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Mapping Degree Theory

http://dx.doi.org/10.1090/gsm/108

Mapping Degree Theory

Enrique OutereloJesús M. Ruiz

American Mathematical Society Providence, Rhode Island

Real Sociedad Matemática Española Madrid, Spain

Graduate Studies in Mathematics

Volume 108

Editorial Board of Graduate Studies in Mathematics

David Cox, Chair

Steven G. Krantz Rafe Mazzeo Martin Scharlemann

Editorial Committee of the Real Sociedad Matematica Espanola

Guillermo P. Curbera, Director

Luis Alıas Alberto ElduqueEmilio Carrizosa Pablo PedregalBernardo Cascales Rosa Marıa Miro-RoigJavier Duoandikoetxea Juan Soler

2000 Mathematics Subject Classification. Primary 01A55, 01A60, 47H11, 55M25, 57R35,58A12, 58J20.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-108

Library of Congress Cataloging-in-Publication Data

Outerelo, Enrique, 1939–Mapping degree theory / Enrique Outerelo and Jesus M. Ruiz.

p. cm. — (Graduate studies in mathematics ; v. 108)Includes bibliographical references and index.ISBN 978-0-8218-4915-6 (alk. paper)1. Topological degree. 2. Mappings (Mathematics). I. Ruiz, Jesus M. II. Title.

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10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09

To Jesus Marıa Ruiz Amestoy

vii

Contents

Preface ix

I. History 1

1. Prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Inception and formation . . . . . . . . . . . . . . . . . . . . . 14

3. Accomplishment . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4. Renaissance and reformation . . . . . . . . . . . . . . . . . . . 35

5. Axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6. Further developments . . . . . . . . . . . . . . . . . . . . . . . 42

II. Manifolds 49

1. Differentiable mappings . . . . . . . . . . . . . . . . . . . . . . 49

2. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . 53

3. Regular values . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4. Tubular neighborhoods . . . . . . . . . . . . . . . . . . . . . . 67

5. Approximation and homotopy . . . . . . . . . . . . . . . . . . 75

6. Diffeotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7. Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

III. The Brouwer-Kronecker degree 95

1. The degree of a smooth mapping . . . . . . . . . . . . . . . . . 95

2. The de Rham definition . . . . . . . . . . . . . . . . . . . . . . 103

3. The degree of a continuous mapping . . . . . . . . . . . . . . . 110

viii Contents

4. The degree of a differentiable mapping . . . . . . . . . . . . . 114

5. The Hopf invariant . . . . . . . . . . . . . . . . . . . . . . . . 119

6. The Jordan Separation Theorem . . . . . . . . . . . . . . . . . 124

7. The Brouwer Theorems . . . . . . . . . . . . . . . . . . . . . . 133

IV. Degree theory in Euclidean spaces 137

1. The degree of a smooth mapping . . . . . . . . . . . . . . . . . 137

2. The degree of a continuous mapping . . . . . . . . . . . . . . . 145

3. The degree of a differentiable mapping . . . . . . . . . . . . . 153

4. Winding number . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5. The Borsuk-Ulam Theorem . . . . . . . . . . . . . . . . . . . . 160

6. The Multiplication Formula . . . . . . . . . . . . . . . . . . . 171

7. The Jordan Separation Theorem . . . . . . . . . . . . . . . . . 176

V. The Hopf Theorems 183

1. Mappings into spheres . . . . . . . . . . . . . . . . . . . . . . . 183

2. The Hopf Theorem: Brouwer-Kronecker degree . . . . . . . . . 191

3. The Hopf Theorem: Euclidean degree . . . . . . . . . . . . . . 196

4. The Hopf fibration . . . . . . . . . . . . . . . . . . . . . . . . . 200

5. Singularities of tangent vector fields . . . . . . . . . . . . . . . 206

6. Gradient vector fields . . . . . . . . . . . . . . . . . . . . . . . 211

7. The Poincare-Hopf Index Theorem . . . . . . . . . . . . . . . . 218

Names of mathematicians cited 225

Historical references 227

Bibliography 233

Symbols 235

Index 239

ix

Preface

This book springs from lectures on degree theory given by the authorsduring many years at the Departamento de Geometrıa y Topologıa at theUniversidad Complutense de Madrid, and its definitive form correspondsto a three-month course given at the Dipartimento di Matematica at theUniversita di Pisa during the spring of 2006. Today mapping degree is asomewhat classical topic that appeals to geometers and topologists for itsbeauty and ample range of relevant applications. Our purpose here is topresent both the history and the mathematics.

The notion of degree was discovered by the great mathematicians ofthe decades around 1900: Cauchy, Poincare, Hadamard, Brouwer, Hopf,etc. It was then brought to maturity in the 1930s by Hopf and also byLeray and Schauder. The theory was fully burnished between 1950 and1970. This process is described in Chapter I. As a complement, at theend of the book there is included an index of names of the mathematicianswho played their part in the development of mapping degree theory, manyof whom stand tallest in the history of mathematics. After the first his-torical chapter, Chapters II, III, IV, and V are devoted to a more formalproposition-proof discourse to define and study the concept of degree andits applications. Chapter II gives a quick presentation of manifolds, withspecial emphasis on aspects relevant to degree theory, namely regular val-ues of differentiable mappings, tubular neighborhoods, approximation, andorientation. Although this chapter is primarily intended to provide a reviewfor the reader, it includes some not so standard details, for instance con-cerning tubular neighborhoods. The main topic, degree theory, is presentedin Chapters III and IV. In a simplified manner we can distinguish two ap-proaches to the theory: the Brouwer-Kronecker degree and the Euclideandegree. The first is developed in Chapter III by differential means, with aquick diversion into the de Rham computation in cohomological terms. Wecannot help this diversion: cohomology is too appealing to skip. Amongother applications, we obtain in this chapter a differential version of theJordan Separation Theorem. Then, we construct the Euclidean degree in

x Preface

Chapter IV. This is mainly analytic and astonishingly simple, especiallyin view of its extraordinary power. We hope this partisan claim will beacknowledged readily, once we obtain quite freely two very deep theorems:the Invariance of Domain Theorem and the Jordan Separation Theorem,the latter in its utmost topological generality. Finally, Chapter V is de-voted to some of those special results in mathematics that justify a theoryby their depth and perfection: the Hopf and the Poincare-Hopf Theorems,with their accompaniment of consequences and comments. We state andprove these theorems, which gives us the perfect occasion to take a glanceat tangent vector fields.

