Mapping Degree Theory

%NRIQUE/UTERELO JesúS-2UIZ

'RADUATE3TUDIES IN-ATHEMATICS 6OLUME

!MERICAN-ATHEMATICAL3OCIETY 2EAL3OCIEDAD-ATEMÉTICA%SPA×OLA http://dx.doi.org/10.1090/gsm/108

Enrique Outerelo Jesús M. Ruiz

Graduate Studies in Mathematics Volume 108

American Mathematical Society Providence, Rhode Island

Real Sociedad Matemática Española Madrid, Spain Editorial Board of Graduate Studies in Mathematics David Cox, Chair Steven G. Krantz Rafe Mazzeo Martin Scharlemann

Editorial Committee of the Real Sociedad Matemática Española Guillermo P. Curbera, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Pablo Pedregal Bernardo Cascales Rosa Mar´ıa Miró-Roig Javier Duoandikoetxea Juan Soler

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

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

Library of Congress Cataloging-in-Publication Data Outerelo, Enrique, 1939– Mapping degree theory / Enrique Outerelo and Jes´us 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, Jes´us M. II. Title. QA612.O98 2009 515.7248—dc22 2009026383

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2009 by the authors. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09 To Jes´us Mar´ıa Ruiz Amestoy

Contents

Preface ix

I. History 1 1.Prehistory...... 1 2.Inceptionandformation...... 14 3.Accomplishment...... 28 4.Renaissanceandreformation...... 35 5.Axiomatization...... 38 6.Furtherdevelopments...... 42

II. Manifolds 49 1. Differentiable mappings ...... 49 2.Differentiablemanifolds...... 53 3.Regularvalues...... 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.ThedeRhamdeﬁnition...... 103 3. The degree of a continuous mapping ...... 110

vii viii Contents

4. The degree of a diﬀerentiable mapping ...... 114 5.TheHopfinvariant...... 119 6. The Jordan Separation Theorem ...... 124 7.TheBrouwerTheorems...... 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.Windingnumber...... 156 5.TheBorsuk-UlamTheorem...... 160 6.TheMultiplicationFormula...... 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.TheHopfTheorem:Euclideandegree...... 196 4.TheHopffibration...... 200 5. Singularities of tangent vector fields ...... 206 6.Gradientvectorfields...... 211 7. The Poincaré-HopfIndexTheorem...... 218

Names of mathematicians cited 225

Historical references 227

Bibliography 233

Symbols 235

Index 239 Preface

This book springs from lectures on degree theory given by the authors during many years at the Departamento de Geometr´ıa y Topolog´ıa at the Universidad Complutense de Madrid, and its definitive form corresponds to a three-month course given at the Dipartimento di Matematica at the Università di Pisa during the spring of 2006. Today mapping degree is a somewhat classical topic that appeals to geometers and topologists for its beauty and ample range of relevant applications. Our purpose here is to present both the history and the mathematics. The notion of degree was discovered by the great mathematicians of the decades around 1900: Cauchy, Poincaré, Hadamard, Brouwer, Hopf, etc. It was then brought to maturity in the 1930s by Hopf and also by Leray and Schauder. The theory was fully burnished between 1950 and 1970. This process is described in Chapter I. As a complement, at the end of the book there is included an index of names of the mathematicians who played their part in the development of mapping degree theory, many of whom stand tallest in the history of mathematics. After the first historical chapter, Chapters II, III, IV, and V are devoted to a more formal proposition-proof discourse to define and study the concept of degree and its applications. Chapter II gives a quick presentation of manifolds, with special emphasis on aspects relevant to degree theory, namely regular val- ues of differentiable mappings, tubular neighborhoods, approximation, and orientation. Although this chapter is primarily intended to provide a review for the reader, it includes some not so standard details, for instance concerning tubular neighborhoods. The main topic, degree theory, is presented in Chapters III and IV. In a simplified manner we can distinguish two ap- proaches to the theory: the Brouwer-Kronecker degree and the Euclidean degree. The first is developed in Chapter III by differential means, with a quick diversion into the de Rham computation in cohomological terms. We cannot help this diversion: cohomology is too appealing to skip. Among other applications, we obtain in this chapter a differential version of the Jordan Separation Theorem. Then, we construct the Euclidean degree in

