Patch Spaces: A Geometric Representation for Poincare ...v1ranick/papers/patches.pdf · Patch spaces were invented by the author for use in his study "Combi- natorial symmetries of

Patch Spaces A Geometric Representation for Poincare Spaces

Lowell Jones

The Annals of Mathematics 2nd Ser Vol 97 No 2 (Mar 1973) pp 306-343

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Patch spaces A geometric representation

for Poincare spaces

TABLEOF CONTENTS

Section Page Preface 306

1 Preliminaries 307 2 Elementary properties 310

sect 3 Surgery on patch spaces 311 sect 4 Changing patch structures 323 sect 5 Obstruction to transversality 329 sect6 The existence and uniqueness of patch space structures on

Poincar6 duality spaces 334 sect 7 Applications 338

(70) Surgery composition formula (71) Surgery product theorem (72) Computing cobordism classes of Poincar6 spaces (73) Representing homology classes by Poincari spaces (74) Characteristic variety theorem and homotopy type of BSF (711) Patch space structures for spaces which are Poincari mod

a set of primes

Preface

Here is the theme of this paper Given any geometric construction C defined for differentiable manifolds there is a single functorial surgery obstruction S which measures the obstruction to extending C to Poincar6 duality spaces

If C = surgery techniques then S = 0 ie surgery techniques extend to Poincar6 spaces (see 5 3) This was first announced by L Jones with J Paulson 1121 Significant results in this direction had previously been published by N Levitt 1161

If C = transversality then S is made explicit in 5 5 below As an application (see 72) let C denote the classical reduction of the differ-

entiable bordism group (oriented or unoriented) to a homotopy group Analy- zing S (which is just a transversality obstruction) one sees that up to an extension or quotient by one of Z Z there are isomorphisms QfP zr j (TBSF) Lj(l) QT E r j (TBF) for all j 2 S Here Q Qz are the oriented unoriented bordism groups of Poincar6 duality spaces TBSF T B F are the associ-

307 PATCHSPACES

ated Thom spectra ~(l)is the surgery group for simply connected surgery This result was first announced by L Jones with J Paulson [12] and for the oriented case by N Levitt [15] F Quinn has recently made an independent study of surgery and transversality in the Poincark category

[19] from which this result follows Our approach to the problem of extending geometric constructions is to

first replace a Poincar6 space by a more geometric object called a patch space In the differentiable category a compact closed manifold can be represented as a collection of closed m-discs DI i E I together with a set of gluing diffeomorphisms gij D I ( j ) -+ Dy(i) defined on codimension zero submanifolds DF(j) c D I and subject to the relations gij = g gij0 gjk = gik A patch space is the analogous object in the homotopy category a collection of compact differentiable manifolds Mi i E I (all of the same dimension) together with a collection of gluing homotopy equivalences gij Mi($ -+ Mj(i) defined on codimension zero submanifolds Mi($ c Mi and subject to the relations g z g~ glj0 gjk z g ( ( 4 = -gt 7 means homotopic to) Further transversality conditions are also stipulated (91) Now for a geometric construction C S is the (global) obstruction to gluing the C(Mi) i euro I together by transversality and up to homotopy equivalence under the maps gij M(j) +

Mj (i) Patch spaces were invented by the author for use in his study Combi-

natorial symmetries of the m-disc There they were needed in showing that a certain functorial surgery problem could be solved I am indebted to Wu- Chung Hsiang for several stimulating conversations when this work had only just gotten under way Also arguments in [lo] were helpful in formulating exactly what a patch space should be To G Cooke I owe the results of Subsection 73 below

Let M be a differentiable closed manifold Ni i E Iis a set of differentiable submanifolds of M indexed by the finite set I which are in transverse position This means that for I = J U J with J n J= Nj inter-sects nNi transversely for each jE J In this case there is for each subset Jc I an open neighborhood UJ of nN in M and a diffeomorphism C U zJ onto a linear bundle z over nlN z for fiber C-+ has R k ~ maps nNi onto the zero-section of T and further satisfies

(11) For each Jc I z splits into a Whitney sum of linear bundles

T E (7 i V J ) where Nj denotes (niNi)

308 LOWELL JONES

Property 11 is a codimension property of transverse submanifolds in a differentiable manifold I want to formalize i t so tha t i t makes sense to talk of transverse subcomplexes in any C W complex This is the purpose of the following

Definition 12 Let X be a locally finite countable CW complex A set of real codimension subspace of X i n transverse position consists of a collection Yi E I of subcomplexes of X indexed by the finite set I so tha t each intersection niYi-hereafter denoted by Y- is also a subcomplex of X for every J c I a collection U Jc I of open neighborhoods in X for the corresponding spaces Y J c I and homeomorphisms C U -z onto the real linear bundles z over Y having Rkas fiber which send Y cU onto the zero section of z The homeomorphisms C U --z must satisfy 11when Ni c I is replaced by Yi E I

An augmentation of the real codimension subspace Yii c I of the C W complex X consists of a set YiE I of real codimension subspaces of X and an imbedding g I-- I of index sets satisfying Y = Y (C U --+

z) = (C U -z) for all J c I

Definition 13 Let X X denote CW complexes having real codimension subspaces in transverse position Yic I Y i E I respectively A mapping from X to Xin transverse position to the real codimension subspaces Yi iE I of X consists of a continuous mapping f X -X an augmentation Yic I of the real codimension subspaces Y iE I of XIwith a given embedding g I -I of index sets and a set isomorphism r ( I - g(I))-I satisfying f -(Y)= Y for all Jc ( I - g(I)) It is required tha t in small enough neighborhoods of the respective zero-sections the composition (from left to right)

is a well-defined linear bundle map which maps a neighborhood of the origin of each fiber in z isomorphically onto a neighborhood of the origin of the corresponding fiber in z

Definition 14 A patch space consists of a finite C W complex P and a set Pii E I of real codimension subspaces of P in transverse position which satisfy

14a If i c I then zi = Pi x (-1 1) and P splits P into a union of sets P = PF u P so tha t

309 PATCH SPACES

P n C(P x (-1 I)) = C(P x (-1 01) It is required that every point in P is contained in the interior of a t least

one of the subspaces Pi+ i c I

14b There exists for each i c I a compact differentiable manifold Mi and a homotopy equivalence hi (Mi dMi) - (P+ Pi) which is in transverse position to the real codimension sets Pif n Pjj E Iof PZ Furthermore the composition (going from left to right)

hhl(~+ n (n P)))) cM~-tP 2P+n ( n j BP))

is a homotopy equivalence for all Jc I-here Pj) denotes any of the three possibilities P j P i Py

The subspaces P+of P in 15a are called the patches of P The mappings hi Mi 4P are called diferentiable charts for the patch space P The definition of a patch space does not give a particular set of differentiable charts to P i t only states tha t differentiable charts do exist

Patch spaces with boundaries can also be defined in this case the index d is added to I of 14 and either d P or P denotes the boundary

The idea tha t guides my further definitions of additional structures on patch spaces (eg subobjects transversality of a map) can be stated as follows the structure must be given in real codimension terms and i t must be copied up to homotopy equivalence in a set of differentiable charts covering the given patch space For example

Definition 15 Let P be a patch space with patches P+i E IA set of patch subspaces in P consists of an augmentation Pii c I of the real codimension subspaces Pii c Ito P satisfying

15a There exists a set of differentiable charts hi Mi -P+i E Ifor P so tha t each hi is in transverse position to the real codimension subspaces Pif n Pii c I of P Furthermore for every Jc I and

i c I hi h(nj Pi+n P)))-(n PZtn P)))

is a homotopy equivalence Here Pj) indicates any one of the two possibilities

Pj Ph -- P - Pj

Elements of the set Pij E I- Iare called patch subspaces of the patch space P

Clearly for any Jc I P inherits a patch structure from P with patches Pi+n P i E I - Jand patch subspaces Pi n P iE I- ( I u J))

Definition 16 Let X be a C W complex containing the real codimension

310 LOWELL JONES

subspaces Yic ILet P be a patch space A mapping from P to X in patch transverse position to Yi i E I) consists of a map f P -+ X which is in transverse position to the subspaces Yi i E Iof X (as in Definition 13) Furthermore the real codimension subspaces fA1(Y)ic Iof P are required to be patch subspaces of the patch space P

2 Elementary properties

Let (P aP) be a patch space with boundary having patches P+i E I The codimension one subspaces Pi i E I u a dice P into 2-3111smaller cubes of the form nisrPi) or ( n i s r Pik) n Pa where Pi) denotes any one of the three possibilities Pi Pf P(- P - Pi+) Of course some of these cubes may be empty as is the case for niPi- since every point of P is con-tained in the interior of some patch PC Each cube niEIP+is uniquely determined from a decomposition of the index set I into a union I=J U J+u J-of pairwise disjoint sets J J J- by the correspondence

(J J + J-) -( n i s J Pi) n (niEJ+pt) n ( n i e J - p ~ )

I will use the symbol A to denote the cube corresponding to (J J + J-) with lJl = k For A (J J f J-) set- (CjEJ+Pj n A) U (Cj-pj n A)(Pa n A) For A+ = A n Pa set

ah+ = (Cj+ Pj n A+) u (Cis-Pj n A+) Then ah is the topological boundary of A in PJand aA+ is the topological boundary of A in PJ n Pa Any subspace K of P which is the union of a set of connected components of cubes is called a cubical subcomplez of P From this definition and 14 above i t is directly verified tha t each aA is a cubical subcomplex of P that each component of a cubical subcomplex is again a cubical subcomplex tha t the intersection of cubical subcomplexes is again a cubical subcomplex

LEMMA21 For every cube A i n P (A aA) i s homotopically equivalent to a diferentiable manifold with boundary

The formal dimension of the patch space P is the topological dimension of the domain Mi of any differentiable chart hi Mi -P for P If P is con-nected then the formal dimension is easily seen to be well-defined A con-nected patch space P is orientable if H(P 2 ) s 2 for p equal the formal dimension of P

The existence of differentiable charts h Mi -P+ for a patch space P

311 PATCH SPACES

imposes something analogous to a local Euclidean structure on P For this reason i t should be suspected that P satisfies Poincark duality This is indeed the case The following theorem is proven by using the piecing together arguments in [31]

THEOREM22 An orientable patch space (PdP) i s a n orientable Poincare d u a l i t y pair

R e m a r k If P is not an orientable patch space i t can be shown exactly as in the oriented case that P is a Poincark duality space The fundamental class for P will lie in a homology group H(P 2)with twisted coefficients (see [31])