We have included an assorted collection of some 180 problems and exer-cises distributed among the sections of Chapters II to V, none for ChapterI due to its nature. Those problems and exercises, of various difficulty, fallinto three different classes: (i) suitable examples that help to seize the ideasbehind the theory, (ii) complements to that theory, such as variations fordifferent settings, additional applications, or unexpected connections withdifferent topics, and (iii) guides for the reader to produce complete proofsof the classical results presented in Chapter I, once the proper machineryis developed.

We have tried to make internal cross-references clearer by adding theRoman chapter number to the reference, either the current chapter numberor that of a different chapter. For example, III.6.4 refers to Proposition 6.4in Chapter III; similarly, the reference IV.2 means Section 2 in Chapter IV.We have also added the page number of the reference in most cases.

One essential goal of ours must be noted here: we attempt the simplestpossible presentation at the lowest technical cost. This means we restrictourselves to elementary methods, whatever meaning is accepted for ele-mentary. More explicitly, we only assume the reader is acquainted withbasic ideas of differential topology, such as can be found in any text on thecalculus on manifolds.

We only hope that this book succeeds in presenting degree theory asit deserves to be presented: we view the theory as a genuine masterpiece,joining brilliant invention with deep understanding, all in the most accom-plished attire of clarity. We have tried to share that view of ours with thereader.

Los Molinos and Majadahonda E. Outerelo and J.M. RuizJune 2009

225

Names of mathematicians citedAleksandrov, Pavel Sergeevich

(1896–1982), 32Amann, Herbert, 39

Bernstein, Sergei Natanovich(1880–1968), 34

Betti, Enrico (1823–1892), 13Bohl, Piers (1865–1921), 13Brouwer, Luitzen Egbertus Jan

(1881–1996), 14Brown, Arthur Barton, 35

Caccioppoli, Renato (1904–1959), 34Cauchy, Augustin Louis

(1789–1857), 2Cech, Eduard (1893–1960), 30

Deimling, Klaus, 40Dyck, Walter Franz Anton

(1856–1934), 31Dylawerski, Grzegorz, 46

Elworthy, K. David, 43

Fuhrer, Lutz, 38Fuller, F. Brock, 45

Gauss, Karl-Friedrich (1777–1855), 2Geba, Kazimierz, 43

Hadamard, Jacques Salomon(1865–1963), 14

Heinz, Erhard, 38Hermite, Charles (1822–1901), 7Hopf, Heinz (1894–1971), 28Hurewicz, Witold (1904–1956), 30

Ize, Jorge, 43

Jacobi, Carl Gustav Jacob(1804–1851), 7

Jodel, Jerzy, 46Jordan, Camille (1838–1922), 15

Kronecker, Leopold (1823–1891), 7

Leray, Jean (1906–1998), 33Liouville, Joseph (1809–1882), 2

Marzantowicz, Waclaw, 46Massabo, Ivar, 43Miranda, Carlo, 28

Nagumo, Mitio, 35Nirenberg, Louis, 43

Ostrowski, Alexander (1893–1986), 2

Picard, Charles Emile (1856–1941),12

Poincare, Jules Henri (1854–1912),11

Pontryagin, Lev Semenovich, 43

de Rham, Georges (1903–1990), 37Riemann, Friedrich Bernhard Georg

(1826–1866), 13Romero Ruiz del Portal, Francisco,

45Rothe, E., 29Rybicki, Slawomir, 48

Sard, Arthur (1909–1980), 35Schauder, Juliusz Pawel

(1899–1943), 33Schmidt, Erhard (1876–1959), 28

226 Names

Schonflies, Arthur Moritz(1853–1928), 15

Schwartz, Laurent (1915–2002), 38Smale, Stephen, 42Sturm, Jacques Charles Francois

(1803–1855), 2Sylvester, James Joseph

(1814–1897), 7

Thom, Rene (1923–2002), 42Tromba, Anthony J., 43

Veblen, Oswald (1880–1960), 15Vignoli, Alfonso, 43

Weierstrass, Karl Theodor Wilhelm(1815–1897), 10

Weiss, Stanley A., 39

227

Historical references

We list below the references mentioned in Chapter I. We also include threebasic papers on the history of degree theory: [Siegberg 1980a], [Siegberg1980b], and [Mawhin 1999].

[Adams 1969] J.F. Adams: Lectures on Lie Groups. New York-Amsterdam: W.A.Benjamin, Inc., 1969.

[Aleksandrov-Hopf 1935] P. Aleksandrov, H. Hopf: Topologie. Berlin: Verlag vonJulius Springer, 1935.

[Amann-Weiss 1973] H. Amann, S.A. Weiss: On the uniqueness of topologicaldegree. Math. Z. 130 (1973), 39–54.

[Banach 1922] S. Banach: Sur les operations dans les ensembles abstraites et leursapplications aux equations integrales. Fundamenta Math. 3 (1922), 133–181.

[Bohl 1904] P. Bohl: Uber die Bewegung eines mechanischen Systems in der Naheeiner Gleichgerwichtslage. J. Reine Angew. Math., 127 (1904), 179–276.

[Brouwer 1912a] L.E.J. Brouwer: Uber abbildung von Mannigfaltigkeiten. Mat.Ann. 71 (1912), 97–115.

[Brouwer 1912b] L.E.J. Brouwer: On looping coefficients. Konink. Nederl. Akad.von Wet. Proc. 15 (1912), 113–122.

[Brouwer 1912c] L.E.J. Brouwer: Continuous one-one transformations of surfacesin themselves. CNAG Proc. 15 (1912), 352–360.

[Brouwer 1976] L.E.J. Brouwer: Collected works, vol. II. Geometry, analysis,topology and mechanics. H. Freudenthal (ed.). Amsterdam-Oxford: NorthHolland, 1976.

[Brown 1935] A. B. Brown: Functional dependence. Trans. Amer. Math. Soc. 38(1935), 379–394.

[Caccioppoli 1936] R. Caccioppoli: Sulle corrispondenze funzionalli inverse dira-mata: teoria generale e applicazioni ad alcune equazioni funzionali non lin-eari e al problema di Plateau, I & II. Rend. Accad. Naz. Linzei 24 (1936),258–263 & 416–421.

