

By the arithmetic-geometric main inequality ’i ai 1n Âi an

i , ai � 0,we then have:

⇣

Z

Rn|u|

nn�1 dx

⌘

n�1n

n

’i=1

⇣

Z

Rn

�

�

�

∂u∂xi

�

�

�

dx⌘

1n 1

n Âi

⇣

Z

Rn

�

�

�

∂u∂xi

�

�

�

dx⌘

=1n

Z

Rn Âi

�

�

�

∂u∂xi

�

�

�

dx =1nkruk1,

appendix 201

which establishes the Sobolev inequality for p = 1. Now substitute|u|q, with q = n�1

n pn, into the above inequality. This yields

kukqpn = k|u|qk n

n�1 1

nkr|u|qk1

qn

Z

Rn|u|q�1

⇣

Âi

�

�

�

∂u∂xi

�

�

�

⌘

dx

qn

np�1

p k|u|q�1kp0 krukp =qn

np�1

p kukpnp0pn krukp.

Since q� pnp0 = 1 we obtain the desired inequality.

We should point out that the above proof provides a universalconstant C(n, p). This is by no means an optimal constant. It ispossible to optimize the constant C(n, p). A direct consequence ofthe Sobolev inequality are the higher order Sobolev inequalities. Wewill use these later to prove embeddings for the higher order Sobolevspaces.

A.3.7 Morrey’s inequality Let u 2 C•0 (Rn), and let p > n, then there

exists a (universal) constant C(p, n) > 0 such that�

�u(x)� u(y)�

� C(n, p)|x� y|qkrukp,

where q

n = 1n �

1p . Proof: Let Q(R) be an open

cube containing the origin and whose faces are parallel to coordinatehyperplanes. We have

u(x)� u(0) =Z 1

0

ddt

u(tx)dt, x 2 Q(R).

Since ddt u(tx) = x ·ru(tx) we obtain:

|u(x)� u(0)| Z 1

0|x ·ru(tx)|dt

Z 1

0Â

i|xi|

�

�

�

∂u∂xi

(tx)�

�

�

dt

R Âi

Z 1

0

�

�

�

∂u∂xi

(tx)�

�

�

dt

Define the average u = 1|Q|

R

Q u(x)dx. Then we have that

|u� u(0)| =�

�

�

1|Q|

Z

Qu(x)dx� u(0)

�

�

�

1|Q|

Z

Q|u(x)� u(0)|dx

R|Q|

Z

Q

⇣

Âi

Z 1

0

�

�

�

∂u∂xi

(tx)�

�

�

dt⌘

dx

=1

Rn�1

Z 1

0

⇣

Âi

Z

Q

�

�

�

∂u∂xi

(tx)�

�

�

dx⌘

dt

=1

Rn�1

Z 1

0

⇣

Âi

Z

tQ

�

�

�

∂u∂xi

(y)�

�

�

1tn dy

⌘

dt

The gradient term can be estimated as followsZ

tQ

�

�

�

∂u∂xi

(y)�

�

�

dy ⇣

Z

Q

�

�

�

∂u∂xi

(y)�

�

�

pdy⌘

1p t

1p0 |Q|

1p0 .


Combining these estimates yields

|u� u(0)| Rnp0 �n+1 n

p�1p krukp

Z 1

0t

np0 �ndt =

np�1

p

1� np

R1� np krukp.

We can translate the origin to an arbitrary point x, and consequently

|u� u(x)| = np�1

p

1� np

R1� np krukp.

If we do the same for a point y we obtain

|u(x)� u(y)| = 2n

p�1p

1� np

R1� np krukp.

We can now choose a cube Q(R) with R = 2kx� yk so that both x andy are in Q, then

|u(x)� u(y)| 22� np

np�1

p

1� npkx� yk1� n

p krukp,


In the borderline case that p = n we can also prove an analogue ofthe Sobolev inequality — Trudinger-Moser inequality — which yieldsembeddings into generalizations of Lp-spaces; so-called Orlicz spaces.These space are defined via strictly convex functions generalizing tp.

Continuous and compact embeddings

The fundamental inequalities proved in the previous section can beused now to establish various continuous and compact embeddingsof Sobolev spaces. In order to simplify matters let us consider thespaces Wm,p

0 (D). Since C•0 (D) is a dense subset in W1,p

0 we have theSobolev embedding for these spaces and

kukq Ckuk1,p, u 2W1,p0 (D), p q pn.

The latter can be proved using the idea of extension maps. For thespaces Wm,p

0 (D) we can prove the following lemma.

A.3.8 Lemma. Define the map e : Wm,p0 (D)!Wm,p(Rn) by

e(u) = u, x 2 D a.e., e(u) = 0, x 2 Rn\D.

Then ke(u)km,p,Rn = kukm,p, which implies that the map e is continu-ous. Proof: The proof follows from a standard density

argument. Let un 2 C•0 (D) converging to some u 2 Wm,p

0 (D) in the

appendix 203

Wm,p-topology. The definition of e yields that e(un) 2 C•0 (Rn), and

ke(un)km,p,Rn = kunkm,p. This provides Cauchy sequences in bothspaces and thus the map extends to a continuous map from Wm,p

0 (D)

to Wm,p(Rn).

Using the Sobolev inequality from the previous section we cannow prove the above Sobolev inequality for W1,p

0 (D) into Lq(D).From interpolation Lp-spaces we obtain the following interpolationinequality: Let 1

q = q

p + 1�q

pn, then

kukq kukq

pkuk1�q

pn .

A.3.9 Exercise. Prove the above interpolation inequality (Hint: use Hölders inequal-ity). We start with proving the Sobolev embedding for W1,p(Rn).

By definitionkukp kuk1,p.

From the Sobolev inequality we deduce that

kukpn Ckuk1,p.

Combining these two estimates together with the interpolationinequality we obtain:

kukq kukq

pkuk1�q

pn C1�qkukq

1,pkuk1�q

1,p = C1�qkuk1,p.