ix x Preface

Chapter IV. This is mainly analytic and astonishingly simple, especially in view of its extraordinary power. We hope this partisan claim will be acknowledged 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 devoted to some of those special results in mathematics that justify a theory by their depth and perfection: the Hopf and the Poincaré-Hopf Theorems, with their accompaniment of consequences and comments. We state and prove these theorems, which gives us the perfect occasion to take a glance at tangent vector fields. We have included an assorted collection of some 180 problems and exercises distributed among the sections of Chapters II to V, none for Chapter I due to its nature. Those problems and exercises, of various difficulty, fall into three different classes: (i) suitable examples that help to seize the ideas behind the theory, (ii) complements to that theory, such as variations for different settings, additional applications, or unexpected connections with different topics, and (iii) guides for the reader to produce complete proofs of the classical results presented in Chapter I, once the proper machinery is developed. We have tried to make internal cross-references clearer by adding the Roman chapter number to the reference, either the current chapter number or that of a different chapter. For example, III.6.4 refers to Proposition 6.4 in 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 simplest possible presentation at the lowest technical cost. This means we restrict ourselves to elementary methods, whatever meaning is accepted for elementary. More explicitly, we only assume the reader is acquainted with basic ideas of differential topology, such as can be found in any text on the calculus on manifolds. We only hope that this book succeeds in presenting degree theory as it 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 the reader.

Los Molinos and Majadahonda E. Outerelo and J.M. Ruiz June 2009 Names of mathematicians cited

Aleksandrov, Pavel Sergeevich Jacobi, Carl Gustav Jacob (1896–1982), 32 (1804–1851), 7 Amann, Herbert, 39 Jodel, Jerzy, 46 Jordan, Camille (1838–1922), 15 Bernstein, Sergei Natanovich (1880–1968), 34 Kronecker, Leopold (1823–1891), 7 Betti, Enrico (1823–1892), 13 Bohl, Piers (1865–1921), 13 Leray, Jean (1906–1998), 33 Brouwer, Luitzen Egbertus Jan Liouville, Joseph (1809–1882), 2 (1881–1996), 14 Brown, Arthur Barton, 35 Marzantowicz, Waclaw, 46 Massab`o, Ivar, 43 Caccioppoli, Renato (1904–1959), 34 Miranda,Carlo,28 Cauchy, Augustin Louis (1789–1857), 2 Nagumo, Mitio, 35 Cech,ˇ Eduard (1893–1960), 30 Nirenberg, Louis, 43

Deimling, Klaus, 40 Ostrowski, Alexander (1893–1986), 2 Dyck, Walter Franz Anton Picard, Charles Emile (1856–1941), (1856–1934), 31 12 Dylawerski, Grzegorz, 46 Poincar´e, Jules Henri (1854–1912), Elworthy,K.David,43 11 Pontryagin, Lev Semenovich, 43 F¨uhrer, Lutz, 38 Fuller, F. Brock, 45 de Rham, Georges (1903–1990), 37 Riemann, Friedrich Bernhard Georg Gauss, Karl-Friedrich (1777–1855), 2 (1826–1866), 13 Geba, Kazimierz, 43 Romero Ruiz del Portal, Francisco, 45 Hadamard, Jacques Salomon Rothe, E., 29 (1865–1963), 14 Rybicki, Slawomir, 48 Heinz, Erhard, 38 Hermite, Charles (1822–1901), 7 Sard, Arthur (1909–1980), 35 Hopf, Heinz (1894–1971), 28 Schauder, Juliusz Pawel Hurewicz, Witold (1904–1956), 30 (1899–1943), 33 Schmidt, Erhard (1876–1959), 28 Ize, Jorge, 43 225 226 Names