228 Historical references

[Cauchy 1837a] A.-L. Cauchy: Calcul des indices des fonctions. J. EcolePolytechnique XV (1837), 176–226.

[Cauchy 1837b] A.-L. Cauchy: Extrait d’une lettre sur une memoire publie aTurin, le 16 Juin 1833, et relatif aux racines des equations simultanees.Comptes Rendus Acad. Sc. Paris IV (1837), 672–675.

[Cauchy 1855] A.-L. Cauchy: Sur les compteurs logarithmiques. Comptes RendusAcad. Sc. Paris XL (1855), 1009–1016.

[Cech 1932] E. Cech: Hoherdimensionale Homotopiegruppen. Verhandlungen derInternationalen Mathematiken Kongresses, Zurich (1932), 203.

[Dancer 1983] E.N. Dancer: On the existence of zeros of perturbed operators.Nonlinear Anal. 7 (1983), 717–727.

[Dancer 1985] E.N. Dancer: A new degree for S1-invariant gradient mappings andapplications. Ann. Inst. Henri Poincare, Anal. Non-lineaire 2 (1985), 329–370.

[Dancer et al. 2005] E.N. Dancer, K. Geba, S.M. Rybicki: Classification of homo-topy classes of equivariant gradient maps. Fundamenta Math. 185 (2005),1–18.

[Deimling 1985] K. Deimling: Non-linear functional analysis. Berlin: Springer-Verlag, 1985.

[Dyck 1888] W.F.A. Dyck: Beitrage zur Analysis situs I. Aufsatz ein- und zwei-dimensionale Mannigfaltigkeiten. Math. Ann. 32 (1888), 457–512.

[Dyck 1890] W.F.A. Dyck: Beitrage zur Analysis situs II. Aufsatz Mannig-faltigkeiten von n-dimensionen. Math. Ann. 34 (1888), 273–316.

[Dylawerski et al. 1991] G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz: AnS1-equivariant degree and the Fuller index. Ann. Pol. Math. 52 (3) (1991),243–280.

[Elworthy-Tromba 1970a] K.D. Elworthy, A.J. Tromba: Differential structuresand Fredholm maps on Banach manifolds. In Global Analysis (Berkeley,1968) 45–94, Proc. Symp. Pure Math. 15. Providence, R.I.: AmericanMathematical Society, 1970.

[Elworthy-Tromba 1970b] K.D. Elworthy, A.J. Tromba: Degree theory on Banachmanifolds. In Nonlinear Functional Analysis (Chicago, 1968) 75–94, Proc.Symp. Pure Math. 18, part 1. Providence, R.I.: American MathematicalSociety, 1970.

[Fuhrer 1971] L. Fuhrer: Theorie des Abbildungsgrades in endlich-dimensionalenRaumen. Dissertation, Freie Universitat Berlin, 1971.

[Fuhrer 1972] L. Fuhrer: Ein elementarer analytischer Beweis zur Eindeutigkeitdes Abbildungsgrades in Rn. Math. Nachr. 54 (1972), 259-267.

[Fuller 1967] B. Fuller: An index of fixed point type for periodic orbits. AmericanJ. Math. 89 (1967), 133–148.

Historical references 229

[Gauss 1813] K.-F. Gauss: Theoria attractionis corporum sphaeroidicorum el-lipticorum homogeneorum methodo nova tractata. Gottingische gelehrtAnzeigen, April 5 (1813), 545–552.

[Gauss 1833] K.-F. Gauss: Zur mathematischen Theorie der electrodynamis-che Wirkungen. In Werke, 5. Gottingen: Koniglichen Gesellschaft derWissenchaften zu Gottingen (1867), 601–626.

[Geba et al. 1986] K. Geba, I. Massabo, A. Vignoli: Generalized topological de-gree and bifurcation. In Non-linear Functional Analysis and Applications(Proc. NATO Adv. Study Inst., Maratea, Italy, 1985) 55–73, Mathematicaland Physical Sciences 173. Dordrecht: Reidel, 1986.

[Geba et al. 1990] K. Geba, I. Massabo, A. Vignoli: On the Euler characteristicof equivariant gradient vector fields. Boll. Un. Mat. Ital. A(7) 4 (1990),243–251.

[Hadamard 1910] J.-S. Hadamard: Sur quelques applications de l’indice de Kro-necker. In [Tannery 1910], 437–477.

[Heinz 1959] E. Heinz: An elementary analytic theory of the degree of mappingin n-dimensional space. J. Math. and Mech. 8 (1959), 231–247.

[Hopf 1925] H. Hopf: Uber die Curvatura integra geschlossener Hyperflachen.Math. Ann. 95 (1925), 340–367.

[Hopf 1926a] H. Hopf: Abbildungsklassen n-dimensionaler Mannigfaltigkeiten.Math. Ann. 96 (1926), 209–224.

[Hopf 1926b] H. Hopf: Vectorfelder in n-dimensionalen Mannigfaltigkeiten. Math.Ann. 96 (1926), 225–250.

[Hopf 1931] H. Hopf: Uber die Abbildungen der dreidimensionalen Sphare auf dieKugelflache. Math. Ann. 104 (1931), 637–665.

[Hopf 1933] H. Hopf: Die Klassen der Abbildungen der n-dimensionalen Polyederauf die n-dimensionale Sphare. Comment. Math. Helvetici 5 (1933), 39–54.

[Hopf 1935] H. Hopf: Uber die Abbildungen von Spharen auf Spharen niedrigererDimension. Fundamenta Math. 25 (1935), 427–440.

[Hopf 1966] H. Hopf: Ein Abschnitt aus der Entwicklung der Topologie. Jber.Deutsch. Math.-Verein. 68 (1966), 96–106.

[Hurewicz 1935a] W. Hurewicz: Beitrage zur Topologie der Deformationen I.Hoherdimensionalen Homotopiegruppen. Proc. Akad. Wetenschappen 38(1935), 112–119.

[Hurewicz 1935b] W. Hurewicz: Beitrage zur Topologie der Deformationen II.Homotopie und Homologiegruppen. Proc. Akad. Wetenschappen 38 (1935),521–528.

