Equivariant degree theory

Jorge Ize, Alfonso Vignoli3110175509, 9783110175509

This volume presents a degree theory for maps which commute with a group of symmetries. This degree is no longer a single integer but an element of the group of equivariant homotopy classes of maps between two spheres and depends on the orbit types of the spaces. The authors develop completely the theory and applications of this degree in a self-contained presentation starting with only elementary facts. The first chapter explains the basic tools of representation theory, homotopy theory and differential equations needed in the text. Then the degree is defined and its main abstract properties are derived. The next part is devoted to the study of equivariant homotopy groups of spheres and to the classification of equivariant maps in the case of abelian actions. These groups are explicitly computed and the effects of symmetry breaking, products and composition are thoroughly studied. The last part deals with computations of the equivariant index of an isolated orbit and of an isolated loop of stationary points. Here differential equations in a variety of situations are considered: symmetry breaking, forcing, period doubling, twisted orbits, first integrals, gradients etc. Periodic solutions of Hamiltonian systems, in particular spring-pendulum systems, are studied as well as Hopf bifurcation for all these situations.

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Table of contents :
Preface……Page 8
Contents……Page 10
Introduction……Page 12
1.1 Group actions……Page 22
1.2 The fundamental cell lemma……Page 26
1.3 Equivariant maps……Page 29
1.4 Averaging……Page 33
1.5 Irreducible representations……Page 38
1.6 Extensions of Γ-maps……Page 46
1.7 Orthogonal maps……Page 50
1.8 Equivariant homotopy groups of spheres……Page 56
1.9 Symmetries and differential equations……Page 63
1.10 Bibliographical remarks……Page 78
2.1 Equivariant degree in finite dimension……Page 80
2.2 Properties of the equivariant degree……Page 82
2.3 Approximation of the Γ-degree……Page 88
2.4 Orthogonal maps……Page 90
2.5 Applications……Page 93
2.6 Operations……Page 98
2.7 Bibliographical remarks……Page 106
3.1 The extension problem……Page 107
3.2 Homotopy groups of Γ-maps……Page 123
3.3 Computation of Γ-classes……Page 129
3.4 Borsuk–Ulam results……Page 140
3.5 The one parameter case……Page 157
3.6 Orthogonal maps……Page 177
3.7 Operations……Page 186
3.8 Bibliographical remarks……Page 216
4.1 Range of the equivariant degree……Page 218
4.2 Γ-degree of an isolated orbit……Page 232
4.3 Γ-Index for an orthogonal map……Page 266
4.4 Γ-Index of a loop of stationary points……Page 309
4.5 Bibliographical remarks……Page 346
Appendix A Equivariant Matrices……Page 348
Appendix B Periodic Solutions of Linear Systems……Page 353
Bibliography……Page 358
Index……Page 380