Sch¨onﬂies, Arthur Moritz (1853–1928), 15 Schwartz, Laurent (1915–2002), 38 Smale, Stephen, 42 Sturm, Jacques Charles Fran¸cois (1803–1855), 2 Sylvester, James Joseph (1814–1897), 7

Thom, Ren´e (1923–2002), 42 Tromba, Anthony J., 43

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

Weierstrass, Karl Theodor Wilhelm (1815–1897), 10 Weiss, Stanley A., 39 Historical references

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

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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 theory and 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 Mathematical Society, 1964.

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

[3] A. Dold: Teor´ıa de punto fijo (I, II, III). México: Monograf´ıas del Instituto de Matemática, 1986.

[4] J.M. Gamboa, J.M. Ruiz: Iniciaci´on al estudio de las variedades diferenciales (2a edici´on revisada). Madrid: Sanz y Torres, 2006.

[5] A. Granas, J. Dugunji: Fixed Point Theory. Springer Monographs in Mathematics. 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 Mathematics 33. Springer-Verlag, New York-Heidelberg: Springer-Verlag, 1976.

[8] M.A. Krasnosel’skii: Topological methods in the theory of nonlinear integral equations (translated from Russian). New York: Pergamon Press, 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. 233 234 Bibliography

[11] E.L. Lima: Introdu¸caoa ` Topologia Diferencial. Rio de Janeiro: In- stituto de Matem´atica 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 Rham cohomology 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 chapter by E. Zehnder and notes by R.A. Artino). Courant Lecture Notes in Mathematics 6. New York: Courant Institute of Mathematical Sciences, American Mathematical Society, 1974.

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

[17] P.H. Rabinowitz: Théorie du Degré Topologique et applicationsades ` problèmes aux limites non lineaires (redigé par H. Berestycki). Lec- ture Notes Analyse Numérique Fonctionelle. Paris: UniversitéParis VI, 1975.

[18] M. Spivak: Calculus on manifolds: A modern approach to classical theorems of advanced calculus. Boulder: Westview Press, 1971. Symbols y = f(x)1g¯ = ψ ◦ g ◦ ϕ−1 : |K|→|L| 24 n n−1 z + a1z + ···+ an =0 2 d(f)25 − − J x1 (f)3tf(x) (1 t)x x0 F (t, x)= tf(x)−(1−t)x 27 Z(z)=X(x, y)+iY (x, y)3Sn + 28 1 Z (z) 2πi Γ Z(z) dz 3 (K1,K2)28 x1 y1 χ(M)30 N = x0 Jy0 (∆)3 1 −F δ = µ1 − µ2 4 Φh(x)=x h(x)34 2 ∂F ∂F d(Φ, W, 0) = d(Φh,WM , 0) 34 ∂x ∂y w = 5 x −F(x)=0 34 ∂f ∂f ∂x ∂y dist 36 1 dz w(Γ, a)= 2πi Γ z−a 5 d(f,G,a)36 1 dζ w(f(Γ ), 0) = 6 Φ(|x|)dx =1 38 2πi f(Γ ) ζ Rm w(f(Γ ), 0) = w(Γ, ak)αk 6 d(y(x),Ω,z)38 k w(f(Γ ), 0) = α 6 | − | k k Ω Φ( y(x) z )J(y(x))dx 38 E(k, )8M(W)41

A(k, )8M0(Ω)41 #E(k, ) − #A(k, )8∂Ω = Ω \ Ω 41 ∗ χ(F0,F1,...,Fn)8f,f∞,f 43 1 − ∗ 2 #E(k, ) #A(k, ) 8 d (f,U,0) 44 x−x0 d∗(f,U)44 V (z)= x−x03 11 S1 → V,ν dS 11 ρ : GL(V )46 F0=0 A, α =(αr)r≥0 46 w = N1 − N2 17 n D = {x ∈ Rn : x≤1} 20 Deg(f,Ω)46 n ∗ ∂D = Sn−1 20 Deg(f,Ω)=Σ(d (f,Ω)) 47 • S1 ∗ a 47 K, |K|, |K| 23