[Hurewicz 1936a] W. Hurewicz: Beitrage zur Topologie der Deformationen III.Klassen und Homologietypen von Abbildungen. Proc. Akad. Wetenschap-pen 39 (1936), 117–126.

230 Historical references

[Hurewicz 1936b] W. Hurewicz: Beitrage zur Topologie der Deformationen IV.Aspharische Raume. Proc. Akad. Wetenschappen 39 (1936), 215–224.

[Ize 1981] J. Ize: Introduction to bifurcation theory. In Differential Equations(Sao Paulo, 1981), 145–203. Lecture Notes in Math. 957. Berlin-New York:Springer-Verlag, 1982.

[Ize et al. 1986] J. Ize, I. Massabo, A. Vignoli: Global results on continuation andbifurcation for equivariant maps. In Non-linear Functional Analysis andApplications (Maratea, Italy, 1985), 75–111. NATO Adv. Sci. Inst. Ser. CMath. Phys. Sci. 173. Dordrecht: Reidel, 1986.

[Ize et al. 1989] J. Ize, I. Massabo, A. Vignoli: Degree theory for equivariant maps,I. Trans. Amer. Math. Soc. 315 (2) (1989), 433–510.

[Ize et al. 1992] J. Ize, I. Massabo, A. Vignoli: Degree theory for equivariant maps,II: The general S1-actions. Memoirs Amer. Math. Soc. 481 (1992).

[Ize-Vignoli 2003] J. Ize, A. Vignoli: Equivariant Degree Theory. de Gruyter Seriesin Nonlinear Analysis and Applications 8. Berlin: Walter de Gruyter & Co.,2003.

[Jordan 1893] C. Jordan: Cours d’analyse de l’Ecole Polytechnique, I. 1893.

[Kronecker 1869a] L. Kronecker: Uber Systeme von Functionen mehrerer Vari-abeln, I. Monatsberichte koniglich Preuss. Akad. Wissens. Berlin, March 4(1869), 159–193.

[Kronecker 1869b] L. Kronecker: Uber Systeme von Functionen mehrerer Vari-abeln, II. Monatsberichte koniglich Preuss. Akad. Wissens. Berlin, August5 (1869), 688–698.

[Leray-Schauder 1933] J. Leray, J.P. Schauder: Topologie et equations fonction-nelles. Comptes Rendus Acad. Sc. Paris 1197 (1933), 115–117.

[Leray-Schauder 1934] J. Leray, J.P. Schauder: Topologie et equations fonction-nelles. Ann. Sc. Ecole Normale Sup. Paris 51 (1934), 45–78.

[Liouville-Sturm 1837] J. Liouville, J.-Ch.-F. Sturm: Note sur un theoreme deM. Cauchy relatif aux racines des equations simultanees. Comptes rendusAcad. Sc. Paris 40 (1937), 720–724.

[Mawhin 1999] J. Mawhin: Leray-Schauder degree: A half century of extensionsand applications. Topol. Methods Nonlinear Anal. 14 (1999), 195–228.

[Miranda 1940] C. Miranda: Un’osservazione su un teorema di Brouwer. Boll. Un.Mat. Ital. (2) 3 (1940), 5–7.

[Nagumo 1951a] M. Nagumo: A theory of degree of mapping based on infinitesi-mal analysis. American J. Math. 73 (1951), 485–496.

[Nagumo 1951b] M. Nagumo: Degree of mapping in convex linear topologicalspaces. American J. Math. 73 (1951), 497–511.

Historical references 231

[Nirenberg 1971] L. Nirenberg: An application of generalized degree to a classof nonlinear problems. In Troisieme Colloque sur l’Analyse Fonctionnelle(Liege, 1970), 57–74. Louvain: Vander, 1971.

[Picard 1891] Ch.-E. Picard: Sur le nombre des racines communes a plusieursequations simultanees. Comptes Rendus Acad. Sc. Paris 113 (1891), 256–358.

[Picard 1891/1905] Ch.-E. Picard: Traite d’Analyse, I and II. Paris: Gauthier-Villars, 1891 and 1893/1905.

[Poincare 1883] J.H. Poincare: Sur certains solutions particulieres du problemedes trois corps. Comptes Rendus Acad. Sc. Paris 97 (1883), 251–252.

[Poincare 1885a] J.H. Poincare: Sur les courbes definies par les equations dif-ferentielles. J. Math. Pures Appl. 1 (1885), 167–244.

[Poincare 1895b] J.H. Poincare: Analysis Situs. J. Ecole Polytechnique (2) 1(1895), 1–123.

[Poincare 1886] J.H. Poincare: Sur les courbes definies par une equation dif-ferentielle. J. Math. Pures Appl. 2 (1886), 151–217.

[Poincare 1899] J.H. Poincare: Complement a l’Analysis Situs. Rend. Circ. Mat.Palermo 13 (1899), 285–343.

[Poincare 1900] J.H. Poincare: Complement a l’Analysis Situs. Proc. LondonMath. Soc. 32 (1900), 277–308.

[Pontryagin 1955] L.S. Pontryagin: Smooth manifolds and their application inhomotopy theory. Amer. Math. Soc. Transl. (2) 11 (1959), 1–114, fromTrudy Mat. Inst. im Steklov 45 (1955).

[de Rham 1955] G. de Rham: Varietes differentiables. Paris: Hermann, 1955.

[Rothe 1936] E. Rothe: Uber Abbildungsklassen von Kugeln des HilbertschenRaumes. Compos. Math. 4 (1936), 294–307.

[Ruiz del Portal 1991] F.R. Ruiz del Portal: Teorıa del grado topologico gene-ralizado y aplicaciones. Dissertation. Madrid: Universidad Complutense,1991.

[Ruiz del Portal 1992] F.R. Ruiz del Portal: On the additivity property of thegeneralized degree. Math. Japonica 37 (1992), 657–664.

[Rybicki 1994] S. Rybicki: A degree for S1-equivariant orthogonal maps and itsapplications to bifurcation theory. Nonlinear Anal. 23 (1994), 83–102.

[Sard 1942] A. Sard: The measure of critical points of differentiable maps. Bull.Amer. Math. Soc. 48 (1942), 883–897.