Using the extension operator we then find the embedding inequalityfor W1,p

0 (D) for q � p. In the case that D is a bounded domain theinequality holds for 1 q pn which follows from the fact that

kukq |D|p�q

p kukqp on bounded domains. We can now formulate the

following theorem.

Theorem A.3.10 Let D ⇢ Rn be a domain and let m mp < n. Then

kukq Ckukm,p, 8u 2Wm,p0 (D),

with 1p �

1q �

1pn,m

= 1p �

mn . In the case that D is a bounded domain the

result holds for 1 q pn,m.

Proof: The case m = 1 was proved above. We prove this by in-duction. Assume the embedding statement is true for m � 1. Letu 2Wm,p

0 (D), n > mp � m, then u and the partial derivatives vi =∂u∂xi

belong to Wm�1,p0 (D). Consequently

kukpn,m�1 Ckukm�1,p, kvikpn,m�1 Ckvikm�1,p,

which implies that u 2W1,pn,m�10 (D), where 1

pn,m�1= 1

p �m�1

n . We nowcombine this with the embedding for m = 1;

kukpn,mCkuk1,pn,m�1C0kukm,p ,


where 1pn,m

= 1pn,m�1

� 1n = 1

p �mn , which proves the inequality into

Lpn,m(D). The remainder of the proof follows along the same lines asin the case m = 1.

A similar theorem can be proved for the spaces Wm,p(D). We statethis theorem without proof.

Theorem A.3.11 Let D ⇢ Rn be a domain with a C1 boundary ∂D and letm mp < n. Then

kukq Ckukm,p, 8u 2Wm,p(D),

with 1p �

1q �

1pn,m

= 1p �

mn . In the case that D is a bounded domain the

result holds for 1 q pn,m.

A.3.12 Exercise. Prove the Sobolev inequality

kukpn krukp, u 2W1,p0 (D).

Moreover, show that for D a bounded domain, pn can be replaced by 1 q pn.Sobolev inequalities as in the above exercise can be used to find

equivalent norms for Wm,p0 (D) in terms of only the highest order

derivatives.Very important for applications to differential equations is fact that

the embedding operators are sometimes compact mappings.

Theorem A.3.13 Let D ⇢ Rn be a bounded domain with a C1 boundary∂D and let m mp < n. Then the embedding

i : Wm,p(D) ,! Lq(D),

with 1 � 1q > 1

pn,m, is compact

The case mp = n can be considered as a limiting case of theprevious embeddings.

Theorem A.3.14 Let D ⇢ Rn be a domain with a C1 boundary ∂D and letmp = n. Then

kukq Ckukm,p, 8u 2Wm,p(D),

with p q < •. If the domain is bounded then the embedding i :Wm,p(D) ,! Lq(D) is compact.

The final case concerns the embedidngs into Hölder spaces.

Theorem A.3.15 Let D ⇢ Rn be a domain with a C1 boundary ∂D and letmp > n. Then

kukCk�1,q Ckukm,p, 8u 2Wm,p(D),

with k = m�⇥ n

p⇤

, and 0 < q ⇥ n

p⇤

+ 1� np , or 0 < q < 1, when n

p isan integer. If the domain is bounded then the embedding i : Wm,p(D) ,!Ck�1,q(D) is compact.

appendix 205

We will finish this section with a proof of the compactness of theembedding

Wm,p0 (D) ,! Lpn,m(D).

In order to do so we start with a special case. Consider the C =

[�1, 1]n ⇢ Rn. On this domain we can consider functions that extendperiodically, which means we are looking at functions on the periodiclattice Rn/Zn, which we denote by Tn; the n-dimensional torus. Forsuch functions we can use Fourier series. Define

Âk2Zn

akeik·x, x 2 Rn.

Under the condition a�k = ak, the above series represent real functionon Tn, provided that the series converge. From Fourier theory wehave that the space L2(Tn) is characterized as follows; a functionu : TntoR lies in L(Tn) if and only if its Fourier coefficients satisfy

Âk2Zn

|ak|2 < •,

in which case we write u(x) = Âk2Zn akeik·x. In may be clear from theprevious considerations that the space W1,2(Tn) is then characterizedby the convergence criterion

u 2W1,2(Tn), iff Âk2Zn

(1 + |k|2)|ak|2 < •.

A.3.16 Lemma. The embedding

i : W1,2(Tn) ,! L2(Tn),

is compact. Proof: We prove that the embedding i is the limit of

finite rank maps. Define

iN(u) = Â|ki |N

akeik·x.

It is obvious that iN is a finite rank map. We now estimate

kiN(u)� i(u)k22 = Â

|ki |>N|ak|2 = Â

|ki |>N

1 + |k|21 + |k|2 |ak|2

11 + n2N2 Â

|ki |>N|ak|2 =

11 + n2N2 kuk

21,2,

which proves that iN converges to i in the operator norm, establishingthe compactness of the map i.

Let D ⇢ Rn be a bounded domain. As in the case of Rn we cannow embed the space W1,2

0 (D) into W1,2(Tn) by considering the


extension map. We may assume without loss of generality that D iscontained in the interior of C. Then by using the extension map as inLemma A.3.8 we obtain the continuous map:

e : W1,20 (D) �!W1,2(Tn).

The yields the following lemma.

A.3.17 Lemma. Let D ⇢ Rn be a bounded domain, then the embed-ding

i : W1,20 (D) ,! L2(D),

is compact. Proof: We have the composition

W1,20 (D) �!W1,2(Tn) ,! L2(Tn) �! L2(D),

given by i = r � iTn � e, where e is the iTn is the embedding map fromLemma A.3.16, and r the restriction map from L2(Tn) to L2(D). SinceiTn is compact, the so is i. Indeed, for any bounded sequence {un},the sequence {(iTn � e)(un))} has a convergent subsequece, and is thus{(r � iTn � e)(un))} has a convergent subsequence.

We will now give a compactness proof for the case p � 2. Considerthe embeddings

i0 : W1,p0 (D) �!W1,2

0 (D) ,! L2(D) �! L1(D).

Due to Lemma A.3.17 the embedding i0 is compact. We also have thecontinuous embedding from Theorem A.3.10:

i1 : W1,p0 (D) �! Lpn,1(D).