235 236 Symbols

Deg(f,Ω)=(αr)47{(x, u) ∈ νM : u <τ(x)} 72 ∂kf H(t, x),H (x),H 75 ··· 49 t t ∂xi1 ∂xik [M,N]75 ϕ =(ϕ1,...,ϕp)54 k πk(N)=[S ,N]75 x =(x1,...,xm)54 k k − π (M)=[M,S ]75 ψ 1 ◦ ϕ 54 f(x) − g(x) <ε(x)76 dimx(M), dim(M)54 m − ζ 85 U ∩ M = f 1(0) 55 ζM = {ζx : x ∈ M} 85 ∂fi " # 55 ∂xj ∂ ∂ ∂ ,..., ∂ = ζx 85 x1 x xm x h(U ∩ M)=V ∩ (Rm ×{0})55 ζM×N =(ζM ,ζN )86 Hm : λ ≥ 056 signx(f)86 ∂M 56 ν : M → Rm+1 87 ∂([0, 1] × M)=M ∪ M 57 0 1 ∂ζ 91 TxM 58 x∈f −1(a) signx(f)95 ∂ = ∂ϕ (x)58 ∂xi x ∂xi deg(f)95 gradx(f)58 deg(f1 ×···×fr) 102 dxf(u)=grad (f),u 58 x T m = S1 ×···×S1 102 T(x,y)(M × N)=TxM × TyN 58 π : Sm → RPm 103 d f : T M → T N 59 x x y k Γc (M) 104 d g =(d ψ)−1 ◦ d f ◦ d ϕ 59 a b x a k → k+1 d:Γc (M) Γc (M) 104 γ(0) = d γ(1) 60 0 ∧ d(α β) 104 dxf(u)=(f ◦ γ) (0) 60 dω =0,ω =dα 104 RPm, CPm 61 k R Hc (M, ) 104 TM 61 k Z Hc (M, ) 104 VM 61 m Rm R R Hc ( , )= 104 Cf , Rf 62 Hm(Rm, R) = 0 104 Rf = N \ f(Cf )62 dα = 0 104 T f −1(a)=ker(d f)62M x x m R → R : Hc (M, ) 104 O(n)66M m h = ∂hi 105 OM 66 i=1 ∂xi ω → f ∗ω 105 νM 71 ∗ · M f ω =deg(f) N ω 107 y − ρ(y) ⊥ Tρ(y)M 71 ν =grad(f)/ grad(f) 108 dist(y, M)=y − ρ(y) 71 ΩM =det(ν, ·) 108 Symbols 237

dxν : TxM → TxM 108 deg(Σ(g)) = deg(g) 176 K(x)=det(d ν) 109 f(x) = −g(x) 186 x m Z κ = M KΩM 109 π (M)= 193 m m m ΩSm 109 πm(S )=π (S )=Z 193 ∗ ν ΩSm = KΩM 109 deg(f · g)=deg(f)+deg(g) 195 χ(M) 109 πm(X)=Z 198 1 Sm S2m−1 → Sm κ = 2 vol( )χ(M) 109 200 H1(S1 × S1, R)=R2 110 H(f ◦ g)=H(f)deg(g) 200 2 deg(f) 111 π3(S )=Z 205 m+n+1 m+n+1 4 f g: S → S 113 π7(S )=Z ⊕ Z4 ⊕ Z3 205 1 8 exp : R → S 113 π15(S )=Z ⊕ Z8 ⊕ Z3 ⊕ Z5 205 h = −f + f,g g : Sm → Sm 113 H(g ◦ f)=deg(g)2H(f) 205 −1 m ∂ϕ deg(f)=deg(ψ ◦ f ◦ ϕ ) 117 ξ = ξi 206 i=1 ∂xi − ¯ p, p, φ, Φ 119 ξ =(ξ1,...,ξm) 206