3 Surgery on patch spaces

This section shall be concerned with extending the techniques of surgery to patch spaces It is presumed that the reader is familiar with [13] and the first six sections of [32] The strategy followed is one of pointing out where the program of [32] runs into difficulty for patch spaces and discussing how these difficulties are overcome Naturally the notation of [32] will be used wherever possible

Recall that a B F bundle T -X is called a Spivak fibration for the Poincar6 duality pair (X Y) if the top homology class of the quotient of Thom spaces T(~)lT(rl ) is spherical Fundamental results are a Spivak fibration exists for any Poincare duality pair (X Y) any two Spivak fibrations for (X Y) are BF-equivalent For these and other properties of Spivak fibrations the reader should consult [22] for the simply-connected case [31] and [3] for the non-simply-connected case

According to Theorem 22 any oriented patch space (P dP) is a Poincar6 duality pair so (P dP) has a Spivak fibration A surgery problem having the oriented patch space (P aP) as domain is a diagram

f 7 - -z

where f is a degree one map into the oriented Poincark duality pair (X Y) which restricts to a homotopy equivalence f I 3 P 4 Y and f is a BF- bundle mapping from the Spivak fibration r -P to the BF-bundle I---X There are also surgery patch cobordisms

THEOREM34 L e t f P --X be a s in 32 above p = dim (P) If p 2 11 a n d p f 14 t h e n there i s a well-de$ned obstruction o(f) E Li(n(X)) which

312 LOWELL JONES

v a n i s h e s i f a n d o n l y i f t he s u r g e r y p rob l em 31 i s n u l l cobordant

The proof of Theorem 34 is lengthy however the idea behind the proof is simple enough Here i t is in essence Suppose surgery has been done on f (P dP) -+ (X Y) to make f k-connected Let giSik 4P i = 1 2 I represent a set of generators for K(P) Clearly the g(r) 4Skare trivial BF-bundles This property can be used to engulf all the g Sk-+ P in a patch P+cP ie an additional patch Pf can be added to P so that each g factors up to homotopy as

gisk-P

After replacing Pf by the domain of a well-selected differentiable chart h Ma-P+ the problem of doing surgery in P has been reduced to doing surgery in the differentiable part Ma of P Likewise surgery obstructions can be defined in a differentiable part of P

I would first like to take up the engulfing problem alluded to above Let P be a patch space with boundary having patches PC iE I) Let Pf be a closed subspace of P so that its topological boundary Pa in P is a codimen-sion one-patch subspace of P If there is a differentiable chart haM -PI then Pf will be considered as a patch a u g m e n t i n g the patch structure of P

D e f i n i t i o n 35 A patch e n g u l f i n g of the map g N- P consists of a patch isomorphism h P -P a patch Pt augmenting P and a factoriza-tion of h 0 g N-P up to homotopy by

Here i is the inclusion In practice no distinction is made between P P so Pf will be considered as an augmentation of P and the previous factori-zation becomes

In 317 below the requirements are improved to dim (P)2 6 only

PATCH SPACES 313

A necessary condition for g N - P to be engulfed in a patch is that g(r)have a BO reduction where r is a Spivak fibration for (PdP) In some cases this necessary condition is sufficient to engulf g N-- P in a patch a s the following lemma shows

LEMMA36 Let r -P be a Spivak jibration for (PdP) N a jifinite simplicia1 complex satisfying dim ( P )- dim ( N )2 6 I f dim ( P )= 2k and dim ( N )5 k or i f dim ( P )= 2k + 1 and dim ( N )5 k + 1 then g N -P can be engulfed i n a patch if and only if g(r)has a BO reduction

Proof If dP 0 then g N -P can be pushed away from dP so tha t dP never enters into the discussion below Pu t g N -P in transverse posi-tion to the real codimension subspaces Pii E I of P and set Ni r g-(Pi) Nii e I is then a set of real codimension subspaces of N in transverse position (as in (12))and g N - P is a homomorphism of spaces with real codimension structures Note that the hypothesis dim ( P )- dim ( N )2 6 assures that g(N)n DP = 0From this point on the proof divides into two steps

Step 1 In this step a closed subspace Pf of P - dP will be constructed satisfying the following

(37) The topological boundary Pa of Pf in (P - dP) is a real codimen-sion one subspace of P in transverse position to the Pii E I (12)) and divides P into the two halves PfP(- P - P)

(38) For every cube A in P the inclusion A n Pac A n Pf is two-connected

(39) For every cube A in P (An P A A 17P A n Pa) is a Poinear6 duality triple

The construction of Pf is carried out inductively (induction on the num-ber of cubes in P) Begin by writing P as the increasing sequence of cubical subcomplexes 0= KOc Kl c K c c K = P so that each Kj+is obtain-ed from Kj by adding the cube Alj to Kj satisfying dim (A t j )2 dim (Alj i l ) Suppose that the part of P lying in Kj - this will be denoted by Pi$-has already been constructed satisfying 37-39 for every cube in Kj Let

h (MdM) - aAljTgt

be a homotopy equivalence from a differentiable manifold h I splits along the codimension-one subspace PaSjn dA+ (see 37-39 and Theorem 121 in [32])Homotopy

In 317 below the dimension requirements are improved to dim (P)- dim (N)2 3

LOWELL JONES

to a map in general position to itself and let (T T) denote the regular neighborhood of i ts image Replace A by M glued along the split equivalence Ih Extend Pato by adding T t o it corners should be rounded a t d(T)

Step 2 Let P be the subspace just constructed hi Mi -P i E I is a set of differentiable charts for P By using 37-39 in conjunction with 121 of [32] Pt n P2 can be copied cube for cube up to homotopy equivalence in Mi by transversality So Pais actually a codimension one patch subspace of P In particular (P Pa) is a Poincar6 duality pair In order to complete the proof of Lemma 36 i t must be shown tha t P has an associated differentiable chart ha Ma -P as in 14b The first step in this direction is to show tha t the Spivak normal fibration for (P Pa) has a BO reduction If dim (N) lt (112) dim (P) then by the general position construction of each Pan A g gM(A) -Pan A must be a homotopy equivalence Hence g N- Pa is an equivalence so rjPumust have a BO reduction In general g N- Pa is a homotopy equivalence but for a double point set of a t most dimension one But n(SO) -n(SF) is epic for i I_ 1 and monic a t i = 0 so the double point set causes no difficulty

Since the Spivak fibration for (Pa+ Pa) has a BO reduction there is a differentiable surgery problem

where zXais the linear normal bundle of the differentiable manifold Mu By putting ha in transverse position to the real codimension subspaces Pin P+ i e Iof Pa Mu is given a set of patches hll(P n P) i E I ) for which ha Mu -Po is a patch homomorphism I claim tha t surgery can be done on the patch homomorphism ha Mu -Pa+to change i t into a patch isomorphism It suffices to do surgery on the patch homomorphism ha Mu - Pf so tha t every cube of P is copied up to a homotopy equivalence in Mu This is accomplished by induction over the cubical skeleton of Mu using (38) above in conjunction with Theorem 33 in [32]

This completes the proof of 36 QED

Remark 311 The patch Pf engulfing g N- P which has been constructed above does not intersect DP if r s dim (P) - dim (N) - 1

315 PATCH SPACES

Remark 312 PL is homotopy equivalent to a finite CW complex having no cells of dimension greater than dim (N)

We shall need the following two refinements of Lemma 36 N and r -P are as in Lemma 36 N is a subcomplex of N w i t h dim (N) lt dim (N) L i is a codimension zero patch subspace of dP and g = (N N) - (P L+) is a given map of pairs

LEMMA313 Let Pf be a patch i n L + - L engulfing g N -Li Then P+can be extended to a patch PC i n P engulfing f (N N) - (P LA)

LEMMA 314 Let P be a patch i n P engulfing g (N N) (P dP) -+

Suppose that P d P are both connected Then P A a n be enlarged to a patch PC so that Pf PI n d P are connected and the inclusion induced homomor- phisms n(PI) -+ n(P) n(PI n dP) -n(dP) are isomorphisms

All the arguments required to prove Lemma 313 have already been in- troduced in the proof of Lemma 36 The proof of Lemma 314 requires further comment

Proof of Lemma 314 Using the differentiable surgery lemma in the appendix to 5 4 below there is no difficulty in embedding objects in P pro- vided they have dimension 2 2 In particular one and two-handles can be added to P+n d P in d P and to Pf in P until Lemma 314 becomes satisfied To assure tha t only a finite set of two-handles need be added we use 311 in [21] if r G -G is a homomorphism between finitely presented groups then a finite set of elements generate kernel (r) a s their minimal normal subgroup in G QED

Now we return to the problem of completing surgery on

(P dP) --J-- (X Y ) First consider surgery below the middle dimensions Suppose tha t

f P X is k-connected and let g Sk-P represent an element in Kk(P) -+

Since g(y) is the trivial BF bundle g Sk P can be engulfed in a patch -+

P+away from dP After replacing Pf by the domain of a differentiable chart ha Ma -P+ g Sk P can be homotopied to a differentiable embed- -+

ding g Sk (Ma Pa) which is in transverse position to all the patches of -+ -

P Let U+be a closed tubular neighborhood of g(Sk) in Ma intersecting each cube A of Ma transversely in a closed tubular neighborhood for A n g(Sk)

316 LOWELL JONES

in A Because the composition f 0 g Sk X is null homotopic after a homo- 4

topy of f we may assume that f 1- maps U+ into a point z E X Then f Y I+ -z I provides a BF trivialization for the Spivak fibration Y 1 - -U+ of ( U f U) where U is the topological boundary of U in Ma Using 33 of 1321 as was done in Step 2 of the proof of Lemma 36 realize a differentiable chart h M -U+having the following properties h M -U- is covered by a BF bundle map h T -Y I+ so tha t the composition f c h i --+ z has a BO reduction where z is the BO normal bundle for M

Replace the BF trivialization

by the BO trivialization

Now do framed differentiable surgery on g Sk-M By tapering the patches of P the resulting surgery cobordism is given a patchspace structure This completes the discussion of patch surgery below the middle dimensions

Modulo the results of 1321 more difficulties arise in describing surgery on (32) i n the middle dimensions when dim (P) = 2k than when dim (P) =

2k + 1 Accordingly I will prove Theorem 34 when dim (P) = 2k and leave the same task when dim (P)= 2k + 1 to the reader