[Schonflies 1902] A.M. Schonflies: Uber einen grundlegenden Satz der AnalysisSitus. Gott. Nachr. Math. Phys. Kl. (1902), 185–192.

[Siegberg 1980a] H. W. Siegberg: Brouwer degree: History and numerical com-putation. In Numerical solutions of highly nonlinear problems, W. Forster(ed.), 389–411. Amsterdam: North-Holland, 1980.

232 Historical references

[Siegberg 1980b] H. W. Siegberg: Some historical remarks concerning degree the-ory. American Math. Monthly 88 (1981), 125–139.

[Smale 1965] S. Smale: An infinite dimensional version of Sard’s Theorem. Amer-ican J. Math. 87 (1965), 861–866.

[Sylvester 1853] J.J. Sylvester: On a theory of the syzygetic relations of two ratio-nal integral functions, comprising an application to the theory of Sturm’sfunctions, and that of greatest algebraic common measure. Philos. Trans.Roy. Soc. London 143 (1853), 407-562.

[Tannery 1910] J. Tannery: Introduction a la theorie des fonctions d’une variable,t. 2, 2ieme edition. Paris: Hermann, 1910.

[Thom 1954] R. Thom: Quelques proprietes globales des varietes differentiables.Comment. Math. Helvetici 28 (1954), 17–86.

[Veblen 1905] O. Veblen: Theory of plane curves in non-metrical analysis situs.Trans. Amer. Math. Soc. 6 (1905), 83–98.

233

Bibliography

For prerequisites, we recommend the quite ad hoc texts [4], [10], [16], and[18]. On the other hand, there is a wealth of literature on degree theoryand related topics; we suggest as further reading the following books:

[1] J. Cronin: Fixed points and topological degree in nonlinear analysis.Mathematical Surveys 11. Providence, R.I.: American MathematicalSociety, 1964.

[2] K. Deimling: Nonlinear Functional Analysis. Berlin: Springer-Verlag,1985.

[3] A. Dold: Teorıa de punto fijo (I, II, III). Mexico: Monografıas delInstituto de Matematica, 1986.

[4] J.M. Gamboa, J.M. Ruiz: Iniciacion al estudio de las variedadesdiferenciales (2a edicion revisada). Madrid: Sanz y Torres, 2006.

[5] A. Granas, J. Dugunji: Fixed Point Theory. Springer Monographs inMathematics. New York: Springer-Verlag, 2003.

[6] V. Guillemin, A. Pollack: Differential Topology. Englewood Cliffs,N.J.: Prentice Hall, Inc.,1974.

[7] M.W. Hirsch: Differential Topology. Graduate Texts in Mathematics33. Springer-Verlag, New York-Heidelberg: Springer-Verlag, 1976.

[8] M.A. Krasnosel’skii: Topological methods in the theory of nonlinearintegral equations (translated from Russian). New York: PergamonPress, 1964.

[9] W. Krawcewicz, J. Wu: Theory of degrees with applications to bifur-cations and differential equations. New York: John Wiley & Sons,1997.

[10] S. Lang: Differential manifolds. Berlin: Springer-Verlag, 1988.

234 Bibliography

[11] E.L. Lima: Introducao a Topologia Diferencial. Rio de Janeiro: In-stituto de Matematica Pura e Aplicada, 1961.

[12] N.G. Lloyd: Degree Theory. Cambridge Tracts in Mathematics 73.Cambridge: Cambridge University Press, 1978.

[13] I. Madsen, J. Tornehave: From calculus to cohomology: de Rhamcohomology and characteristic classes. Cambridge, New York, Mel-bourne: Cambridge University Press, 1997.

[14] J. Milnor: Topology from the differentiable viewpoint. Princeton Land-marks in Mathematics. Princeton, N.J.: Princeton University Press,1997.

[15] L. Nirenberg: Topics in nonlinear functional analysis (with a chapterby E. Zehnder and notes by R.A. Artino). Courant Lecture Notesin Mathematics 6. New York: Courant Institute of MathematicalSciences, American Mathematical Society, 1974.

[16] E. Outerelo, J.M. Ruiz: Topologıa Diferencial. Madrid: Addison-Wesley, 1998.

[17] P.H. Rabinowitz: Theorie du Degre Topologique et applications a desproblemes aux limites non lineaires (redige par H. Berestycki). Lec-ture Notes Analyse Numerique Fonctionelle. Paris: Universite ParisVI, 1975.

[18] M. Spivak: Calculus on manifolds: A modern approach to classicaltheorems of advanced calculus. Boulder: Westview Press, 1971.

235

Symbols

y = f(x) 1

zn + a1zn−1 + · · ·+ an = 0 2

Jx1x0(f) 3

Z(z) = X(x, y) + iY (x, y) 3

12πi

∫Γ

Z′(z)Z(z) dz 3

N = x1x0Jy1y0(∆) 3

12δ = µ1 − µ2 4

w =

∣∣∣∣∣∣∂F∂x

∂F∂y

∂f∂x

∂f∂y

∣∣∣∣∣∣ 5

w(Γ, a) = 12πi

∫Γ

dzz−a 5

w(f(Γ ), 0) = 12πi

∫f(Γ )

dζζ 6

w(f(Γ ), 0) =∑

k w(Γ, ak)αk 6

w(f(Γ ), 0) =∑

k αk 6

E(k, ) 8

A(k, ) 8

#E(k, )−#A(k, ) 8

χ(F0, F1, . . . , Fn) 8

12

(#E(k, )−#A(k, )