If we now use interpolation we obtain:

i : W1,p0 (D) �! Lq(D),

for 1 q pn,1 with i satsifying the inequality

ki(u)kq ki0(u)k1�q

1 ki1(u)kq

pn,1 Cqkuk1�q

1,p kukq

1,p = Cqkuk1,p.

From the first half of the inequality we see immediately that that anybounded sequence {un} yields a convergent subsequence in Lq(D),provided q < 1, i.e. q < pn,1. This proves that the embedding i iscompact for all 1 q < pn,1. The compactness of this embedding canalso prove the compactness of the embedding W1,p(D) ,! Lq(D) beusing an appropriate extension mapping.

As for the higher order Sobolev spaces Wm,p0 (D) we have now

proved the following result.

appendix 207

Theorem A.3.18 Let D ⇢ Rn be a bounded domain, then the embedding

i : Wm,p0 (D) ,! Lq(D), p � 2

is compact for all 1 q < pn,m.

Proof: We have that the embedding i : Wm,p0 (D) ! W1,p

0 (D) ,!L1(D) is compact. Now we use the interpolation again.

There are many ways we can extend these results. For example,by more advanced interpolation theory, the result of Lemma A.3.17

also yields compactness in Theorem A.3.18 for 1 p < 2. Alsointerpolation can be used to prove that the fractional space Wm,p

0 (D),m > 0, have the following compact embeddings:

Wm,p0 (D) ,!Wm�k,p

0 (D),

with 0 < k < m.

A.4 Partitions of unity

We start with introduce the notion partition of unity for smoothmanifolds. We should point out that this definition can be used forarbitrary topological spaces.

A.4.1 Definition. Let M be smooth m-manifold with atlas A = {(ji, Ui)}i2I .A partition of unity subordinate to A is a collection for smooth func-tions {li : M! R}i2I satisfying the following properties:(i) 0 li(p) 1 for all p 2 M and for all i 2 I;(ii) supp(li) ⇢ Ui;(iii) the set of supports {supp(li)}i2I is locally finite;(iv) Âi2I li(p) = 1 for all p 2 M.

Condition (iii) says that every p 2 M has a neighborhood U 3 p

such that only finitely many li’s are nonzero in U. As a consequenceof this the sum in Condition (iv) is always finite.

Theorem A.4.2 For any smooth m-dimensional manifold M with atlas Athere exists a partition of unity {li} subordinate to A.

In order to prove this theorem we start with a series of auxiliaryresults and notions.

A.4.3 Lemma. There exists a smooth function h : Rm ! R such that0 h 1 on Rm, and h|B1(0) ⌘ 1 and supp(h) ⇢ B2(0).

Proof: Define the function f1 : R ! R as follows

f1(t) =

8

<

:

e�1/t, t > 0,

0, t 0.


One can easily prove that f1 is a C•-function on R. If we set

f2(t) =f1(2� t)

f1(2� t) + f1(t� 1).

This function has the property that f2(t) ⌘ 1 for t 1, 0 < f2(t) < 1for 1 < t < 2, and f2(t) ⌘ 0 for t � 2. Moreover, f2 is smooth. Toconstruct f we simply write f (x) = f2(|x|) for x 2 Rm\{0}. Clearly,f is smooth on Rm\{0}, and since f |B1(0) ⌘ 1 it follows that f issmooth on Rm.

3 We use the convention krukp =⇣

R

Rn Âi�

�

∂u∂xi

�

�

pdx⌘

1p .An atlas A gives an open covering for M. The set Ui in the atlas

need not be compact, nor locally finite. We say that a covering U =

{Ui} of M is locally finite if every point p 2 M has a neighborhoodthat intersects only finitely Ui 2 U . If there exists another coveringV = {Vj} such that every Vj 2 V is contained in some Vj ⇢ Ui 2U , then V is called a refinement of U . A topological space X forwhich each open covering admits a locally finite refinement is calledparacompact.

A.4.6 Lemma. Any topological manifold M allows a countable, lo-cally finite covering by precompact open sets.

Proof: We start with a countable covering of open balls B = {Bi}.We now construct a covering U that satisfies

Figure A.1: The functions f1 and f2.

(i) all sets Ui 2 U are precompact open sets in M;(ii) Ui�1 ⇢ Ui, i > 1;(iii) Bi ⇢ Ui.We build the covering U from B using an inductive process. LetU1 = B1, and assume U1, · · · , Uk have been constructed satisfying(i)-(iii). Then

Uk ⇢ Bi1 [ · · · [ Bik ,

where Bi1 = B1. Now set

Uk+1 = Bi1 [ · · · [ Bik .

Choose ik large enough so that ik � k + 1 and thus Bk+1 ⇢ Uk+1.From the covering U we can now construct a locally finite covering

V by setting Vi = Ui\Ui�2, i > 1.

appendix 209

Next we seek a special locally finite refinement {Wj} = W of Vwhich has special properties with respect to coordinate charts of M;

(i) W is a countable and locally finite;

(ii) each Wj ⇢ W is in the domain of some smooth coordinate mapjj :j! Rm for M;

(iii) the collection Z = {Zj}, Zj = j

�1j (B1(0)) covers M.

The covering W is called regular.

A.4.8 Lemma. For any open covering U for a smooth manifold Mthere exists a regular refinement. In particular M is paracompact.

Proof: Let V be a countable, locally finite covering as describedin the previous lemma. Since V is locally finite we can find a neigh-borhood Wp for each p 2 M that intersects only finitely many setVj 2 V . We want to choose Wp in a smart way. First we replace Wp

by Wp \ {Vj : p 2 Vj}. Since p 2 Ui for some i we then replace Wp

by Wp \Vi. Finally, we replace Wp by a small coordinate ball Br(p) sothat Wp is in the domain of some coordinate map jp. This providescoordinate charts (Wp, jp). Now define Zp = j

�1p (B1(0)).

Figure A.2: A locally finite covering[left], and a refinement [right].