φa,b 119 dzξ : TzM → TzM 207 −1 × −1 → S2m−2 f (a) f (b) 119 det(dzξ) 207

H(f)=deg(φa,b) 119 ind (ξ)=signdet(d ξ) 207 z z −1 −1 (f (a),f (b)) 121 ξ¯ deg ξ¯ 209 Hm(S2m−1, R) = 0 123 S Ind(ξ)= ind (ξ) 209 ∧ z z (f)= S2m−1 α dα 123 grad(f) 211 deg2(f) 124 Gxϕ 212 w (M,p) 124 2 ¯ Jxξ 212 w2(p) 127 Hx(f ◦ ϕ) 212 f 137 Qz(f) 213 d(f,D,a) 138 Qz(f)=(f ◦ γ) (t) 213 w(f,a)=d(f,D,a¯ ) 156 indz(f) 214 f(x)=±f(−x) 160 − indz (f) − Ind(ξ)= z( 1) 214 f(x) = ± f( x) 163 f(x) f(−x) − k Ind(ξ)= z( 1) αk 214 − f(x) f( x) 1 = ∓ − 166 f(x) f( x) dist(x, H)= a x, a 215 | | ∞ limx→+∞ f(x) =+ 169 SO(3) 217 s(X),s(X) 171 Ind(ξ)=deg(η) 219 ◦ d(g f,D,a) 172 daf =IdE ⊕daξ 219 Σ(g) 176 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 − k χ(M)= k( 1) αk 222 1 deg(ν)= 2 χ(M) 222 Index

Accessible points, 16 Bump function, 51 Additivity, 37, 39–41, 45, 46, 149 Admissible class of mappings, 41 Canonical orientation, 85 Admissible extension, 44 Cauchy Argument Principle, 5, 6, Alexandroff compactification, 43 160 Amann-Weiss degree, 42 Cauchy index of a function, 3 Analysis Situs, 13 Cauchy index of a planar curve, 5, Antipodal extension, 187 11 Antipodal images, 186 Cauchy index with respect to a Antipodal preimages, 167 planar curve, 15 Approximation and homotopy, 78 Cell, 14 Approximation by smooth Center, 208 mappings, 76 Change of coordinates, 54 Arcs do not disconnect, 15, 180 Change of variables for integrals, 38, Attractor, 207 107 Axiomatization of Euclidean degree, Characteristic of a regular function 149 system, 8 Circulation, 208 Balls do not disconnect, 180 Classification of compact Barycentric subdivision, 25 differentiable curves, 63 Bolzano Theorem, 167 Closed differential form, 104 Borsuk Theorem, 195 Cobordism, 42 Borsuk-Ulam Theorem, 163 Cohomotopy group, 75 Boundaries are not retractions, 13, Coincidence index of two mappings, 133 29 Boundary, 14 Colevel of a space with involution, Boundary of a manifold, 56 171 Boundary of a simplicial complex, Combinatorial topology, 13, 23 23 Completely continuous mapping, 33 Boundary Theorem, 19, 96, 101, Complex multiplication, 100 111, 150, 157 Complex projective space, 61 Brouwer Fixed Point Theorem, 13, Converse Boundary Theorem, 194, 20, 28, 29, 134, 152, 169 199 Brouwer-Kronecker and Euclidean Convolution kernel, 144 degrees, 157, 158 Counterclockwise orientation, 85 Brouwer-Kronecker degree, 31, 99, Critical point, 62 111 Critical value, 35, 62 239 240 Index