)8

V (z) = x−x0

‖x−x0‖3 11∫F0=0

〈V, ν〉dS 11

w = N1 −N2 17

Dn= x ∈ Rn : ‖x‖ ≤ 1 20

∂Dn= Sn−1 20

K, |K|, |•K| 23

g = ψ g ϕ−1 : |K| → |L| 24

d(f) 25

F (t, x) = tf(x)−(1−t)x‖tf(x)−(1−t)x‖ 27

Sn+ 28

(K1,K2) 28

χ(M) 30

Φh(x) = x−Fh(x) 34

d(Φ,W, 0) = d(Φh,WM , 0) 34

x−F(x) = 0 34

dist 36

d(f,G, a) 36∫Rm Φ(|x|)dx = 1 38

d(y(x), Ω, z) 38∫ΩΦ(|y(x)− z|)J(y(x))dx 38

M(W) 41

M0(Ω) 41

∂Ω = Ω \Ω 41

f, f∞, f∗ 43

d∗(f, U, 0) 44

d∗(f, U) 44

ρ : S1 → GL(V ) 46

A, α = (αr)r≥0 46

Deg(f,Ω) 46

Deg(f,Ω) = Σ(d∗(f,Ω)) 47

S1 ∗ a 47

236 Symbols

Deg(f,Ω) = (αr) 47

∂kf∂xi1

··· ∂xik49

ϕ = (ϕ1, . . . , ϕp) 54

x = (x1, . . . , xm) 54

ψ−1 ϕ 54

dimx(M), dim(M) 54

U ∩M = f−1(0) 55( ∂fi∂xj

)55

h(U ∩M) = V ∩ (Rm × 0) 55

Hm : λ ≥ 0 56

∂M 56

∂([0, 1]×M) = M0 ∪M1 57

TxM 58

∂∂xi

∣∣x= ∂ϕ

∂xi(x) 58

gradx(f) 58

dxf(u) = 〈gradx(f), u〉 58

T(x,y)(M ×N) = TxM × TyN 58

dxf : TxM → TyN 59

dag = (dbψ)−1 dxf daϕ 59

γ′(0) = d0γ(1) 60

dxf(u) = (f γ)′(0) 60

RPm,CPm 61

TM 61

VM 61

Cf , Rf 62

Rf = N \ f(Cf ) 62

Txf−1(a) = ker(dxf) 62

O(n) 66

OM 66

νM 71

y − ρ(y) ⊥ Tρ(y)M 71

dist(y,M) = ‖y − ρ(y)‖ 71

(x, u) ∈ νM : ‖u‖ < τ (x) 72

H(t, x), Ht(x), Ht 75

[M,N ] 75

πk(N) = [Sk, N ] 75

πk(M) = [M, Sk] 75

‖f(x)− g(x)‖ < ε(x) 76

ζm 85

ζM = ζx : x ∈ M 85[∂

∂x1

∣∣∣x, . . . , ∂

∂xm

∣∣∣x

]= ζx 85

ζM×N = (ζM , ζN ) 86

signx(f) 86

ν : M → Rm+1 87

∂ζ 91∑x∈f−1(a) signx(f) 95

deg(f) 95

deg(f1 × · · · × fr) 102

Tm = S1 × · · · × S1 102

π : Sm → RPm 103

Γkc (M) 104

d : Γkc (M) → Γk+1

c (M) 104

d(α ∧ β) 104

dω = 0, ω = dα 104

Hkc (M,R) 104

Hkc (M,Z) 104

Hmc (Rm,R) = R 104

Hm(Rm,R) = 0 104∫M

dα = 0 104∫M

: Hmc (M,R) → R 104

h =∑m

i=1∂hi

∂xi105

ω → f∗ω 105∫M

f∗ω = deg(f) ·∫Nω 107

ν = grad(f)/‖ grad(f)‖ 108

ΩM = det(ν, ·) 108

Symbols 237

dxν : TxM → TxM 108

K(x) = det(dxν) 109

κ =∫M

KΩM 109

ΩSm 109

ν∗ΩSm = KΩM 109

χ(M) 109

κ = 12vol(S

m)χ(M) 109

H1(S1 × S1,R) = R2 110

deg(f) 111

f g : Sm+n+1 → Sm+n+1 113

exp : R → S1 113

h = −f + 〈f, g〉g : Sm → Sm 113

deg(f) = deg(ψ f ϕ−1) 117

p,−p, φ, Φ 119

φa,b 119

f−1(a)× f−1(b) → S2m−2 119

H(f) = deg(φa,b) 119

(f−1(a), f−1(b)) 121

Hm(S2m−1,R) = 0 123

(f) =∫S2m−1 α ∧ dα 123

deg2(f) 124

w2(M,p) 124

w2(p) 127

‖f‖ 137

d(f,D, a) 138

w(f, a) = d(f , D, a) 156

f(x) = ±f(−x) 160

f(x)‖f(x)‖ = ± f(−x)

‖f(−x)‖ 163

f(x)‖f(x)‖ = ∓ f(−x)

‖f(−x)‖ 166

lim‖x‖→+∞ |f(x)| = +∞ 169

s(X), s′(X) 171

d(g f,D, a) 172

Σ(g) 176

deg(Σ(g)) = deg(g) 176

f(x) = −g(x) 186

πm(M) = Z 193

πm(Sm) = πm(Sm) = Z 193

deg(f · g) = deg(f) + deg(g) 195

πm(X) = Z 198

S2m−1 → Sm 200

H(f g) = H(f) deg(g) 200

π3(S2) = Z 205

π7(S4) = Z⊕ Z4 ⊕ Z3 205

π15(S8) = Z⊕ Z8 ⊕ Z3 ⊕ Z5 205

H(g f) = deg(g)2H(f) 205

ξ =∑m

i=1 ξi∂ϕ∂xi

206

ξ = (ξ1, . . . , ξm) 206

dzξ : TzM → TzM 207

det(dzξ) 207

indz(ξ) = sign det(dzξ) 207

deg(

ξ‖ξ‖

∣∣∣S

)209

Ind(ξ) =∑

z indz(ξ) 209

grad(f) 211

Gxϕ 212

Jxξ 212

Hx(f ϕ) 212

Qz(f) 213

Qz(f) = (f γ)′′(t) 213

indz(f) 214

Ind(ξ) =∑

z(−1)indz(f) 214

Ind(ξ) =∑

z(−1)kαk 214

dist(x,H) = 1‖a‖ 〈x, a〉 215

SO(3) 217

Ind(ξ) = deg(η) 219

daf = IdE ⊕daξ 219

Ind(−ξ) = (−1)m Ind(ξ) 220

238 Symbols

Ind(ξ) = χ(M) 220

χ = F − E + V 221

χ = 2− 2g 221

χ(S2) = 2 221

χ(M) =∑

k(−1)kαk 222

deg(ν) = 12χ(M) 222

239

IndexAccessible points, 16Additivity, 37, 39–41, 45, 46, 149Admissible class of mappings, 41Admissible extension, 44Alexandroff compactification, 43Amann-Weiss degree, 42Analysis Situs, 13Antipodal extension, 187Antipodal images, 186Antipodal preimages, 167Approximation and homotopy, 78Approximation by smooth

mappings, 76Arcs do not disconnect, 15, 180Attractor, 207Axiomatization of Euclidean degree,