For every k, {Zp : p 2 Vk} is an open covering of Vk, whichhas a finite subcovering, say Z1

k , · · · , Zmkk . The sets Zi

k are sets of theform Zpi for some pi 2 Vk. The associated coordinate charts are(W1

k , j

1k), · · · , (Wmk

k , j

mkk ). Each Wi

k is obtained via the constructionabove for pi 2 Vk. Clearly, {Wi

k} is a countable open covering of Mthat refines U and which, by construction, satisfies (ii) and (iii) in thedefinition of regular refinement. It is clear from compactness that{Wi

k} is locally finite. Let us relabel the sets {Wik} and denote this

covering by W = {Wi}. As a consequence M is paracompact.

Proof of Theorem A.4.2: From Lemma A.4.8 there exists a regularrefinement W for A. By construction we have Zi = j

�1i (B1(0)), and

we define

bZi := j

�1i (B2(0)).


Using Lemma A.4.3 we now define the functions µ: M! R;

µi =

8

<

:

f2 � ji on Wi

0 on M\Vi.

These functions are smooth and supp(µi) ⇢Wi. The quotients

b

l =µj(p)

Âi µi(p),

are, due to the local finiteness of Wi and the fact that the denomina-tor is positive for each p 2 M, smooth functions on M. Moreover,0 b

lj 1, and Âj blj ⌘ 1.Since W is a refinement of A we can choose an index kj such that

Wj ⇢ Ukj . Therefore we can group together some of the function b

lj:

li = Âj : kj=i

b

lj,

which give us the desired partition functions with 0 li 1,Âi li ⌘ 1, and supp(li) ⇢ Ui.

Some interesting byproducts of the above theorem using partitionsof unity are. In all these case M is assumed to be an smooth m-dimensional manifold.

Theorem A.4.10 4 For any close subset A ⇢ M and any open set U � A,

Figure A.3: Constructing a nestedcovering U from a covering with ballsB [left], and a locally finite coveringobtained from the previous covering[right].

there exists a smooth function f : M! R such that(i) 0 f 1 on M;(ii) f�1(1) = A;(iii) supp( f ) ⇢ U.Such a function f is called a bump function for A supported in U.

By considering functions g = c(1� f ) � 0 we obtain smooth functionsfor which g�1(0) can be an arbitrary closed subset of M.

Theorem A.4.11 5 Let A ⇢ M be a closed subset, and f : A ! Rk4 See Lee, Prop. 2.26.

be a smooth mapping.6 Then for any open subset U ⇢ M containing A 5 See Lee, Prop. 2.26.

there exists a smooth mapping f † : M ! Rk such that f †|A = f , andsupp( f †) ⇢ U.

appendix 211

Figure A.4: The different stages ofconstructing the sets Wp colored withdifferent shades of grey.

A.5 Posets, lattices and Boolean algebras

A.5.1 Definition. A partially ordered set or poset (P,) is a set Pwith a binary relation , called a partial order, which satisfies thefollowing axioms:

(i) (reflexivity) p p, for all p 2 P,

(ii) (anti-symmetry) p q, and q p, then p = q,

(iii) (transitivity) if p q, and q r, then p r.

If two elements p, q are not related, or incomparable we write p k q.A relation < which satisfies only the transitivity property and p 6< pfor all p 2 P is called a strict partial order and is denoted by (P,<).

Posets and strict poset are equivalent. A partial order definesa strict order via: p < q if and only if p q and p 6= q. Similarly, astrict order < defines a partial order via: p q if and only if p < q,or p = q. The notations (P,) and (P,<) are used interchangeablyto emphasize order or strict order. When there is no ambiguityabout the partial order, we refer to partially ordered sets simply bydenoting the set P. A partial order is called a total order if any twoelements can be compared.

A chain is a subset which totally ordered. An anti-chain is anunordered set. A poset P has length `(P) = n if every chain in P

has at most n + 1 elements. A poset has width w(P) = n if everyanit-chain has at most n elements. Note that an infinite poset canhave finite length. An anti-chain has length ` = 1, and a chain haswidth w = 1.

A least element in a poset, if it exists, is denoted by ? and has theproperty that ? p for all p 2 P. A greatest element in a poset, if itexists, is denoted by > and has the property that > � p for all p 2 P.Least and greatest elements need not exist in a poset. A minimalelement p 2 P satisfies: p0 p, implies p0 = p. Similarly, a maximalelement p 2 P satisfies: p0 � p, implies p0 = p. Minimal and maximal


elements always exists and are the least and greatest element, whenunique.

A.5.2 Definition. Let P and P0 be posets. A mapping f : P ! P0

is called order preserving if f (p) f (q) for all p q and orderreversing if f (p) � f (q) for all p q.

The set of order-preserving maps, or order homomorphismsis denoted by Ord(P,P0) and the set of order reversing maps, ororder anti-homomorphisms is denoted by Ord⇤(P,P0). The set ofall posets as objects and order-preserving, or order-reversing mapsas morphisms form the small categories and ⇤ respectively (seeAppendix ?? for details on categories).

Consider two elements p, q 2 (P,). The infimum of p and q (if itexists) is the greatest lower bound with respect to the partial ordering and is denoted by inf(p, q). Similarly, if there exists a least upperbound with respect to the partial ordering it is called the supremumand is denoted by sup(p, q).

We can regard the infimum and supremum as binary operationson a poset P if for any two elements p, q 2 P the infimum andsupremum exist:

p _ q = sup(p, q), p ^ q = inf(p, q). (A.2)

A poset for which infimum and supremum exist for all pairs p, q 2 P

is called a lattice, and introduces an algebraic structure to P. The op-eration _ is called ‘vee’ or join and the operation ^ is called ‘wedge’or meet. A lattice can also be defined independently as an agebraicstructure.