First I recall a few facts about the normal bundle invariants in the middle dimension Let 7 -Skdenote a k-plane bundle which is a stably

free BO bundle The isomorphism type of 7 -Skis uniquely determined by the Euler characteristic of 7 if 2k = 0 (mod 4) and by the Kervaire invariant of -r if 2k = 4m + 2 -both of which are fiber homotopy invariants (see 83 in [13]) For 2k = 4m + 2 k 1 3 7 there are exactly two such isomorphism classes while if 2k = 0 (mod 4) the isomorphism classes are in one-one correspondence with the integers via (7 -Sk)++~ ( z ) 2 where ~ ( z ) is the Euler characteristic of z The same set of invariants are available for stably trivial (k - 1)-spherical fibrations E -Skover Sk and they determine the fiber homotopy type of E -Sk

LEMMA315 Let E -Skbe a stable trivial (k - 1)-spherical Jibration

PATCH SPACES 317

Then E -Sk is fiber homotopically equivalent to a (k - 1)-sphere bundle t -Skassociated to a k-plane bundle z -Skwhich is a stably free BO bundle The isomorphism class of z -Skis uniquely determined by the fiber homo-topy class of E -Sk

PIProof If k = 0 l then E -Skis the trivial fibration Skx Sk-l-Sk

If k = 2 then the fiber homotopy type of E -Skis uniquely determined by the Euler characteristic of E -Sk SO we can assume k 2 3 Let E be the mapping cylinder for E -Sk Because E -Skis a stably trivial BF bundle there is a homotopy equivalence

Complete codimension one surgery on t t-((E E)x 0) - (E E)x 0 (k 2 3) and use Whitneys trick to choose a differentiable embedding Skc t+(E x 0) which is a homotopy equivalence Then the normal bundle z of Skin t-(E x 0) is a stably free k-plane bundle The H-cobordism theorem shows tha t z can be chosen to engulf all of t-(E x 0) Then t t -E is a fiber homotopy equivalence

Suppose that E is fiber homotopy equivalent to the (k - 1)-sphere bundles associated to the stably free k-plane bundles z z Then the Euler charac-teristic or Kervaire invariants for z z (depending on whether k is even or odd) are identical So z z are isomorphic k-plane bundles This completes the proof of Lemma 315

Now we can define the obstruction to completing surgery on (32) when dim (p) = 2k Begin by doing surgery on

to make f k-connected Since (P dP) ( X Y) are both Poincar6 duality pairs Kk(P) must be a stably free finitely generated Z(n(X))-module (see 51 in [32] and recall tha t P aP X Y a r e all finite CW complexes) So after doing differentiable surgery in the interior of a cube A on a finite number of trivial k - 1 spheres we may assume tha t Kk(P) is a finitely generated free Z(n(X))-module having a basis represented by gi Sk-P i = 1 2 1 Let Pf c P - dP be a patch engulfing Ui=gi

Choose homotopy classes gi Sk-Pf homotopic to gi Sk-P in P and replace P by the domain of a differentiable chart ha Ma -Pf Choose differentiable immersions g Sk-+Ma - Pa which are in general position to

318 LOWELL JONES

each other homotopic in Ma to fli Sk-+ Ma and which have fiber homotopically trivial normal bundles T ~ If k + 1 3 7 these properties uniquely aetermine the differentiable immersion class of 3 Sk-Ma - Pa In the case when k is even this is seen as follows according to 82 in [9] the correspondence which sends the immersion class represented by g Sk RZk to -+

its normal bundle in R is a one-one correspondence between immersion classes of Skin R and stably free k-plane bundles over SkSO the existence and uniqueness of the immersion class g Sk-+Ma with the properties described above follows from Lemma 315 together with the fact tha t each g(r) is the trivial BF bundle If k is odd and k 1 3 7 then there are precisely two distinct stably free k-plane bundles r over SkUsing the proposition in Chapter 1of [32] there are immersions of each r in R each of these immersions restrict to an immersion g Sk RZk on Sk which has r-+

for normal bundle in R Since there are a t most two immersion classes g Sk-RZk (see [9]) i t must be that the correspondence which assigns to g Sk-RZk its normal bundle is a one-one correspondence between immersion classes of Skin R and stably free k-plane bundles over SkSO the existence and uniqueness of 3 Sk Ma when k is odd and k f 1 3 7 follows as-+

before Now an obstruction o(f) E Lzhk(Z(n(X)))is determined by the intersection numbers h(gi(Sk) j j (Sk)) and self-intersection numbers p(si(Sk)) and the restriction to Ma of the orientation for P (see Chapter 5 in [32]) Note that the hypothesis dim (P) 2 11 dim (P) f 14 assures that k f 1 3 7 so this definition of a(f) can always be made

I claim that a(f) is a surgery patch cobordism invariant The only step in the verification of this claim which has not either been described in the prior discussion of this section or does not appear in $ 5 of [32] is the following lemma

LEMMA316 o(f) is independent of the particular diferentiable chart ha Ma -+ Pf whose domain replaced Pf above

Proof Let hb ML -P+be a second differentiable chart P P will be the patch spaces obtained by replacing P+by Ma M respectively and h P -P will be a patch space isomorphism o(f) e Lhk(Z(n(X))) denotes the surgery obstruction computed from the homotopy elements (h-log) Sk-ML i = 1 2 1 in the manner described above

Let S be the finite set of double points of the immersion

U=l3 UL1Sk-Ma Choose a differentiable regular neighborhood U+for U= gi(Sk) in Ma so that

PATCH SPACES

any x e S is the origin of a disc DZk = Dt x D5 satisfying

Here (12)Dk are all the points in D$ a t most half the radius of D5 from the origin I t is required that the Dk are pairwise disjoint Make h ML -Ma transverse regular to each D so that h h-1(D2k)-+Dik is a diffeomorphism Pu t h ML -Ma in transverse position to U mod h I h-l (U S =k) and complete codimension one surgery on h h-(U) -+ U away from the diffeomorphism

h h-(U D) -USESDik

so tha t the homotopy equivalence h Mi -Ma splits along U The diffeo-morphism

h-I u(u+n D ~ )-h-(U (U+n D2))

can be extended to a homotopy equivalence

which is the homotopy inverse to h h-l(U(+))-U(+) By using the Whitney embedding theorem the embedding

can be extended to an embedding

which is homotopic to

U L 1( E - l o g ) ((Ui=S k )- (UiE1gi1(UXes(D5 x 0) U (0 x D)))

Ui=s i l ( (UxEs(aD x 0) U (0 x do))-(h-(u+) - h-l(u D ~ ) h-l(u+)n h-(U d ~ ~ ) ) Then

-(U=lTi I ( ~ = l ~ k ) - l ~ = l ~ i l ( d x E S( D ~ O ) U ( O ~ D ) I= e 7

(Uf=lTi I I I = ~ S ~ ~ ( ~ ~ D ~ ) -- U=lh-o 9i

defines an immersion Ui=ri U=S b h-l(U) homotopic to (Uf=E - o g) Ui=S k -Mi which satisfies for all ij= 1 2 1

320 LOWELL JONES

Clearly h-(U) is a regular neighborhood for U=ri(Sk) in Mi (use the H-cobordism theorem) so h h-(U(+) U(+) provides a fiber homotopy equiva- -+

lence between the normal bundles of the immersions ri 3 showing tha t the normal bundle for each ri is fiber homotopically trivial Thus r i Sk ML-+

are the immersion classes with which or(f) is computed and (316) shows tha t a(f) = a(f) This completes the proof of Lemma 316

To complete the proof of Theorem 34 i t remains to show tha t if a( f ) = 0 then surgery can be completed Use Lemma 314 to represent a subkernel for the middle dimensional kernel group as immersions in a connected patch having the correct fundamental group Now Whitneys trick applies as usual to move these immersions to disjoint embeddings on which surgery can be completed This completes the proof of Theorem 34 QED

Remark 317 The dimension restrictions of Theorem 34 can be improved to the single requirement tha t dim ( P ) h 5

First the restriction that dimension ( P ) - dim (N) 2 6 of the engulfing Lemmas 36 313 314 must be improved to dim ( P ) - dim (N) 2 3 Under these circumstances g N 4P may intersect with the 3 4 and 5-dimensional cubes of P but Theorem 121 of 1321 does not apply to copy by transversality a regular neighborhood of these intersections nor can Theorem 33 of [32] be applied as in Step 2 of the proof of Lemma 36 to change a patch homomorphism to a patch isomorphism on the 3 4 and 5-cubes of these regular neighborhoods The first difficulty is overcome for 3 4-cubes by requiring tha t image (g N-+ P ) have a linear tubular neighborhood in each 3 or 4-dimensional cube of P and then applying the lemma of the appendix to 9 4 below in place of Theorem 121 from [32] for a five-cube A5we may have to enlarge the tubular neighborhood for g(N) n AS in A5 by adding a finite number of 2-handles along trivial 1-spheres in i t s boundary then the recent 4-dimensional surgery results 171 151 extend Theorem 121 of 1321 to handle the 5- cube situation The second difficulty is handled in a similar fashion

Now as an illustration we consider surgery a t dimension six Let

(P 8P) ( X Y )

PATCH SPACES 321

be as in (32) After doing surgery below the middle dimension f will be 3-connected and K3(P)will be an fg-free Z(7rl(X))-module Represent a basis for K(P) by mappings g S3-PI and engulf these mappings in a patch P+c P Replace P by the domain of a differentiable chart ha Ma 4P Homotopy the ai i = 1 2 I to differentiable immersions (l S34Ma i = l 2 I and choose a differentiable regular neighborhood U for UL1ji(S3) in Ma f can be homotopied so that f(UL)c X (X = one skele-ton of X) Choose a framing T~ I r X x STNext choose a differentiable chart h M 4 U which is covered by a BF-bundle mapping h fJr--+ Y IU+ where z is the linear normal bundle for the differentiable manifold M so that the composition

has BO reduction h f -X x STReplace

I foh I

M -- X1 As in Theorem 11 of [32] the BO framing if 4X 1 x STdetermines unique framed immersion classes g S3x D3-M i = 1 2 I repre-senting the homotopy classes g S3--+ P i = 1 2 I Use the restric-tion of these immersion classes to S3x 0 -S7to calculate a special Hermi-tian form

(K3(P) AP ) -- ~ ( fL(E(X))

We can stabilize this surgery problem by the factor X(CP2)3and as in the proof of Theorem 99 in [32] compute the new surgery obstruction a(f) in the differentiable chart Mx(CP~)~ But a(fl) is a patch-surgery cobordism invariant and a(f) can be reconstructed from a(f ) hence o(f) is a patch surgery cobordism invariant I t is easy to see that a(f ) = 0 allows patch surgery to be completed