149

Balls do not disconnect, 180Barycentric subdivision, 25Bolzano Theorem, 167Borsuk Theorem, 195Borsuk-Ulam Theorem, 163Boundaries are not retractions, 13,

133Boundary, 14Boundary of a manifold, 56Boundary of a simplicial complex,

23Boundary Theorem, 19, 96, 101,

111, 150, 157Brouwer Fixed Point Theorem, 13,

20, 28, 29, 134, 152, 169Brouwer-Kronecker and Euclidean

degrees, 157, 158Brouwer-Kronecker degree, 31, 99,

111

Bump function, 51

Canonical orientation, 85Cauchy Argument Principle, 5, 6,

160Cauchy index of a function, 3Cauchy index of a planar curve, 5,

11Cauchy index with respect to a

planar curve, 15Cell, 14Center, 208Change of coordinates, 54Change of variables for integrals, 38,

107Characteristic of a regular function

system, 8Circulation, 208Classification of compact

differentiable curves, 63Closed differential form, 104Cobordism, 42Cohomotopy group, 75Coincidence index of two mappings,

29Colevel of a space with involution,

171Combinatorial topology, 13, 23Completely continuous mapping, 33Complex multiplication, 100Complex projective space, 61Converse Boundary Theorem, 194,

199Convolution kernel, 144Counterclockwise orientation, 85Critical point, 62Critical value, 35, 62

240 Index

Curve, 54Curve germs, 60, 90Cusp, 60

Degree for mappings between spacesof different dimensions, 43

Degree of a complex polynomial, 151Degree of a complex rational

function, 190Degree of a composite mapping, 27,

100, 111Degree of a differentiable mapping,

114, 154Degree of a symmetry, 100Degree of the antipodal map, 99Degree on manifolds with boundary,

193Deimling Axiomatization, 149Deimling axiomatization, 40Derivative of a differentiable

mapping, 59Diffeomorphism, 54Diffeotopies of Euclidean spaces, 81Diffeotopies of manifolds, 82Diffeotopies preserve orientation, 86Diffeotopy, 80Differentiable function, 49Differentiable manifold, 54Differentiable mappings, 49Differentiable mappings are

homotopic to theirderivatives, 153

Differentiable retraction, 67Differential form, 104Dimension of a manifold, 54Dipole, 210Directional derivation, 58Divergence of a vector field, 105

Easy Sard Theorem, 63, 187Electric field flow, 11, 160Equivariant degree, 46Equivariant mapping, 46Equivariant mapping of spaces with

involution, 171Equivariant mapping of spheres, 171Equivariant topological degree, 45Euclidean degree, 138, 143, 145

Euclidean degree is locally constant,139, 146

Euler characteristic, 30, 109, 220Euler formula, 221Even and odd extension lemmas,

161Even and odd mappings, 160Exact form, 104Excision, 37, 44, 46Exhaustion by compact sets, 54Existence of solutions, 1, 19, 27, 34,

37, 40, 44, 46, 149Extension by zero, 53Exterior differentiation, 104Exterior of a compact hypersurface,

129Exterior of a planar curve, 15

Face, 23Fibrations of spheres, 202Finite representation, 46First homotopy group, 30Fixed Point Problem, 1Fixed Point Theorem for odd

mappings, 167Fixed points of mappings of spheres,

21, 27Flow of a tangent vector field, 207Flow through a surface, 11Framed cobordism, 43Fredholm operator, 34Fubini Theorem, 65, 216Fuhrer Axiomatization, 149Fuller index, 45Fundamental group, 30Fundamental Theorem of Algebra,

2, 102, 150Fuhrer axiomatization, 39

Gauss curvature, 109Gauss mapping, 30, 108Gauss Theorem on electric flows, 11,

160Gauss-Bonnet Theorem, 31, 109,

222Geba-Massabo-Vignoli degree, 43Genus of a compact surface, 30, 221Global equations, 55, 129

Index 241

Global normal field, 215Good simplicial approximation, 25Gradient of a smooth function, 58Gradients on arbitrary manifolds,

211Gramm matrix, 212Green’s Theorem, 16

Hadamard index, 36Hadamard Integral Theorem, 17,

159Hedge-hog Theorem for manifolds,

221Hedge-hog Theorem for spheres, 135Height function, 215Heinz Integral Formula, 38, 144Hessian of a function at a critical

point, 213Hirsch Theorem, 166Homology groups, 222Homomorphisms on cohomology,

105Homotopic mappings, 75Homotopy, 75Homotopy and smooth homotopy,

79Homotopy for manifolds with

boundary, 193Homotopy for non-compact

manifolds, 193Homotopy for non-orientable

manifolds, 194Homotopy group, 30, 75Homotopy invariance, 25, 27, 37,

39–41, 44, 46, 98, 101, 107,112, 122, 140, 142, 144,147, 156

Homotopy of mappings into spheres,186

Hopf fibration, 202, 205Hopf invariant, 30, 119Hopf invariant in odd dimension,

122Hopf invariant of a composite

mapping, 200Hopf Theorem, 29, 191, 196Hypersurface, 54

Identity off a set, 80Impulsor, 207Incoming (eingang) point, 8Index of a center, 208Index of a dipole, 210Index of a function at a critical

point, 214Index of a function system on a

hypersurface, 19Index of a non-degenerate zero, 207Index of a quadratic form, 214Index of a saddle, 207Index of a sink, 207Index of a source, 207Index of an isolated zero, 209Integral curvature, 30, 109Integral index, 3Integral lines, 207Interior of a compact hypersurface,

129Interior of a planar curve, 15Intermediate Value Theorem in

arbitrary dimension, 12Invariance of dimension, 28, 169Invariance of Domain Theorem, 28,

56, 167, 169, 181Invariance of interiors, 20, 169Invariance of the characteristic

under continuousdeformations, 12

Invariant set, 46Inverse image of a regular value, 24,

35, 62Inverse Mapping Theorem for

manifolds, 59Inward tangent vectors at a

boundary point, 90Isolated coincidence point of two

mappings, 29

Jacobian, 4, 9, 58Join of two mappings, 113Jordan hypersurface, 31Jordan Separation Theorem, 6, 14,