A.5.3 Definition. A lattice (L,_,^) _^ is a set L with the binaryoperations _,^ : L⇥ L! L satisfying the following axioms:(i) (idempotent) a ^ a = a _ a = a for all a 2 L,(ii) (commutative) a ^ b = b ^ a, and a _ b = b _ a for all a, b 2 L,(iii) (associative) a ^ (b ^ c) = (a ^ b) ^ c and a _ (b _ c) = (a _ b) _ c

for all a, b, c 2 L,(iv) (absorption) a ^ (a _ b) = a _ (a ^ b) = a for all a, b 2 L.A distributive lattice satisfies the additional axiom(v) (distributive) a ^ (b _ c) = (a ^ b) _ (a ^ c) and a _ (b ^ c) =

(a _ b) ^ (a _ c) for all a, b, c 2 L.A lattice is bounded if there exist neutral elements 0 and 1 withproperty that(vi) 0 ^ a = 0, 0 _ a = a, for all a 2 L, and 1 ^ a = a, 1 _ a = 1, for all

a 2 L.A set K ⇢ L is a sublattice if a ^ b 2 K and a _ b 2 K for all a, b 2 K. Asublattice that contains 0 and 1 is called a (0, 1)-sublattice.

appendix 213

A.5.4 Remark. For the distributivity axiom (v) only one of the iden-tities is required, i.e. the first one implies the second one and viceversa.

When clear from context, we write L and omit the explicit refer-ence to the lattice operations. We should emphasize that posets andlattices need not be finite sets. Finite lattices are always bounded.

A.5.5 Remark. Observe that any lattice (L,_,^) has a natural partialorder given by

a b if a ^ b = a or, equivalently, a b if a _ b = b,

which makes (L,) a poset.

We emphasize that distributivity and boundedness are not auto-matic for posets with inf and sup. If a lattice satisfies the axiom (v0)(modular) a � c =) a ^ (b _ c) = (a ^ b) _ c, it is called a modular.

A.5.6 Proposition. A poset (P,), with _ and ^ as defined in (A.2) isan algebraic lattice, i.e. the binary operation _ and ^ satisfy (i)-(iv) ofDefinition A.5.3.

A.5.7 Definition. Let (L,^,_) and (L0,^0,_0) be lattices. A functionh : L! L0 is a lattice homomorphism if

h(a ^ b) = h(a) ^0 h(b), h(a _ b) = h(a) _0 h(b),

a lattice anti-homomorphism if

h(a ^ b) = h(a) _0 h(b), h(a _ b) = h(a) ^0 h(b).

The set of lattice homomorphisms is denoted by Hom(L, L0) and theset of lattice anti-homomorphisms is denoted by Hom⇤(L, L0). Lat-tice homomorphisms for which h(0) = 0, and h(1) = 1 (h(0) =

1, and h(1) = 0 for anti-homorphisms) are called (0, 1)-(anti-)homomorphisms. These sets are denoted by Hom0,1(L, L0) andHom⇤0,1(L, L0) respectively.

With lattices as objects and with lattice (anti-)homomorphismsas morphisms we obtain the small categories and ⇤ respectively. Ifdistributivity or modularity holds the categories are denoted by Dand M respectively.

Observe that (Hom(L, L0),) and (Hom⇤(L, L0),) are posets withthe order defined by h k if and only if h(a) k(a), for all a 2 L.An anti-homomorphism h : L ! L0 can be regarded as a latticehomomorphism by redefining the operations ^ = _0 and _ = ^0 onL0. Observe that the new order on L0 is the dual of the original order.

The algebraic structure of lattices does not accommodate inversion.In some of the structures we encounter some inversion relations


are satisfied. In this context of lattices this is descibed by Booleanalgebras.

A.5.8 Definition. A Boolean algebra is a bounded distributive lattice(L,_,^,c ) with the additional complementation relation a 7! ac 2 L,which satisfies the axiom:

(vii) a ^ ac = 0 and a _ ac = 1, for all a 2 L.

A Heyting algebra is a bounded distributive lattice (L,_,^,c ) withthe additional pseudo-complementation relation a 7! ac 2 L, whichsatisfies the axiom:

(vii0) a ^ ac = 0, for all a 2 L.

A.5.9 Remark. To visualize partial orders and lattices in small exam-ples we adopt the following conventions. Let (P,) be a poset. Forp, q 2 P we say q covers p if p < q and p r < q implies that p = r.This relation is denoted by p ⌃ q. Observe that if P is finite, then <

determines ⌃ and vice versa.As mentioned in Example 5.4.10, a partial order relation can

be represented by a directed graph, which is often useful for visu-alization. It is a common convention to depict these graphs in aHasse diagram. This diagram is a planar representation which isconstructed as follows. In a diagram of (P,<) we associate a circleto each element of P, and an edge is drawn between the circles forp, q 2 P if p ⌃ q. The circles are placed so that if p ⌃ q, then the circlefor p is vertically lower than the circle for q, and every edge intersectsno circles other than its endpoints. We note without proof that such adiagram is always possible.

A.6 Homology and cohomology

In this section we will give an elementary exposition of the basichomology and cohomology theories that are used in this text. Thistreatment is self-contained and at elementary level. For more detailedaccounts of homology and cohomology theory we refer to varioustexts in algebraic topology. Before defining the homology introducethe concept of simplicial complex.

Simplicial homology

Simplicial homology is the simplest homology theory from a con-ceptual point view and is used for simplicial complexes in Rn. Acollection of q + 1 points, or vertices {v1, · · · , vq+1} in Rn is said to bean independent set of points if the vectors v2 � v1, · · · , vq+1 � v1 form

appendix 215

a linearly independent system. Note that this holds relative to anychosen vertex vi.

A.6.1 Definition. A q-simplex (q n) is a (convex) subset sq of Rn

generated by q + 1 independent vertices v1, · · · , vq+1 in the followingway:

sq :=n

x 2 Rn �

� x = Âi

livi, Âi

li = 1, li � 0o

.

The notation for a specific q-simplex is sk = [v1 · · · vk+1]. By choosingan ordering of the vertices one obtains oriented k-simplices., whichare devided into two orientation classes. For a given orientated q-simplex sq permuting the vertices by a odd permutation yields anoriented q-simplex of opposite orientation which we denote by �sq.