Remark 318 Let h (P aP) -(P aP) be a patch isomorphism between oriented patch spaces covered by the B F bundle mapping h Y 4Y If

322 LOWELL JONES

dim (P) 2 5 then surgery obstructions a(f ) a ( f 0 h) are defined for

(PlP ) ---L (PI 8P1) -(XY) ( X Y )

I t is clear from the proof of Theorems 34 and 317 that a ( f ) = o(f o h) As a consequence of this we can deduce the following composition formula Let

be (Poincar6) surgery problems the second of which is assumed to have a fixed differentiable reduction In addition suppose (M dM) is a differentiable pair Then the surgery obstructions a(g) o(g) a(g 0 g) are related by the

Surgery composition formula a(g) + o(g) = a(g 0 g)

This is verified as follows Note that the surgery obstruction for the composition

is o(g 0 9) - a(g2)) because this surgery problem is just the sum of

Complete surgery on

changing g(-g) into an isomorphism of patch spaces the domain and range of g(-g) each have only one patch since they are differentiable manifolds so when g (-g) becomes a homotopy equivalence i t also becomes a patch space isomorphism But then

PATCH SPACES 323

o((311-+rI)O (a (-92))) = o(giI-) = ~ ( g i )+ ~(I -M)= oki )

It follows that a(gl 0 9) - o(g) = o(g) which gives the desired composition

formula a 1 9) = 43) + o(g2)

4 Changing patch structures

The key metatheorem for patch space theory is this Any geometric construction tha t can be performed on differentiable manifolds has an exten-sion to patch spaces but for a single surgery obstruction I ts meta-proof is this Perform the geometric construction in each patch then use trans-versality and surgery to glue these operations together patch by patch Only a single surgery obstruction occurs because any patchspace is (in a homological sense which this section shall make explicit) no more complicated than a patch space having only two patches It would be most instructive for the reader to derive the results in 5 5 below for the two-patch case

Consider the increasing filtration for P of the cubical subcomplexes DP c D I Pc DP c cDP c cDimpjP = P where DP is the union of all cubes in P having codimension greater than or equal to dim (P) - k This sequence gives rise to the cubical cha in complex DC(P) of P

--H(DPD-P 2) H-(D-PD-P 2)2 I l l I l l

DC(P) DCk-l(P)

where 8 is just the composition

If P is an orientable patch space then DC(P) is a free 2-module having as 2-basis all the connected components of cubes having codimension (dim (P) - k) in P DH(P A) denotes the homology of DC(P) with coef-ficients in A

THEOREM41 There i s a patch cobordism ( W V ) of (P 8P) sa t i s fy ing

(a) V = 8 P x [0 11 (W P ) has o n l y one-handles (b) DH ((a- W 8 V) 2)= DH((a+ W 8 V) 2)r 2 where p = dim (P) (c) DH((d+ W d+ V) 2)= 0 in d imens ions gt 1 where (8 W a+V)

ranges over the patches of (a+W 8 V)

Theorem 41 is proven in the two Lemmas 43 and 47 below There is

324 LOWELL JONES

also a mapping and framed version of Theorem 41 which we shall use with-out further to-do in $5 below

DC(PdP) contains a subchain complex FC(PaP) defined as follows Choose an orientation [PI E DHp(P(dP2))for P and an ordering i lt i lt lt iof the elements in the index set I The ordering of I gives a framing T r P x RIJ to the normal bundle of P in P ( J c I)in fact T = $i(~ilvJ) (11) becomes an ordered summation each summand of which has a framing T r Pi x (- 1 l ) determined uniquely by P+n T~ r Pi x [O 1) The composition

3 P x RIJ1 +Hp((PJx (R1J - D J )) u (P x R~~~n 8 ~ )1

sends [PI to an orientation [P] for the Poincar6 duality pair (Pj P n dP) where 7 is a subdivision map 8 is the composition of the collapsing and ex-cision maps 9 is the Thom isomorphism and DlJ is the unit disc in RIJ If ~ ( l= p - 1 J I ) is a cube in Po f the form A = P n (niG+p) n ( n iGJ -P i ) where I is the union of pairwise disjoint sets J JTlJ- then an orientation is obtained by restricting [P] to A If A a A a A are the con-nected components of A then [A] restricts to the orientations [Alpj]for Aj Since the set of oriented connected components [Aj] of (p - 1)-dimensional cubes in P - 8 P are a free 2-basis for DC-(PdP) [A] = xj[Aj] can be regarded as an element in DC-(PaP) D~-(P~P)is defined to be the subgroup of DC-(PaP) generated by all these oriented (p - 1)-dimensional cubes [A] Clearly ~ 7 - ( P ~ P )is free 2-module with the set of oriented (p - 1)-cubes [A] in P aP as a 2-basis Our method of deducing the orienta-tions [A] from the global orientation [PI for P assures that 8 DCk(PaP)-Dck-(PdP) sends D T ~ ( P ~ P )into DX~-(P~P)S O DC(P~P) is a sub-chain complex of DC(PaP) The homology of the chain complex E ( P ~ P ) is denoted by DH(P~P 2 ) Note that if every cube of P is connected then D ~ ( P ~ P )= DC(PoP) and DH(P~P 2 ) = DH(PoP 2 )

LEMMA43 There i s a patch H-cobordism (W V) from (P aP) =

a_(W V) to a patch space (P aP) satisfying

P DT(amp- 2) = DT(= 2) r Z(p = dim (P))

325 PATCH SPACES

and DH~((P)+((P~) n aP) 2)= 0 for each patch (Pi)+ of P r j gt 1

Proof Add the extra patches (Pi)+ - Pi u Pi x [-120] i E I ) to P where Pi x ( -1 l ) is the collaring of P in P satisfying Pi x (-11) n Pi+=

Pi x [Ol) Note that every point in P is contained in the interior of some augmenting patch (Pi)+Each set (Pi)+ i~I i 5 i) dices P into 23 cubes of the form A -= nisi(P))() A n aP Although these cubes do not necessarily have a differentiable structure (as in Lemma 21) each is a patch space with boundary having a hierarchy of corners so each is a Poinear6 duality space with boundary Just as before the given ordering of the index set I and the global orientation [PI for P induce an orientation [A] for each cube A (Pi)() I shall refer to the cubical structure given P by= nisi

as the qth restricted cubical structure

are qth restricted cubes and a collection of qth restricted cubes K is a qth restricted cubical sub-complex of P P has the increasing filtration DOqPc DjqPc DiqPc c DBP = P where DP is the union of all qth restricted cubes having dimension less than or equal j The qth restricted cubical chain complex D(qC(P8P) is defined as before with homology groups D H (PdP 2 ) D(qC (PoP) contains the sub-chain complexmC having groups D()H (PaP Z) where(PaP) homology D(qCp-(PaP) is the subgroup of D(q)Cp-(P8P) generated by all (p - I ) -dimensional qth restricted oriented cubes [A] in PdP Evidently both

Z) and D(g1H(pi)+ n d~

are defined From here on the proof divides into two steps

Step 1 I claim that by adding and subtracting closed p-balls to each augmenting patch (P)+ (but leaving the patches P i E Iunaltered) the (Pi)+can be made to satisfy the following for every q (= I - 1

(44) (i) =(P~P 2)= D-(P~P Z ) r2 (ii) n 8P) Z) = 0 for each augmenting patch (Pi)+ DH((P)+((P~)+

of P with i 5 i in dimensions gt 1 (iii) Every point in P is contained on the interior of some (Pi)+

This will be proven by induction on q Suppose that (44) is satisfied for some integer r 5 1 11- 2 Adding (Pi)+ to (Pi)+i 5 i) has the effect of subdividing the r th restricted cubical structure of P each r t h restricted

326 LOWELL JONES

cube A is subdivided into the three (r + 1)st restricted cubes A n P A n (P) A n (Pi+)-There is a well-defined subdivision homomorphism

given by S~([A]) = [A n (PL+)+]+ [A n (PiT+)-1 a desuspension homomorphism

d ~ r )c(P)--( pi-+)C~P P n a~

defined by d2([A1]) = [A n PT] and an excision homomorphism

defined by e([A]) = [A n (P)] Now I want to add and subtract closed p-balls from and (Pv)+

to make q = r + 1 satisfy (44) I shall say that an r th restricted cube A of P i s i n P8P if (A - OA)c ( P - 8P) For each r t h restricted cube A in PIOPchoose a closed (p - 1)-disc Dp-1 in a differentiable part of A - ah subject to the following conditions Dp-1 n Pi = 0 if irlt if Dp-1n Pj= 0 for j E I and in the case (PL)+n A A then Dp- c (Pi+)- n A If I = p then set Dp-1 = A1

Let zD-1 be the normal disc bundle to Dp-L in P small enough so tha t the p-balls zDp-1 are pairwise disjoint and zDP-1n Pl = 0 zDp-1 n Pj = 0 are satisfied for i lt i and j E I If Dp- c(PlV+)- or 1 = 0 then add the closed p-ball zDp-1 to (PLl)+ and leave the other (P)+ unaltered If Dp-1 c (PIT_)+ and I p then subtract the interior of (12)zDp-1 from (Pl_)+ -where (12)zDp-1 is the p-ball in zD-z having one-half the radius of zDP-1 add zD-1 to (Plv+)+ and leave the remaining augmenting patches (P)+ unaltered I t is clear tha t after this maneuvering (44) (iii) is still satisfied and A n PC+ 0 for each r t h restricted cube A in P8P with 1 p and Piv+3A when I = p This assures tha t the subdivision map induces an isomorphism

for every r t h restricted cube A in PaP These isomorphisms piece together to form an isomorphism

PATCH SPACES

for each r th restricted cubical subcomplex K of P In particular

and

(in dimensions gt 1)for all i 5 i showing tha t (44) (i) (ii) are satisfied for q = r + 1 with the possible exception of the equality

in dimensions gt 1 Arguing as before the excision map e is an isomorphism and the desuspension map dz is an isomorphism in range dimensions 2 1 From the first fact and an exact sequence argument we deduce tha t

D T - l ~ ( ~ _ l n a ~ ) ) - n ap)(P G D ~ - l ) ~ - l ( ~ + ~ T - l

in dimensions lt p - 1 And then from the second fact

D ~ - l l ~ ( ~ ~ l ~ - + ~ ~ ~ l T + + ln a p ) = o in dimensions gt 1

Step 2 Set (P )- = 8 and carry out the process described above of adding and subtracting closed p-balls TP-1 to (P)+ and to (P+)+ so that (44) is satisfied for q I I I In doing so P - is changed from 8 to a disjoint union of closed p-balls