Curve, 54 Euclidean degree is locally constant, Curve germs, 60, 90 139, 146 Cusp, 60 Euler characteristic, 30, 109, 220 Euler formula, 221 Degree for mappings between spaces Even and odd extension lemmas, of different dimensions, 43 161 Degree of a complex polynomial, 151 Even and odd mappings, 160 Degree of a complex rational Exact form, 104 function, 190 Excision, 37, 44, 46 Degree of a composite mapping, 27, Exhaustion by compact sets, 54 100, 111 Existence of solutions, 1, 19, 27, 34, Degree of a differentiable mapping, 37, 40, 44, 46, 149 114, 154 Extension by zero, 53 Degree of a symmetry, 100 Exterior differentiation, 104 Degree of the antipodal map, 99 Exterior of a compact hypersurface, Degree on manifolds with boundary, 129 193 Exterior of a planar curve, 15 Deimling Axiomatization, 149 Deimling axiomatization, 40 Face, 23 Derivative of a differentiable Fibrations of spheres, 202 mapping, 59 Finite representation, 46 Diffeomorphism, 54 First homotopy group, 30 Diffeotopies of Euclidean spaces, 81 Fixed Point Problem, 1 Diffeotopies of manifolds, 82 Fixed Point Theorem for odd Diffeotopies preserve orientation, 86 mappings, 167 Diffeotopy, 80 Fixed points of mappings of spheres, Differentiable function, 49 21, 27 Differentiable manifold, 54 Flow of a tangent vector field, 207 Differentiable mappings, 49 Flow through a surface, 11 Differentiable mappings are Framed cobordism, 43 homotopic to their Fredholm operator, 34 derivatives, 153 Fubini Theorem, 65, 216 Differentiable retraction, 67 Führer Axiomatization, 149 Differential form, 104 Fuller index, 45 Dimension of a manifold, 54 Fundamental group, 30 Dipole, 210 Fundamental Theorem of Algebra, Directional derivation, 58 2, 102, 150 Divergence of a vector field, 105 Führer axiomatization, 39

Easy Sard Theorem, 63, 187 Gauss curvature, 109 Electric field flow, 11, 160 Gauss mapping, 30, 108 Equivariant degree, 46 Gauss Theorem on electric flows, 11, Equivariant mapping, 46 160 Equivariant mapping of spaces with Gauss-Bonnet Theorem, 31, 109, involution, 171 222 Equivariant mapping of spheres, 171 Geba-Massabò-Vignoli degree, 43 Equivariant topological degree, 45 Genus of a compact surface, 30, 221 Euclidean degree, 138, 143, 145 Global equations, 55, 129 Index 241

Global normal field, 215 Identity off a set, 80 Good simplicial approximation, 25 Impulsor, 207 Gradient of a smooth function, 58 Incoming (eingang) point, 8 Gradients on arbitrary manifolds, Index of a center, 208 211 Index of a dipole, 210 Gramm matrix, 212 Index of a function at a critical Green’s Theorem, 16 point, 214 Index of a function system on a Hadamard index, 36 hypersurface, 19 Hadamard Integral Theorem, 17, Index of a non-degenerate zero, 207 159 Index of a quadratic form, 214 Hedge-hog Theorem for manifolds, Index of a saddle, 207 221 Index of a sink, 207 Hedge-hog Theorem for spheres, 135 Index of a source, 207 Height function, 215 Index of an isolated zero, 209 Heinz Integral Formula, 38, 144 Integral curvature, 30, 109 Hessian of a function at a critical Integral index, 3 point, 213 Integral lines, 207 Hirsch Theorem, 166 Interior of a compact hypersurface, Homology groups, 222 129 Homomorphisms on cohomology, Interior of a planar curve, 15 105 Intermediate Value Theorem in Homotopic mappings, 75 arbitrary dimension, 12 Homotopy, 75 Invariance of dimension, 28, 169 Homotopy and smooth homotopy, Invariance of Domain Theorem, 28, 79 56, 167, 169, 181 Homotopy for manifolds with Invariance of interiors, 20, 169 boundary, 193 Invariance of the characteristic Homotopy for non-compact under continuous manifolds, 193 deformations, 12 Homotopy for non-orientable Invariant set, 46 manifolds, 194 Inverse image of a regular value, 24, Homotopy group, 30, 75 35, 62 Homotopy invariance, 25, 27, 37, Inverse Mapping Theorem for 39–41, 44, 46, 98, 101, 107, manifolds, 59 112, 122, 140, 142, 144, Inward tangent vectors at a 147, 156 boundary point, 90 Homotopy of mappings into spheres, Isolated coincidence point of two 186 mappings, 29 Hopf fibration, 202, 205 Hopf invariant, 30, 119 Jacobian,4,9,58 Hopf invariant in odd dimension, Join of two mappings, 113 122 Jordan hypersurface, 31 Hopf invariant of a composite Jordan Separation Theorem, 6, 14, mapping, 200 28, 55, 88, 108, 124, 126, Hopf Theorem, 29, 191, 196 158, 176, 181 Hypersurface, 54 Jordan sets separate the space, 180 242 Index