28, 55, 88, 108, 124, 126,158, 176, 181

Jordan sets separate the space, 180

242 Index

Kneser-Glaeser Theorem, 63Kronecker Existence Theorem, 9Kronecker index, 14Kronecker integral, 11–13Kronecker Integral Theorem, 10, 159

Lemniscate, 171Leray-Schauder degree, 34Letter from Brouwer to Hadamard,

21Level of a space with involution, 171Lifting, 113Linearization of antipodal

extensions, 188Link coefficient, 28, 121Liouville-Sturm Theorem, 5Local coordinate system, 54Local coordinate system compatible

with an orientation, 85Local diffeomorphism, 54Local differentiable extension, 49Local equations, 55Local extrema of a differentiable

function, 59Local finiteness, 50Localization, 24, 59Locally closed set, 55

Manifold, 23Manifold with boundary, 56Manifolds are homogeneous, 83Manifolds as retractions of open

Euclidean sets, 70Mappings into spheres, 99, 183, 185Mappings into toruses, 186Measure zero set, 64Mod 2 degree theory, 34, 124Mod 2 winding number, 124Models of a manifold, 31Morse function, 214, 222Morse inequalities, 221Multiplication Formula, 172Multiplicities of roots, 151Multiplicities of zeros, 6

Nagumo Axiomatization, 149Nagumo axiomatization, 37Noether operator, 34

Non-degenerate critical point of afunction, 212

Non-degenerate zero of a tangentvector field, 207

Non-embedded manifolds, 31Non-homotopic mappings with the

same degree, 112Norm of a function, 137Normal vector field, 87, 88, 108, 218Normal vector field compatible with

an orientation, 87Normal vector field on a tube, 219Normality, 27, 37, 39–41, 139, 149Normalization of antipodal

extensions, 188

Opposite orientations, 85Order of the origin with respect to a

surface, 17Orientation at two antipodal points

of a sphere, 88Orientation in a linear space, 85Orientation of a cylinder, 91Orientation of a manifold, 85Orientation of hypersurfaces, 87Orientation of inverse images, 89Orientation of spheres, 88Orientation of the boundary, 91Oriented distance, 215Orthogonal group, 66Outgoing (ausgang) point, 8Outward normal vector field, 88Outward tangent vectors at a

boundary point, 90

Parallelizable manifold, 210Parametrization, 54Partial derivative, 58Perron-Frobenius Theorem, 134Poincare-Bohl Theorem, 19, 159Poincare-Hopf Index Theorem, 21,

30, 109, 219Poincare-Miranda Theorem, 28Polyhedron, 13Positive atlas, 85Positive basis, 85Positive change of coordinates, 85

Index 243

Preserving and reversing orientationmappings, 86

Principal curvatures, 109Product of manifolds, 57Proper homotopy, 76Proper mapping, 75Proper subset of spheres do not

disconnect Euclideanspaces, 180

Properly homotopic mappings, 76Pseudomanifold, 23Pull-back of a differential form, 105

Quadratic transform, 144

Radial retraction, 61Real projective hyperplane, 67Real projective hypersurface, 67Real projective space, 61Real roots of a polynomial, 2, 7Regular function system, 7Regular point, 62Regular value, 62Residual set, 63Retraction, 67de Rham cohomology, 37, 104Riesz Representation Theorem, 211Rotation group, 217

Saddle, 207Sard Theorem, 35Sard-Brown Theorem, 35, 63Schonflies Theorem, 16, 20Screwdriver orientation, 85Segre embedding, 189Semi-cone, 60Set topology, 15Sign of a mapping at a point, 86Simplex, 23Simplicial approximation, 25Simplicial complex, 23Simplicial homology, 13Singular cohomology, 104Singularity of a tangent vector field,

206, 212Sink, 207Smooth function, 49Smoothness, 57, 117

Solid torus, 60Source, 207Spheres are not homeomorphic to

proper subsets, 180Splitting of an isolated zero, 208Stereographic projection and

orientation, 88Stereographic projections, 55Stiefel bundle, 61, 66Stokes’ Theorem, 13, 38, 104Surface, 54Surface in the Euclidean space, 16Suspension, 44Suspension of a mapping, 176, 200Symmetric set, 161

Tangent bundle, 61Tangent space, 58Tangent vector, 58Tangent vector field, 135, 206Tangent vector fields via flows, 221Tangent vector fields via

triangulations, 220Tangent vector fields without zeros,

21, 27Tietze Extension Theorem

(differentiable), 52Topological beast, 16Topological circle, 183Topological degree, 24Topological degree at the origin, 5Topological sinus, 113, 145Torus of dimension m, 102Torus with g holes, 221Total index of a tangent vector field,

209Total index of the solutions of an

equation in a Banachspace, 34

Total number of zeros, 6Transversality, 62Triangulation, 13, 220Tubular retraction, 71

Unitary normal vector field, 87, 88,108, 218

Uryshon separating function, 51

244 Index

Variation of the argument, 14Vector product, 87Volume element, 108

Weingarten endomorphism, 108Whitney Embedding Theorems, 56Winding number, 5, 156, 163, 166Winding number is locally constant,

156

Zero of a tangent vector field, 206

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For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/.

GSM/108

This textbook treats the classical parts of mapping degree theory, with a detailed account of its history traced back to the first half of the 18th century. After a historical first chapter, the remaining four chapters develop the mathematics. An effort is made to use only elementary methods, resulting in a self-contained presentation. Even so, the book arrives at some truly outstanding theorems: the classification of homotopy classes for spheres and the Poincaré-Hopf Index Theorem, as well as the proofs of the original formulations by Cauchy, Poincaré, and others.

Although the mapping degree theory you will discover in this book is a classical subject, the treatment is refreshing for its simple and direct style. The straight-forward exposition is accented by the appearance of several uncommon topics: tubular neighborhoods without metrics, differences between class 1 and class 2 mappings, Jordan Separation with neither compactness nor cohomology, explicit constructions of homotopy classes of spheres, and the direct computation of the Hopf invariant of the first Hopf fibration.

The book is suitable for a one-semester graduate course. There are 180 exer-cises and problems of different scope and difficulty.

For additional information and updates on this book, visit

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