Now choose for each I I I th restricted cube A in PdPa ball Dp-2 c A -dA subject to the conditions Dp-Ln PzIl+l and if A n (P -)+ + A=

then Dp- c A n (P If 1 = p then set Dp- = A Choose normal disc bundles TDP- to each Dp- in P small enough to assure that TP-L n P = holds Add to (P _)- all the p-balls TP-L and denote this new patch by (P+)+ as well I claim that (44) is now satisfied for q = I I1 + 1 with the augmenting patches (Pk)+ j Ii I i + 1) The straight-forward com-putational proof of this fact can be supplied by the reader

Let P denote the patch space having the same underlying CW complex as P and patches (P)+ i 5 i-) A patch H-cobordism ( W V) from (P oP) to (P 3P) is obtained as follows the underlying CW complex for ( W V) is the same as tha t for the product patch cobordism (Px [0 I ] oPx [0 I ] ) the patches for W are PC x [O a] i E I ) u (P)- x [E11i 5 i -) with corners P E x P x 6 rounded away Here Ei 5 i -) U (6 IE Iare

328 LOWELL JONES

distinct numbers in (0 1) with Elt lt a j for all ij QED

LEMMA47 Let f (P dP)-(X Y ) be a given map from the patch space Pinto the connected C W complex X By doing zero- and one-surgery along the cubes of PIoP away from dP a patch cobordism F ( W dP x [0 11) - ( X Y ) o f f (P oP) -- ( X Y )is constructed so that for each cube A i n o WoPx 1 A is connected and f nl(A) -- n l ( X ) is an isomorphism when dim (A)2 4

Proof The meaning of the phrase surgery along the cubes of PdP will be clear from the following proof

The proof is by induction on the number of cubes in PIoP Order the cubes of PoP so that All lt A =gt 1 5 11 and assume tha t f (P oP) -( X Y ) has already been varied through a cobordism so tha t Lemma 47 is satisfied for the first k cubes of PdP Let A be the (k + 1)st cube in PoP Do differentiable zero- and one-surgery on f (P dP) -( X Y ) along framed zero- and one-spheres contained in (A- oA)to make A connected and fn(A) -n (X) an isomorphism (if need be replace (A dA) by an equivalent differentiable manifold pair)

The normal bundle of A in P splits as a sum of trivial line bundles where each line bundle comes from the collaring of a Pj in P with A c Pj Let W be the surgery cobordism just constructed These same normal bundle split- tings extend to the normal bundles to the handles of ( W d-W) giving to W the correct real codimension structure Let hj Mj -Pj be any chart which copies the cube A The lemma in the appendix to $ 4 shows tha t any zero or one-surgery performed in A can be exactly copied by differentiable surgeries performed along the cubes of Mj Hence the chart hj Mj -Pj extends to a chart hjMj -W j for W and these charts make W a patch cobordism QED

The following refinement of Lemma 47 will also be needed

LEMMA48 Suppose f (PoP)-( X Y ) is covered by a BF bundle mapping f y -z where Y -P is a Spivak Jibration for (P dP) Then the cobordism F (wP x [0 I ] ) -( X Y ) i n Lemma 47 can be chosen so that f y -- z extends to a BF bundle mapping F7 -z Here 7 is the Spivak Jibration for W

Proofl Proceed as in the proof of 47 use the fact that the natural maps

~ ~ ( 0 )-- n(F) n l (0)-n(F) are onto to insure tha t surgery framings can be chosen which allow the extension of Y f to 7 F respectively

Appendix to $ 4

LEMMALet f (MI dM) - (M oM) be a degree one map from the dif-

329 PATCH SPACES

ferentiable manifold M to the Poincare duality space M if dim (M) 5 2 then f is assumed a homotopy equivalence between manifolds N c M i s a submanifold having real normal bundle If dim (M) 3 and dim ( N ) 5 1 or if dim (N) lt 1 then there i s a homotopy f so that fl i s i n transverse position to N and f f(N) -+ N is a diffeomorphism If dim (M) = 3 and dim (N) = 1 there is a normal cobordisln 7(W Wa) -+ (M 3M2) off with W = oMl x I so that f f -[+(N) -N i s a difeomorphism

Proof If dim (M) I2 then h is homotopic to a diffeomorphism Suppose dim (M) 13 We may suppose N is connected P u t h in trans-

verse position to N perform zero surgeries to make h-(N) connected deno- t ing by E W - M the resulting surgery cobordism Now d+ W equals the connected sum of M with a finite number of copies of Sm-lx S Add two- handles along the s x S (soE s-) away from amp-(+ (N) This changes E W -M to a product cobordism The degree one assumption on f assures tha t I E-+ ( N ) N is homotopic to a diffeomorphism -+

For dim (M) = 3 and dim ( N ) = 1 proceed as in the last paragraph up to the point where two handles are added to 3 W At dimension 3 this cant in general be accomplished without intersecting N with the core of a two-handle QED

5 Obstruction to transversality

Let fl (P dP) -X be a map from the oriented patch space pair (P oP) t o a connected C W complex X containing the connected real codimension subspace Y (12) such tha t f I is in patch transverse position to Y

The obstructions to transversality lie in a surgery group L (~(X Y)) which I wish to discuss now n(X Y) denotes all the relevant fundamental group data for

Any cr E Lt(n(X Y)) corresponds to a map f(N dN)-+X from the i-dimensional Poinear6 space N in transverse position to Y c X together with a homotopy equivalence g (M oM) (N aN) from the differentiable manifold pair -+

(M oM) such tha t g l a is split along f - I ( Y )c dN a = 0 if and only if there are cobordisms (gM) (7 so tha t iJ restricted to any of o ~ M N)

ddM oMis a homotopy equivalence and iJia_a-~ is split along 7-1-i(Y) When i-codim(Y) 5 3 we slightly alter the allowed form of the geo-

metric representation Let R = N dN f -(Y) f - I (Y) N - f -(Y) or

330 LOWELL JONES

dN - f-(Y) If dim (R) = 3 and we required in the last paragraph tha t -+g g-(R) R be a homotopy equivalence then this is now weakened to

requiring that it only be a Z(r(X Y))-homological equivalence this same weakening is understood for any R c8 with dim (R) = 3 Moreover any of R R having dimension = 3 must have a differentiable structure

To each group Lt(r (X Y)) is associated a splitting theorem If no sub- Poinear4 spaces of dimension (= 4 are involved then they just state tha t if a splitting problem vanishes in Lh(r(X Y)) and the homotopy groups are isomorphic to r (X Y) then i t splits homotopically (see [32]) In lower dimensions only more restricted versions can be proven (see the appendix to this section) fTHEOREM50 There i s a single surgery obstruction t(f) e Lt-(r(X Y)) (p = dim (P)) which vanishes if and only if there i s a n orienged patch cobordism F ( W V) -- X of f (P oP) --X so that FI-is i n patch transverse position to Yc X

COROLLARYTO 50 Ijt(f) = 0 then (W V) can be chosen so that V is the product cobordism and ( W P) has no handles i n dimensions 2 3

There is also a framed version of this result when (P oP) --X is covered by a B F bundle mapping f y --r where r 4 X is an oriented B F bundle and y -- P is a Spivak fibration for the pair (P 8P)

THEOREM51 Y)) be as i n 50 Let t(f) E LfP- ( n ( ~ I n addition suppose that f (P dP) -+ X is covered by f y -+ r If t(f) = 0 the patch cobordism F ( W V) -+ X i n Theorem 50 can be chosen so that f y -+ z extends to a BF bundle mapping F 7 z covering F ( W V) -X Here 7 i s a-+

Spivak ji bration for ( W o W)

Remark 52 Because of the low dimensional surgery anomalies discussed in the appendix to $ 5 below the reader will see tha t if t(f) = 0 then in the proof of Theorem 50 the transverse inverse image of Y captured under f P- X is not fully a patch space it is equipped with differentiable charts hi Mi -+ ( ~ - I ( Y ) ) ~ (Y) and the k-cubes which copy the cubes of f - I of f -(Y) with Ic + 3 up to homotopy equivalence but only copies the 3-cubes of fP1(Y) up to Z(n(Y))-homological equivalence Such an object is called a pseudo-patch space So to be precise Theorems 50-51 should be restated with this in mind Pseudo-patch spaces are just as good as patch spaces in tha t all the results of 591-4 are easily seen to hold for them as well Moreover if in Theorem 50 we begin with a pseudo patch space P we can compute t(f) and if t(f) = 0 then we can recover a pseudo patch space f -(Y) as in

PATCHSPACES 331

the proof of Theorem 50 given below For this reason we will make no distinctions between patch spaces and pseudo-patch spaces

Proof of Theorem 50 First the 3-dimensional cubical subcomplex DP has a differentiable structure so transversality extends to f

Suppose transversality has been extended to f lapJDrcpfor some T 2 3 We define an ( r + 1)-dimensional obstruction to transversality cocycle t as follows For each (r + 1)-cube A-- in P choose a homotopy equivalence h (p dp) - (A-- oA--) from a differentiable manifold pair Let t-(A--) denote the image in Lh(n(X Y)) to splitting hi along f - I s --( Y) Let h (p op) - (A-- dA--) be any other equivalence Then t-(A --) + P = t-(A --) holds by the surgery composition formula (317) where P is the image in Lh(n(X Y)) of the obstruction to splitting h- 0 h dM--dM along (f 0 h I)-( Y) But h 0 h- M-M and (f 0 hf)-(Y) provides a null cobordism for P hence the cochain

tr+l DC+ (PIdP) Lh(n(X Y)) +

is well defined A standard argument shows that t- is in fact a cocycle Let u DC(PdP) -Lt(n(X Y)) be an arbitrary homomorphism For

each cube A- of P split A- into two connected manifolds A- U At-_with A- cA- - oA- unioned along a connected boundary It can always be arranged tha t

are epimorphisms and isomorphisms on anything of dimension 2 4 It might be necessary to perform zero and one-surgeries along the r-cubes of P to achieve this (94) Now vary the splitting of lb-r oAb- -O+A~- along f l a + l b l ( Y) through the surgery cobordism h W- dA- representing -u(A-) E Lt(n(X Y)) Change the transversality of f l D r ( P - O - ( P ) t o Y by gluing each W U a--rAa_ along the split equivalence a+ W -- A straight-forward argument shows this maneuvering has succeeded in changing the obstruction to transversality cacycle from t- to (t_ + u0 a) where d DC+ -- DC