Kneser-Glaeser Theorem, 63 Non-degenerate critical point of a Kronecker Existence Theorem, 9 function, 212 Kronecker index, 14 Non-degenerate zero of a tangent Kronecker integral, 11–13 vector field, 207 Kronecker Integral Theorem, 10, 159 Non-embedded manifolds, 31 Non-homotopic mappings with the Lemniscate, 171 same degree, 112 Leray-Schauder degree, 34 Norm of a function, 137 Letter from Brouwer to Hadamard, Normal vector field, 87, 88, 108, 218 21 Normal vector field compatible with Level of a space with involution, 171 an orientation, 87 Lifting, 113 Normal vector field on a tube, 219 Linearization of antipodal Normality, 27, 37, 39–41, 139, 149 extensions, 188 Normalization of antipodal Link coefficient, 28, 121 extensions, 188 Liouville-Sturm Theorem, 5 Local coordinate system, 54 Opposite orientations, 85 Local coordinate system compatible Order of the origin with respect to a with an orientation, 85 surface, 17 Local diffeomorphism, 54 Orientation at two antipodal points Local differentiable extension, 49 of a sphere, 88 Local equations, 55 Orientation in a linear space, 85 Local extrema of a differentiable Orientation of a cylinder, 91 function, 59 Orientation of a manifold, 85 Local finiteness, 50 Orientation of hypersurfaces, 87 Localization, 24, 59 Orientation of inverse images, 89 Locally closed set, 55 Orientation of spheres, 88 Orientation of the boundary, 91 Manifold, 23 Oriented distance, 215 Manifold with boundary, 56 Orthogonal group, 66 Manifolds are homogeneous, 83 Outgoing (ausgang) point, 8 Manifolds as retractions of open Outward normal vector field, 88 Euclidean sets, 70 Outward tangent vectors at a Mappings into spheres, 99, 183, 185 boundary point, 90 Mappings into toruses, 186 Measure zero set, 64 Parallelizable manifold, 210 Mod 2 degree theory, 34, 124 Parametrization, 54 Mod 2 winding number, 124 Partial derivative, 58 Models of a manifold, 31 Perron-Fröbenius Theorem, 134 Morse function, 214, 222 Poincaré-Bohl Theorem, 19, 159 Morse inequalities, 221 Poincaré-Hopf Index Theorem, 21, Multiplication Formula, 172 30, 109, 219 Multiplicities of roots, 151 Poincaré-Miranda Theorem, 28 Multiplicities of zeros, 6 Polyhedron, 13 Positive atlas, 85 Nagumo Axiomatization, 149 Positive basis, 85 Nagumo axiomatization, 37 Positive change of coordinates, 85 Noether operator, 34 Index 243