The boundary of a simplex is given by its faces. The faces of sq arethe sets {x 2 sq | li = 0

which are (q� 1)-simplices. The ith faceis give by [v1 · · · bvi · · · vq+1] where the notation bvi indicates removingthe vertex vi. This brings us to considering collections of simplices.

A.6.2 Definition. A finite collection of simplices K is called q-dimensionalsimplicial complex if;

(i) s` 2 K, ` q then all its faces are contained in K;

(ii) two simplices s` and s`0 can only intersect along common faces.

The dimension q of a simplicial complex K is determined by themaximal dimension of the simplices in K.

The morphisms between simplicial complexes K and K0 are calledsimplicial mappings f : K ! K0 and are defined by the property thateach simplex in K is mapped onto a simplex in K0 of the same dimen-sion or lower. This makes simplicial complexes with the simplicialmappings a category.

For oriented simplicial complexes we can build free groups overthe complex as follows. Let K be a simplicial complex of dimension q,then define

Ck(K; Z) :=n

sk = Â aisik | s

ik 2 K, ai 2 Z

o

, 8k q,

which are finite linear combinations of the k-simplices in K. Anelement sk 2 Ck(K) is called a k-chain in K. A k-chain sk is carriedby a subcomplex L ⇢ K if sk = Â ais

ik with s

ik 2 L. We set Ck = 0

for all k < 0 and k > q. By allowing coefficient in Z we consideroriented simplices and an oriented complex K. A simplicial mappingf : K ! K0 induces a homomorphism f# : Ck(K)! Ck(K0) as follows:

f#([v1 · · · vk+1]) =

8

<

:

[ f (v1) · · · f (vk+1)] if f (vi) 6= f (vj) 8i 6= j,

0 otherwise


and f#(�sk) = � f#(sk).For an oriented simplex we can define a operation that assign to a

simplex its boundary. Let sq = [v1 · · · vq+1] be an oriented q-simplex,then

∂q[v1 · · · vq+1] :=q+1

Âi=1

(�1)i[v1 · · · bvi · · · vq+1],

which is called the q-boundary operator.

A.6.3 Example. Let s2 = [0 e1 e2] ⇢ R2, where ei are the standardbasis vectors in R2. Then ∂2s2 = [e1 e2]� [0 e2] + [0 e1] which is a 1-complex which represents the boundary of s2 with counter clockwiseorientation.

Since the boundary operator is defined for all simplices the defin-tion extends to all elements in q-simplicial complex K, and

∂k : Ck(K; Z)! Ck�1(K; Z), 8k q.

The following lemma about the boundary operator is completelyclear from an intuitive point of view. If we apply the boundaryoperator twice, i.e. the boundary of an oriented boundary, then theanswer is the empty set.

A.6.4 Lemma. Let ∂k = 0 for all k 0 and k > q. Then ∂k�1 � ∂k = 0.

A.6.5 Exercise. Prove Lemma A.6.4.

The groups Ck(K) are called chain groups and the combination{Ck(K), ∂k}k2Z is called a chain complex (due to the property of ∂kgiven in Lemma A.6.4). For a chain complex in general we can defineits homology. Let

Zk(K; Z) = {sk 2 Ck(K) | ∂ksk = 0} = ker(∂k),

Bk(K; Z) = {sk 2 Ck(K) | 9esk+1 2 Ck+1(K) 3 sk = ∂kesk+1} = im(∂k+1).

The elements in Zk are called k-cycles and the elements in Bk arecalled k-boundaries. From Lemma A.6.4 it follows that Bk(K) ⇢Zk(K) as linear subgroups.

A.6.6 Definition. The kth simplicial homology group Hk(K; Z) ofa q-simplicial complex K is defined as the quotient Hk(K; Z) :=Zk(K; Z)/Bk(K; Z). The homology classes are equivalence

classes under the equivalence relation: sk ⇠ s0k, sk, s0k 2 Zk(K) ifsk � s0k 2 Bk(K). Therefore an homology class [sk] can be representedby cycles of the form sk + ∂k+1esk+1, esk+1 2 Ck+1(K). Computingthe homology of a q-simplicial complex becomes a linear algebraproblem with coefficients in Z. The addition in the abelian groupsHk(K) is given by: [sk] + [s0k] = [sk + s0k] by the above considerations.

appendix 217

The above construction can also be carried out over different groupsF, e.g. F = R, F = Q, or F = Zp. Simplicial homology Hk(K; F) maybe different for different groups. As a matter of fact Hk(K, Z) maycontain more detailed information than Hk(K; Q).

A.6.7 Exercise. Let K be a 2-dimensional complex consisting of s2 = [0 e1 e2] ⇢R2 and all its lower dimensional faces. Compute the homology groupsHk(K; Z), k = 0, 1, 2.

A.6.8 Exercise. Let K be a q-simplicial complex. Show that dim H0(K; Z) isequal to the number of connected component of K.

A.6.9 Exercise. Figure below is an example of a simplicial Möbius stripM. Compute the homology H⇤(M; Z) and H⇤(M; Q). Do these homologycalculations give the same answer?

A.6.10 Exercise. Let K = {s0}, a simplicial complex consisting of a single0-simplex. Show that H0(K; Z) ⇠= Z.

Simplicial mapping induce homomorphisms in homology:

A.6.11 Lemma. Let f : K ! K0 be a simplicial mapping then f#∂k =

∂k f# for all k and f# induces a homomorphism f⇤ : Hk(K; Z) !Hk(K0; Z).

Proof: For a single simplex we need to verify the relation f#(∂k[v1 · · · vk] =

∂k f#([v1 · · · vk]). If f#([v1 · · · vk]) is a simplex of dimension k, then therelation follows from the defintion.

A.6.12 Lemma. For f⇤ the following properties hold:(i) if id : K ! K is the identity mapping, then id⇤ : Hk(K) ! Hk(K)

is the identity homomorphism;(ii) Let K, K0, K00 be simplicial complexes. If f : K ! K0 and g : K0 !

K00 are simplicial mappings, then (g � f )⇤ = g⇤ � f⇤,where f⇤ : Hk(K)! Hk(K0) and g⇤ : Hk(K0)! Hk(K00).