Now suppose t- = 0 as cochain If dim (f (Y)) 5 3 then f - l s p - r - l (Y) has a differentiable structure otherwise f - l p - r - l (Y) is a Poincark space of dimension 2 4 In any event we can use the lemma of the appendix to 5 5 below to perform zero- and one-surgeries simultaneously on the image and domain of

From our inductive construction of transversality each (A- f-lll-r(Y)) is a dif-ferentiable manifold pair away from its boundary

332 LOWELL JONES

h (aM f o h- 1811 ( Y)) -(3Ap-T-l f la+v-l (Y))

-where h (M dM) -(A-- dA--) is a differentiable copy of this cube- thereby obtaining a surgery cobordism h ( W V) -( W V) with El split along d+V and h W- W an equivalence There is no difficulty in assuming that W lies in M (see the appendix to 54) Replace A-- by W U M - W and extend the transversality of f l differentiably to f I = Doing this for each ( r + 1)-cube extends transversality to f I a p U D _ I ( P )

Now here is how to define the obstruction to transversality t(f) E

Lh-(n(X Y)) Using 54 there is no loss in assuming

(56) DH (PdP) = DH(PdP) E Z

DH(P-PI- n bP) = 0 in dimensions gt 1 for every patch Pi+of P So every obstruction t vanishes but for t which is identified with an element t(f) E Lh-(n(X Y)) under the isomorphism DH(PbP Lk-(n(X Y))) G

L-l(n(X Y)) The cobordism invariance of t(f) is now an easy exercise

This completes the discussion of extending transversality in the real- codimension sense Now suppose t(f) = 0 so real codimension transver-sality has been extended to all of f 1 We have to copy this in every single patch This is done by combining (56) with an Eilenberg (surgery) obstruction argument Possibly surgery must be performed along zero and one-spheres in the cubes of P and its differentiable charts to make fundamental groups correct and account for low dimensional surgery anomalies (see the appendix to 9 5)

This completes the proof of Theorem 50 QED

Appendix to 55

fLet (M dM) - (N bN) - (X Y) represent a E L(n(X Y)) with a=O fLet (M dM) -

-(N bN) -- (X Y) denote a surgery null cobordism For

R = N f-(Y) or N - f-(Y) let E denote the corresponding cobordism in N

LEMMAg can be chosen to satisfy the following (a) ij = g I x l (b) if dim (R) 2 4 then a i s obtained by adding

handles to R x [0 11along zero o r one-sphees i n Ro x 1 (c) if dim (R) =

3 then a is obtained by first adding one and two-handles to R x [Ol] a t R 9 1 and then performing surgeries along trivial one-spheres on the interior of the resulting cobordism (d) if dim (R) 5 2 ais obtained by adding one-handles to R x [Ol] a t Ro x 1

333 PATCH SPACES

Proof I t is discussed in [31] how to do zero and one-surgeries and surgeries on trivial spheres of any dimension on a Poincark duality space of dimension 2 4 But this is not known for 0-3-dimensional Poincar6 duality spaces this is one of the reasons we require the differentiability of low dimensional spaces in our definition of Lh(n(X Y))

Now (a) + (b) is documented in the literature (see [32] [ 5 ] [7]) (a) + (c) and (a) + (d) require an examination of low dimensional (5 3)

surgery problems having differentiable manifolds both as domain and range Since I am unable to locate such a discussion in the literature here it is

In dimensions O1 surgery can always be completed Let f (M aM) - (N aN) be a normal map a t dimension two restricting

to an equivalence f I oM-aM Complete surgery on the inverse image of each zero or one-simplex in M there is an obstruction o(f) E H(N dN) 2) to choosing a completion that extends to all the two-simplices of N o(f) is a normal cobordism invariant and we can always arrange that a(f ) = 0 by varying f I through a normal cobordism having oN x [Ol] for range

Let f (M oM) --t (N oN) be a three-dimensional normal map with f I a homotopy equivalence and g N - K(n 1) for fundamental group There is a surgery obstruction in o(f) E Lh(n) and if a(f ) = 0 then zero and one surgeries can be done on M - oM and simultaneously on N - bN if need be to make f M -N a Z(n)-homological equivalence (see Chapter 6 in [32])

Now we examine which obstructions a(f ) E L(n) can be realized by a 3-dimensional map f (M aM) -(N dN) having differentiable manifolds both for domain and range We show how to realize these obstructions by the standard plumbing techniques

Define a set of splitting invariants for f as follows represent each element of a given 2-basis of H(K(n I ) 2) by a map ri K(z1) K(Z I ) -+

the composites

truncate to ti in transverse position to R P 4 c RP homotopy f I t o a diffeomorphism then put f in transverse position to the t(RP4) and let Si(f) E Zz denote the obstruction to completing surgery on f f -l(t(RP4)) - t(RP4) Despite their possible ambiguity the usefulness of the Si(f) is this if

The Si(f) are not necessarily surgery cobordism invariants

334 LOWELL JONES

S ( f )= 0 for all i then o ( f )= 0 Here is how to see this Let a ( f )E

H2((NbN)2) denote the obstruction to extending a completion of surgery on f f - (N) -N to the two-skeleton S ( f )= 0 assures that cap product o ( f )n t(RP4) vanishes Extend t(RP4)to a basis t(RP4)U b for H2((N dN) Z) and choose embeddings C j S x D2-N to represent classes dual to the b j which when composed with g N - Kjn 2) are null homotopic possibly zero-surgeries may first have to be done on N to make g n l (N)-n surjective Use the lemma in the appendix to $ 4 to put f in transverse position to the Cj(S1x D making f f-(Cj(S1x D7)-C(S1 x D2) a diffeomorphism for all j Each of these transversalities can be varied through a two-dimensional surgery obstruction if need be so tha t after surgeries are performed simultaneously on the Cj(S1x D and f -(Cj(S1 x D) the obstruction b ( f )E H2((N dN) 2)becomes zero Since Lh(l)= 0 there is no obstruction now to copying all simplices of N by f (M bM) -(N dN) up to 2-homology equivalence

In summary I t has been shown that the subgroup A of Lk(n) repre-sented by three dimensional surgery problems having differentiable manifolds for domain and range is a 2-vector subspace of Hl(K(nI ) 2) If H1(K(nI ) 2) has no 2-torsion then it is isomorphic with Hl(K(nI ) 2)

Here is how to realize all such surgery obstructions Choose a diffeo-morphism f (M bM) --+ ( N bN) so tha t dN- N induces a surjection Hl(bN2)-Hl(N2)then the t(RP4)n bN) form a basis for a Z-vector subspace of Hl(dN2) Vary f dM -dN through any normal cobordism

with oN = dN x [0 11 so that f are diffeomorphisms and f captures the S( f ) on the (t(RP4)n b ~ )[ O I ] Extend this to a normal x cobordism for all of f

6 The existence and uniqueness of patch space structures on Poincark duality spaces

Definition 61 A patch space structwre on the Poinear6 duality pair ( X Y ) is a homotopy equivalence h (P dP) -( X Y ) from a patch space pair (P dP) Two patch space structures

(P PI h

gt(x Y )

h (P bP)

are equivalent if there is a patch H-cobordism h (W V) - ( X Y ) from h to h

335 PATCH SPACES

THEOREM62 Let ( X Y ) be an oriented Poincare duality pair so that X has formal dimension 2 5 Then any patch space structure on Y extends to a patch space structure for X I f dim ( X ) = 4 then any patch space structure on Y extends to one on

r-fold

for some finite r

COROLLARY Let ( X Y ) be as in Theorem 62 so that Y also has TO 62 formal dimension 2 5 Then ( X Y ) has a patch space structure unique u p to equivalence class

Before proving Theorem 62 we derive the following consequences

THEOREM63 Surgery in the Poincare duality category makes sense for spaces having formal dimension 1 6 or for spaces having formal dimension 2 5 which have no boundary I f one i s willing to stabilize with connected sums of S 2 x S 2 then these dimensions can be improved to 5 4 respectively

Let (M bM) be a Poinear6 pair and X a space containing the subspace Y c X with the spherical bundle 4 -Y for normal bundle in X Suppose fM -X restricted to dM is in Poinear6 transverse position to Y c X Finally suppose either dim ( A )1 4 or A has a patch space structure holds for A = M aM or f- la ( Y )

THEOREM 64 There is an obstruction to transversality t ( f )E

Lk-(n(X Y ) ) which is a cobordism invariant t ( f ) = 0 i f and only i f there is a cobordism 7(WV )-X of fi (M bM) -X with f l a _ J in Poincare transverse position to c X

Proof of Theorem 64 Consider first the case when has a BO reduction Apply Theorem 62 and 5 5 To see that the transversality obstruction t ( f ) computed in this way is independent of the BO-reduction it suffices to consider the case where t ( f ) = 0 for one reduction y and t l ( f )+ 0 for another y (t( ) is additive under connected sum) There is the commutative diagram

where N is the normal bundle to f - (Y ) in M and Na its boundary h and h are mappings of spherical fibrations g a fiber homotopy equivalence and g

336 LOWELL JONES

the pull back of g Note that t(f) = 0 -- t(h) = 0 and t(f) 0 - t(h) 0 But

(t(r) = 0 for any homotopy equivalence r) In general we proceed as follows If 6 -Y is an I-spherical fibration for

I 5 1it has a BO-reduction So suppose I 2 2 Let ( 2 ~ ) denote the regular neighborhood with boundary for Y in X corresponding to its normal disc bundle We consider only the case bM = 8 with no loss of generality Pu t f M -X in transverse position to 2 Let Y -Z denote the stable inverse E to 5 pulled back to Z and y -f- (2) its f-pull back Now there is an

f obstruction t(f ) to putting (y dy) -(r dy) in transverse position to Y c y Set t(f) = t(f ) If t(f ) = 0 let g (W V) -+ (7 bhl) denote a cobordism with g la- a mapping of spherical fibrations The splitting r =

s E pulls back to 8- W = P 6 Now gl__ is in transverse position to 5 c y a stabilization by inverse trick as used before extends g g-la-li ((E6)) - (5 [) to a connecting cobordism a (C C)4(E [) Now there is a cobordism f of f with M x [0 11 Ua- C x 1 for domain such tha t d-f is in transverse position to Y QED

Proof of Theorem 62 We consider only the case where Y = 8 and leave for the reader the task of filling in the remaining details