Preserving and reversing orientation Solid torus, 60 mappings, 86 Source, 207 Principal curvatures, 109 Spheres are not homeomorphic to Product of manifolds, 57 proper subsets, 180 Proper homotopy, 76 Splitting of an isolated zero, 208 Proper mapping, 75 Stereographic projection and Proper subset of spheres do not orientation, 88 disconnect Euclidean Stereographic projections, 55 spaces, 180 Stiefel bundle, 61, 66 Properly homotopic mappings, 76 Stokes’ Theorem, 13, 38, 104 Pseudomanifold, 23 Surface, 54 Pull-back of a differential form, 105 Surface in the Euclidean space, 16 Suspension, 44 Quadratic transform, 144 Suspension of a mapping, 176, 200 Symmetric set, 161 Radial retraction, 61 Real projective hyperplane, 67 Tangent bundle, 61 Real projective hypersurface, 67 Tangent space, 58 Real projective space, 61 Tangent vector, 58 Real roots of a polynomial, 2, 7 Tangent vector field, 135, 206 Regular function system, 7 Tangent vector fields via flows, 221 Regular point, 62 Tangent vector fields via Regular value, 62 triangulations, 220 Residualset,63 Tangent vector fields without zeros, Retraction, 67 21, 27 de Rham cohomology, 37, 104 Tietze Extension Theorem Riesz Representation Theorem, 211 (differentiable), 52 Rotation group, 217 Topological beast, 16 Topological circle, 183 Saddle, 207 Topological degree, 24 Sard Theorem, 35 Topological degree at the origin, 5 Sard-Brown Theorem, 35, 63 Topological sinus, 113, 145 Schönflies Theorem, 16, 20 Torus of dimension m, 102 Screwdriver orientation, 85 Torus with g holes, 221 Segre embedding, 189 Total index of a tangent vector field, Semi-cone, 60 209 Set topology, 15 Total index of the solutions of an Sign of a mapping at a point, 86 equation in a Banach Simplex, 23 space, 34 Simplicial approximation, 25 Total number of zeros, 6 Simplicial complex, 23 Transversality, 62 Simplicial homology, 13 Triangulation, 13, 220 Singular cohomology, 104 Tubular retraction, 71 Singularity of a tangent vector field, 206, 212 Unitary normal vector field, 87, 88, Sink, 207 108, 218 Smooth function, 49 Uryshon separating function, 51 Smoothness, 57, 117 244 Index

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

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

Zero of a tangent vector ﬁeld, 206 Titles in This Series

108 Enrique Outerelo and Jesús M. Ruiz, Mapping degree theory, 2009 107 Jeffrey M. Lee, Manifolds and differential geometry, 2009 106 Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, 2009 105 Giovanni Leoni, A first course in Sobolev spaces, 2009 104 Paolo Aluffi, Algebra: Chapter 0, 2009 103 Branko Grünbaum, Configurations of points and lines, 2009 102 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009 101 Ward Cheney and Will Light, A course in approximation theory, 2009 100 I. Martin Isaacs, Algebra: A graduate course, 2009 99 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to Schrödinger operators, 2009 98 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable structure, 2008 97 David C. Ullrich, Complex made simple, 2008 96 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008 95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, 2008 93 Peter W. Michor, Topics in differential geometry, 2008 92 I. Martin Isaacs, Finite group theory, 2008 91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato, A course on the web graph, 2008 88 Nathanial P. Brown and Narutaka Ozawa, C∗-algebras and finite-dimensional approximations, 2008 87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007 86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 82 Serge Alinhac and Patrick Gérard, Pseudo-differential operators and the Nash–Moser theorem (translated by Stephen S. Wilson), 2007 81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 TITLES IN THIS SERIES

70 Seán Dineen, Probability theory in finance, 2005 69 Sebastián Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Körner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 Cédric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 HershelM.FarkasandIrwinKra, Theta constants, Riemann surfaces and the modular group, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. 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 exercises and problems of different scope and difficulty.

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