A.6.13 Exercise. Prove Lemma A.6.12.

A second important ingredient of homology theory is the relativehomology of a pair of simplicial complexes (K, L) with L ⇢ K,where L is a subcomplex of K. The groups Ck(L; Z) are subgroups ofCk(K; Z). Define

Ck(K, L; Z) := Ck(K; Z)/Ck(L; Z),

and equivalence classes are given by sk + Ck(L; Z), sk 2 Ck(K\L; Z)

(a chain consisting of simplices that are not in L). The boundaryoperator ∂k : Ck(K, L) ! Ck�1(K, L) is defined by ∂k(sk + Ck(L)) =

∂ksk + Ck�1(L). From the properties of ∂k it follows that b∂k�1 � b∂ksk =


∂k�1 � ∂ksk + Ck�2(L) = Ck�2(L), which is zero in Ck�2(K, L) andtherefore b∂k�1 � b∂k = 0.

As before set Zk(K, L; Z) = ker(b∂k) and Bk(K, L; Z) = im(b∂k+1)

and are called relative k-cycles and relative k-boundaries respectively.It follows that elements in Zk(K, L) are of the form sk + Ck(L) with∂ksk carried by L and elements in Bk(K, L) are of the form sk =

∂k+1esk+1 + Ck(L) with sk � ∂k+1esk+1 carried by L. Relative homologyclasses [sk] are represented by relative k-cycles of the form sk +

∂k+1esk+1 + Ck(L).

A.6.14 Definition. The kth relative simplicial homology groupHk(K, L; Z) of a simplicial pair (K, L) is defined as the quotientHk(K, L; Z) := Zk(K, L; Z)/Bk(K, L; Z).

Consider the short exact sequence of maps:

0 �! L i�! Kj�! (K, L) �! 0,

where i is the embedding and j(K) = (K,?). It holds that ker(i) = 0and ker(j) = im(i) = L and thus the sequence is exact.

A.6.15 Lemma. The above short exact sequence yields a long exactsequence in homology:

· · · ∂⇤�! Hk(L) i⇤�! Hk(K)j⇤�! Hk(K, L) ∂⇤�! Hk�1(L) i⇤�! · · · ,

where i⇤, j⇤ and ∂⇤ are the induced homomorphisms on homology.

Proof: The homomorphism i⇤ is induced by the simplicial mappingi as described above. Define j# : Ck(K) ! Ck(K, L) by j#(sk) =

sk + Ck(L), where sk 2 Ck(K). Now j#(∂ksk) = ∂ksk + Ck�1(L) andb

∂k(j#sk) = b

∂k(sk + Ck�1(L)) = ∂ksk. Therefore, j#∂k = b

∂k j#. Let[k] 2 Hk(K), then b

∂k j#([sk]) = j#∂k([sk]) = 0 and thus j#([sk]) 2Zk(K, L). The latter defines the homology class [j#([sk])] 2 Hk(K, L)and thereby the mapping j⇤.

Under construction.

Simplicial cohomology

Definition of De Rham cohomology

In the previous chapters we introduced and integrated m-formsover manifolds M. We recall that k-form w 2 Gk(M) is closed ifdw = 0, and a k-form w 2 Gk(M) is exact if there exists a (k� 1)-forms 2 Gk�1(M) such that w = ds. Since d2 = 0, exact forms are closed.We define

Zk(M) =�

w 2 Gk(M) : dw = 0

= Ker(d),

Bk(M) =�

w 2 Gk(M) : 9 s 2 Gk�1(M) 3 w = ds

= Im(d),

appendix 219

and in particularBk(M) ⇢ Zk(M).

The sets Zk and Bk are real vector spaces, with Bk a vector subspaceof Zk. This leads to the following definition.

A.6.16 Definition. Let M be a smooth m-dimensional manifold then the de

Rham cohomology groups are defined as

HkdR(M) := Zk(M)/Bk(M), k = 0, · · ·m, (A.3)

where B0(M) := 0. It is immediate from this

definition that Z0(M) are smooth functions on M that are constanton each connected component of M. Therefore, when M is connected,then H0

dR(M) ⇠= R. Since Gk(M) = {0}, for k > m = dim M, we havethat Hk

dR(M) = 0 for all k > m. For k < 0, we set HkdR(M) = 0.

A.6.17 Remark. The de Rham groups defined above are in fact realvector spaces, and thus groups under addition in particular. Thereason we refer to de Rham cohomology groups instead of de Rhamvector spaces is because (co)homology theories produce abeliangroups.

An equivalence class [w] 2 HkdR(M) is called a cohomology class,

and two form w, w

0 2 Zk(M) are cohomologous if [w] = [w0]. Thismeans in particular that w and w

0 differ by an exact form, i.e.

w

0 = w + ds.

Now let us consider a smooth mapping f : N ! M, then we havethat the pullback f ⇤ acts as follows: f ⇤ : Gk(M) ! Gk(N). FromTheorem ?? it follows that d � f ⇤ = f ⇤ � d and therefore f ⇤ descends tohomomorphism in cohomology. This can be seen as follows:

d f ⇤w = f ⇤dw = 0, and f ⇤ds = d( f ⇤s),

and therefore the closed forms Zk(M) get mapped to Zk(N), and theexact form Bk(M) get mapped to Bk(N). Now define

f ⇤[w] = [ f ⇤w],

which is well-defined by

f ⇤w0 = f ⇤w + f ⇤ds = f ⇤w + d( f ⇤s)

which proves that [ f ⇤w0] = [ f ⇤w], whenever [w0] = [w]. Summariz-ing, f ⇤ maps cohomology classes in Hk

dR(M) to classes in HkdR(N):

f ⇤ : HkdR(M)! Hk

dR(N),


Theorem A.6.18 Let f : N ! M, and g : M! K, then

g⇤ � f ⇤ = ( f � g)⇤ : HkdR(K)! Hk

dR(N),

Moreover, id⇤ is the identity map on cohomology.

Proof: Since g⇤ � f ⇤ = ( f � g)⇤ the proof follows immediately.