Let R denote the boundary of a differentiable regular neighborhood of X in S- (r = large) Let c R -X be the natural collapsing map Then T -+ X pulls back to a spherical fibration c(z) -R where z -X is the BF-bundle which is the stable inverse for the Spivak fibration z -+ X c(z) is a Poinear6 duality space Denote its Spivak fibration by 7 -+ c(z) Note that c(z) is homotopy equivalent to (D x X ) U L where (L L) is a Poincark duality pair and g dD x X- L is a homotopy equivalence Since c(z) is a spherical fibration over the differentiable manifold R a handle body filtration for R induces a filtration for c(T) Arguing by induction on this filtration we can construct a surgery problem with patch space for domain and c(z) for range Using the transversality lemmas of $5 this surgery problem could be put in patch transverse position to the real codimension subspace (o x X c D x X U L) -- c(z) if a certain obstruction to transversality vanished giving a surgery problem with patch space for domain and o x X -- X for range Then but for a surgery obstruction surgery could be completed to give a patch space structure to X

337 PATCH SPACES

There are two points to be elaborated Firstly what is the process for extending the patch space structure step by step over the handles of R Let R-enote the k-handle body of R Suppose C(zl) l R k has a patch space structure (P bP) Pu t the equivalence bP -- hC(z)l a R k in patch transverse position to each C(zl) ISkxaDl where Skx Dz is the framed core sphere for some (k + 1)-handle of R Using $4 there is no loss in assuming that

D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r

So an Eilenberg (surgery) obstruction argument applies to copy the cubes of h-(C(zl) I s k x ( D ~ a n l ) ) under

Glue the differentiable manifolds c(zl)lk-lXl to P along these h-I and taper the various patches near c(zl) ISkx l to obtain a patch space structure for the resulting space (P dP) There is a natural map h (P dP) -- C(Z)I R k - l a R k + ~ ) which can be taken as a patch space normal map So after completing surgery we have a patch space structure for ~ ( z ) l ~ k + ~ This argument works for k 5 dim (R) - 4 c(zl)1 -diln has a differentiable R-3)

structure so any patch space structure for c(zl) IRdim (R ) -3 can be joined with this patch space as before to obtain a patch space normal map P-C(zl) After varying the joining map through a surgery cobordism if need be surgery can be completed to give c(z) a patch space structure

The second point which needs elaboration is the vanishing of the trans- h

versality obstruction for the equivalence P --+ D x X U L If the inclusion dX- X induced an isomorphism n(bX) E r l (X) then t(f) = 0 so transversality yields a normal map from a patch space pair to (X dX) For the case a t hand bX = 8there is a normal map of Poinear6 spaces X -X where X = X U X with XI a differentiable manifold and dXc X in-ducing an isomorphism n(dX) = r1(X1) (See 233 in [31]) Thus there is normal map from a patch space to X this joins with the manifold X along ax= dX as described in the previous paragraph to give a patch space normal map with X for target Combining with X -X we obtain the normal h P-+X If need be a codimension zero submanifold in a top dimensional cube of P can be picked out its boundary joining map varied through a surgery obstruction and then this new space glued back in along the resulting boundary equivalence after which the surgery obstruction a(h)will vanish


LOWELL J O N E S

(70) S u r g e r y composi t ion f o r m u l a A Poinear6 blocked space E is finite CW complex partitioned by subcomplex pairs (Mi aM)) such tha t each (M bM) is a Poincar6 duality pair and dM equals the union of elements in (Mi aMi)) There is a surgery group L(E n) every element of which is represented by a blocked Poincar6 normal map modeled geometrically on the partitioning of 5 having n for global fundamental group (see [ I l l for more details)

If a a are blocked normal maps representing elements of Lh(En) such that range (a)= domain (a)then let a 0 a denote the composition of these blocked normal maps

Proof Proceed as in Remark 318 above

(71) SURGERY PRODUCT THEOREM F o r each Poincare blocked space p modeled o n 5 hav ing global f u n d a m e n t a l g r o u p p -K(n I ) there i s a n i n d e x I) E Lh(5 n ) defined a s in the last a p p e n d i x of [32] I ( p ) i s represented by the product of p w i t h a n 8-dimensional n o r m a l m a p g M + S 8 where i n d e x ( M ) = 8

If f p -pz i s a n o r m a l m a p represen t ing a E Lh(E n ) t h e n

I(b-1)- I(B2) = a 1

holds in L$(E n ) where a denotes the i m a g e of a u n d e r the h o m o m o r p h i s m x M

LXE n)-Lk+(ETI Following Wall this formula is proven from (70) above and commuta-

tivity of

p x ~ 1 8 p x s8

Note that x M equals eight times the periodicity isomorphism (see 12 in [4]) Let p be modeled on E and have global fundamental group p--

K(n 1)

PROPOSITION71 I f I) = 0 t h e n the i m a g e of

x 8L(E n ) -L(E x E n x n) h a s exponen t 8

Many of the applications given here were independently noted by N Levitt [El N Levitt with J Morgan or F Quinn [19]

PATCH SPACES 339

Proof Let f p -p represent x E Lh(5n) Since I) = 0 we have I(pi x 6) = 0 ( i = 1 2) as well QED

(72) Computing cobordism classes of Poincare duality spaces Results of the form given here were first noticed by Norman Levitt [15] for the simply connected case Levitt reduced the following proposition to a plausible conjecture

PROPOSITION72 There i s a long exact sequence t -L)(a(X))-IFQsF(X)lnj(X A TSF) -2L-(n(~)) -

beginning a t the term Lk(n(~)) Proof tj(y) is the obstruction to putting y in transverse position to

X A B S F c X A TSF cj is the usual Thom construction map Let f M- X be a null cobordant map from a differentiable j-manifold

inducing an isomorphism of fundamental groups Split M as M U + M-i

so that n(bM+) -n(M) is surjective Vary the equivalence bM -- bM-through a surgery cobordism representing a E L(n(X)) then glue the ex-tended M+ to M- along the new equivalence to obtain P Set r j(a) = P

QED

(73) Representing homology classes by Poincare duality spaces

PROPOSITION73 Any y E H(X Z ) i s the image of some element i n Q(X)

Proof It is established in [2] [18] that the spectrum T S F has the homotopy type of a wedge of Eilenberg-Maclane spaces Let fK(Z n) -TSF(n) induce an isomorphism of n th homology groups Represent a homo-Iogyc lassy inX by gSXAK(Zn) Put (1A f ) o g = S k - X A TSF(n) in transverse position to X x BSF(n) c X A TSF(n) to obtain a Poincark bordism element P- X representing y the possible obstruction to trans-versality can always be localized in a disc of X so it causes no homological difficulties QED

(74) Characteristic variety theorem and homotopy type of BSF Here

A slightly weaker result was announced a t the same time by L Jones with J Paulson [121

If each B is replaced by the group of patch space cobordism classes then this exact sequence can be completed to the right The low-dimensional surgery groups involved in the completed exact sequence have been discussed in the appendix to sect 5 above

For spaces not simply connected the same representation works for pairs ( X Y )with m(Y)zm ( X )

340 LOWELL JONES

we follow the ideas of Thom [28] and Sullivan [25]~[26] It is known that there is a natural equivalence of graded functors

H( MS) Z sH ( QSO Z)

which commutes with the Hurewicz maps H ( S) -H ( MSO)

H( S) -H( QS) (see [6] or p 209 in [24]) There is a mapping

H(X MSO) Z x [X BSF] A L(l))

defined as follows choose a Poincark thickening of X X which has y E [XBSF] for Spivak fibration Let bXaX -+ MSO(n) be the Poincar6 dual of a E H(X MSO) Z Then 9(a y) denotes the obstruction to putting 62 in transverse position to BSO(n) cMSO(n) In a like manner pairings

H(X MSO A M) x ~(l))[X BSF] (2)

are defined A (BSF) characteristic variety for X (mod odd-torsion) is a finite subset

Vc U H(X MSO A M) U H(X MSO) Z

satisfying y E [X B S F ] is odd-torsion in [X BSF][X TOP] if and only if g(x y) = 0 for all x E V Here M is the co-More space obtained by gluing a 2-disc to S along a degree-2 map

PROPOSITION Any finite CW complex X has a characteristic variety 75

PROPOSITION76 For each n 13 there exists a map

(BSF(n) BSTOP(n)) ---+ X P K(Z 4 i + 1)x X I_K(Z4j+ 3)

inducing a n isomorphism of homotopy groups modulo odd-torsion

COROLLARY There is a universal class 77 u4+l u4+3 (BSF 2) H4-3(BSF 2) ~ 4 - c l

satisfying any f X - B S F has a BSTOP reduction modulo odd-torsion if and only if f (u4+)= 0 = f ( u ~ - ~ )

Proof of Proposition 75 Essentially the same as the proof given in [25] [26] for the (FPL) -characteristic variety theorem

7 5 gives a geometric criterion modulo odd-torsion for determining when a Poincari duality pair (P 8P) has the homotopy type of a manifold namely if and only if all maps M Z ~ A ( P aP)+ T ( f )into the Thom space of linear bundles can be put in transverse position to the zero section There is also a relative version of this criterion A criterion a t the odd primes has previously been given [27] namely that (P aP) satisfy FPL-Poincari duality

PATCH SPACES

Proof of Proposition 76 The mapping 9defined above gives

H((BSF BSTOP) Z) L1-l(l)) OzZ) A computation shows that s composes with the Hurewicz map to give an isomorphism

n(BSF BSTOP) E Lk-(l)OZZ modulo odd-torsion Choose

f (BSF BSTOP) -+ x K(Lk-1(1)OzZ2) k )

representing X s Then f satisfies Proposition 76 QED

Proof of Corollary 77 This is so because of Proposition 75 and the following observation Let (X a x ) be a Poinear6 pair The obstruction to transversality t(y) for any y E [(XdX) x M MSO(j)] has order a divisor of 23 if dim (X) - j = -1(4) The point being that t(y) is the diference of the indices of two Z-manifolds both of which bound Z-Poincar6 spaces although this difference vanishes mod 2 by varying through a framed cobordism the surgery obstruction can only be changed by a multiple of P3

QED

(711) Patch space structures f o r spaces which are Poincare mod a set of primes Let S denote a set of prime integers and Sthe complement to S We define the category C W Objects in C W are finite C W complexes maps are all continuous maps A homotopy in CW from h X -+ Y to h X -+ Y consists of a commutative diagram

where each vertical arrow must induce a mod Shomotopy equivalence - in the C W category - and an isomorphism of fundamental groups

The following is a straightforward deduction from the Serre-Hurewicz theorem

342 LOWELL JONES

h i s a C W( homotopy equivalence t h e n h i s C W homotopic t o a m a p w h i c h pu l l s back to X T h e d o m a i n of both t h i s homotopy a ~ z d i t s posit ive end c a n be chosen w i t h d i m e n s i o n 5 dim (L)

X is a C W Poincar6 space if it satisfies Poincar6 duality for the coefficients Z(II(X)) where Z denotes the integers localized a t S There is also the notion of Z-homology spherical fibrations (see [ I l l for a complete discussion) X is a CW Poinear6 space if and only if the fibration associated to it by the Spivak construction is a Z-homology spherical fibration

A C W(s patch space P is defined as in $ 1with the following variations (1) The differentiable charts hi Mi -+Pi when restricted to each cube

need only be C W homotopy equivalences (2) If zi is a BO reduction for the C W Spivak fibration z and gi =

Sk-f T(zi) is a degree one map mod S then surgery can be completed on gil(P) -Pi to give a C W differentiable chart for Pi

The reason (2) above is hypothesized is this Let W be the C W patch H-cobordism connecting P to P where P is obtained from P by replacing a patch with an associated differentiable chart To see that W has a set of differentiable charts we use (2) Let Wi be the cobordism of P Then some positive multiple 1 zi extends to a BO reduction ti for z where 1 is prime to S Let

h (Skx [07 11 Skx 0 SkX 1)-(T(Ei) T(Ei IPi) T(Ei

be a degree one map mod S Then h lk yields a differentiable chart for Pi (see (2)) which -since the cubical homology of ( Wi Pi) vanishes -extends by surgery to a chart for all of Wi

The patch engulfing lemma (see 9 3 above) holds in C W so surgery can be performed on C W Poinear6 normal maps which have patch spaces for domain surgery obstructions lie in Lh(Z(I3)) Note that (2) above is pre- served under the surgery process described in $3 above

The results of 994 and 5 go through in CW Note that a patch space coming from real codimension transversality as described in the proof of 51 above can always be chosen to satisfy Theorem 41 (b) (c) Such patch spaces also satisfy (2) above

712 homotopy equivalent to a C W patch space the patch H-cobordism class of w h i c h i s u n i q u e l y determined by the homotopy t y p e of X

THEOREM A n y CW Poincare space X of d i m e n s i o n 2 5 i s C W

I t is just an exercise now to show that C W version of Theorems 63 64 and 70-72 hold

PATCH SPACES 343

BIBLIOGRAPHY

[ I ] W BROWDER Homotopy type of differentiable manifolds Colloq on Alg Top Aarhus 1962 42-46

[ 2 ] -- A LIULEVICIUS Cobordism theories Ann of Mathand F P PETERSON 84 (1966) 91-101

[ 3 1 - PoincarC spaces their normal fibrations and surgery Princeton 1971 (mimeographed)

[ 4 ] ------ and G ERUMFIEL A note on cobordism of Poincare duality spaces Bull Amer Math Soc 77 (1971) 400-403

[ 5 ] S E CAPPEL and J L SHANESON On four dimensional surgery and applications Comm Math Halv (to appear)

[ 6 ] P CONNER and E FLOYD Diferentiable periodic maps Springer-Verlag (1964) Berlin [ 7 ] T FARRELLand W C HSIANG Manifolds with l=GxaT (to appear) [ 8 ] A HAEFLIGER and V POENARULa classification des immersions combinatories Inst

des Hautes Etudes Sci No 23 (1964) [ 9 ] M W HIRSCH Immersions of manifolds Trans Amer Math Soc 93 (1959) 242-276 [lo] W C HSIANG A splitting theorem and the Kzinneth formula in algebraic K-theory

algebraic K-theory and its applications Springer-Verlag (1969) 72-77 [ l l ] L E JONES Combinatorial symmetries of the m-disc Notes a t Berkeley (1971) [12] ------ and J Paulson Surgery on Poincari spaces and applications Notices of Amer

Math Soc 17 (June 1970) 688 [13] M A KERVAIRE and J W MILNOR Groups of homotopy spheres I Ann of Math 77

(1963) 504-537 [14] R KIRBY and L SIEBENMANOn the triangulation of manifolds and the Haupt vermutung

Bull of the Amer Math Soc 75 (1969) 742-749 [15] N LEVITT Generalized Thom spectra and transversality for spherical fibrations Bull

Amer Math Soc 76 (1970) 727-731 [16] ----- On the structure of Poincari duality spaces Topology 7 (1968) 369-388 [17] S P N o v ~ ~ o v Homotopy equivalent smooth manifolds I Izv Akad Nauk SSSR Ser

Mat 28 (2) (1964) 365-474 [18] F P PETERSON and HIROSI TODA On the structure of H(BSF Z p ) J Math Kyoto

Univ 7 (1967) 113-121 [19] F QUINN Surgery on Poincarii and normal spaces Bull Amer Math Soc 78 (1972)

262-267 [20] C P ROURKEand E J SANDERSON of Math 87Block bundles I 11 Ann (1968) 1-28

256-278 [21] L SIEBENMANN Thesis Princeton University (1965) [22] MSPIVAK Spaces satisfying Poincarii duality Topology 6 (1967) 77-102 [23] J STASHEFF A classification theory for fiber spaces Topology 2 (1963) 239-246 [24] R STONG Notes on cobordism theory Princeton University Press (1968) [25] D SULLIVANThesis Princeton University (1966) [26] ------ Triangulating and smoothing homotopy equivalences and homeomorphisms

Princeton (1967) (mimeographed) [27] - Proceedings of the conference on topology of manifolds held August 1969 a t

University of Georgia [28] R THOM Les classes caract6ristiques de Pontrjagin des variitamps trianguliies Symposium

Internacional de Topologia Algebraica Mexico 1958 [29] J B WAGONER Smooth and PL surgery Bull Amer Math Soc 73 (1967) 72-77 [30] ----- Thesis Princeton University (1966) [31] C T C WALL Poincari complexes I Ann of Math 86 (1967) 77-101 [32] - Surgery on compact manifolds Academic Press New York [33] E-G ZEEMAN Seminar on combinatorial topology Inst des Hautes Etudes Sci (1963)

(Received January 1971) (Revised January 1978)

Patch spaces A geometric representation

for Poincare spaces

TABLEOF CONTENTS

Section Page Preface 306

1 Preliminaries 307 2 Elementary properties 310

sect 3 Surgery on patch spaces 311 sect 4 Changing patch structures 323 sect 5 Obstruction to transversality 329 sect6 The existence and uniqueness of patch space structures on

Poincar6 duality spaces 334 sect 7 Applications 338

(70) Surgery composition formula (71) Surgery product theorem (72) Computing cobordism classes of Poincar6 spaces (73) Representing homology classes by Poincari spaces (74) Characteristic variety theorem and homotopy type of BSF (711) Patch space structures for spaces which are Poincari mod

a set of primes

Preface

Here is the theme of this paper Given any geometric construction C defined for differentiable manifolds there is a single functorial surgery obstruction S which measures the obstruction to extending C to Poincar6 duality spaces

If C = surgery techniques then S = 0 ie surgery techniques extend to Poincar6 spaces (see 5 3) This was first announced by L Jones with J Paulson 1121 Significant results in this direction had previously been published by N Levitt 1161

If C = transversality then S is made explicit in 5 5 below As an application (see 72) let C denote the classical reduction of the differ-

entiable bordism group (oriented or unoriented) to a homotopy group Analy- zing S (which is just a transversality obstruction) one sees that up to an extension or quotient by one of Z Z there are isomorphisms QfP zr j (TBSF) Lj(l) QT E r j (TBF) for all j 2 S Here Q Qz are the oriented unoriented bordism groups of Poincar6 duality spaces TBSF T B F are the associ-

307 PATCHSPACES









308 LOWELL JONES









309 PATCH SPACES













Pj Ph -- P - Pj




310 LOWELL JONES










311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















307 PATCHSPACES









308 LOWELL JONES









309 PATCH SPACES













Pj Ph -- P - Pj




310 LOWELL JONES










311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















308 LOWELL JONES









309 PATCH SPACES













Pj Ph -- P - Pj




310 LOWELL JONES










311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















309 PATCH SPACES













Pj Ph -- P - Pj




310 LOWELL JONES










311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















310 LOWELL JONES










311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















311 PATCH SPACES








f 7 - -z



312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















312 LOWELL JONES



gisk-P






PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 313









h (MdM) - aAljTgt



LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















LOWELL JONES







315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















315 PATCH SPACES















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















316 LOWELL JONES











PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 317









318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















318 LOWELL JONES










PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES



h h-(U D) -USESDik


h-I u(u+n D ~ )-h-(U (U+n D2))










320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















320 LOWELL JONES








(P 8P) ( X Y )

PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 321



I foh I


(K3(P) AP ) -- ~ ( fL(E(X))



322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















322 LOWELL JONES


(PlP ) ---L (PI 8P1) -(XY) ( X Y )






Complete surgery on


PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 323



formula a 1 9) = 43) + o(g2)





DC(P) DCk-l(P)







324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















324 LOWELL JONES








325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















325 PATCH SPACES












326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















326 LOWELL JONES



d ~ r )c(P)--( pi-+)C~P P n a~






PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES


and










328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















328 LOWELL JONES










Appendix to $ 4


329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















329 PATCH SPACES












330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















330 LOWELL JONES









PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCHSPACES 331










332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















332 LOWELL JONES










Appendix to 55







333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















333 PATCH SPACES









the composites



334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















334 LOWELL JONES








(P PI h

gt(x Y )

h (P bP)


335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















335 PATCH SPACES


r-fold

for some finite r









336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















336 LOWELL JONES








337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















337 PATCH SPACES


D H ( ~ - I ( C ( ~ ~ ( p t Z) I~z)) r







LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















LOWELL J O N E S






I(b-1)- I(B2) = a 1



tivity of

p x ~ 1 8 p x s8


K(n 1)




PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 339








QED








340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















340 LOWELL JONES



















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES







QED




342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















342 LOWELL JONES














PATCH SPACES 343

BIBLIOGRAPHY





















PATCH SPACES 343

BIBLIOGRAPHY





















Documents

Patch Spaces: A Geometric Representation for Poincare ...v1ranick/papers/patches.pdf · Patch spaces were invented by the author for use in his study "Combi- natorial symmetries of