As a direct consequence of this theorem we obtain the invarianceof de Rham cohomology under diffeomorphisms.

Theorem A.6.19 If f : N ! M is a diffeomorphism, then HkdR(M) ⇠=

HkdR(N).

Proof: We have that id = f � f�1 = f�1 � f , and by the previoustheorem

id⇤ = f ⇤ � ( f�1)⇤ = ( f�1)⇤ � f ⇤,

and thus f ⇤ is an isomorphism.

Homotopy invariance of cohomology

We will prove now that the de Rham cohomology of a smooth man-ifold M is even invariant under homeomorphisms. As a matter offact we prove that the de Rham cohomology is invariant under homo-topies of manifolds.

A.6.20 Definition. Two smooth mappings f , g : N ! M are said to behomotopic if there exists a continuous map H : N ⇥ [0, 1]! M such that

H(p, 0) = f (p)

H(p, 1) = g(p),

for all p 2 N. Such a mapping is called a homotopy from/between fto/and g. If in addition H is smooth then f and g are said to be smoothly

homotopic, and H is called a smooth homotopy.

Using the notion of smooth homotopies we will prove the follow-ing crucial property of cohomology:

Theorem A.6.21 Let f , g : N ! M be two smoothly homotopic maps.Then for k � 0 it holds for f ⇤, g⇤ : Hk

dR(M)! HkdR(N), that

f ⇤ = g⇤.

appendix 221

A.6.22 Remark. It can be proved in fact that the above results holdsfor two homotopic (smooth) maps f and g. This is achieved byconstructing a smooth homotopy from a homotopy between maps.

Proof of Theorem A.6.21: A map h : Gk(M) ! Gk�1(N) is called ahomotopy map between f ⇤ and g⇤ if

dh(w) + h(dw) = g⇤w� f ⇤w, w 2 Gk(M). (A.4)

Now consider the embedding it : N ! N ⇥ I, and the trivialhomotopy between i0 and i1 (just the identity map). Let w 2 Gk(N⇥ I),and define the mapping

h(w) =Z 1

0i

∂

∂twdt,

which is a map from Gk(N ⇥ I) ! Gk�1(N). Choose coordinates sothat either

w = wI(x, t)dxI , or w = wI0(x, t)dt ^ dx I0.

In the first case we have that i∂

∂tw = 0 and therefore dh(w) = 0. On

the other hand

h(dw) = h

⇣

∂wI∂t

dt ^ dxI +∂wI∂xi

dxi ^ dxI⌘

=⇣

Z 1

0

∂wI∂t

dt⌘

dxI

=�

wI(x, 1)�wI(x, 0)�

dxI = i⇤1w� i⇤0w,

which prove (A.4) for i⇤0 and i⇤1, i.e.

dh(w) + h(dw) = i⇤1w� i⇤0w.

In the second case we have

h(dw) = h

⇣

∂wI∂xi

dxi ^ dt ^ dxI0⌘

=Z 1

0

∂wI∂dxi

i∂

∂xi

�

dxi ^ dt ^ dxI0�dt

= �⇣

Z 1

0

∂wI∂xi

dt⌘

dxi ^ dxI0 .

On the other hand

dh(w) = d⇣⇣

Z 1

0wI0(x, t)dt

⌘

dxI0⌘

=∂

∂xi

⇣

Z 1

0wI0(x, t)dt

⌘

dxi ^ dxI0

=⇣

Z 1

0

∂wI∂xi

dt⌘

dxi ^ dxI0

= �h(dw).


This gives the relation that

dh(w) + h(dw) = 0,

and since i⇤1w = i⇤0w = 0 in this case, this then completes theargument in both cases, and h as defined above is a homotopy mapbetween i⇤0 and i⇤1.

By assumption we have a smooth homotopy H : N ⇥ [0, 1] ! Mbewteen f and g, with f = H � i0, and g = H � i1. Consider thecomposition e

h = h � H⇤. Using the relations above we obtain

e

h(dw) + deh(w) = h(H⇤dw) + dh(H⇤w)

= h(d(H⇤w)) + dh(H⇤w)

= i⇤1 H⇤w� i⇤0 H⇤w

= (H � i1)⇤w� (H � i0)⇤w

= g⇤w� f ⇤w.

If we assume that w is closed then

g⇤w� f ⇤w = dh(H⇤w),

and thus

0 = [dh(H⇤w)] = [g⇤w� f ⇤w] = g⇤[w]� f ⇤[w],

which proves the theorem.

A.6.23 Remark. Using the same ideas as for the Whitney embeddingtheorem one can prove, using approximation by smooth maps, thatTheorem A.6.21 holds true for continuous homotopies betweensmooth maps.

A.6.24 Definition. Two manifolds N and M are said to be homotopy

equivalent, if there exist smooth maps f : N ! M, and g : M ! N suchthat

g � f ⇠= idN , f � g ⇠= idM (homotopic maps).

We write N ⇠ M., The maps f and g are homotopy equivalences are eachother homotopy inverses . If the homotopies involved are smooth we saythat N and M smoothly homotopy equivalent.

A.6.25 Example. Let N = S1, the standard circle in R2, and M =

R2\{(0, 0)}. We have that N ⇠ M by considering the maps

f = i : S1 ,! R2\{(0, 0)}, g = id/| · |.

Clearly, (g � f )(p) = p, and ( f � g)(p) = p/|p|, and the latter ishomotopic to the identity via H(p, t) = tp + (1� t)p/|p|.

appendix 223

Theorem A.6.26 Let N and M be smoothly homotopically equivalentmanifold, N ⇠ M, then

HkdR(N) ⇠= Hk

dR(M),

and the homotopy equivalences f g between N and M, and M and N re-spectively are isomorphisms.

As before this theorem remains valid for continuous homotopyequivalences of manifolds.

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TOPOLOGICAL METHODS FOR NONLINEAR DIFFER- ENTIAL …vdvorst/lectures-2013.pdf · A.5 Posets, lattices and Boolean algebras 211 A.6 Homology and cohomology 214 B Solutions 225 Bibliography