Degree Theory for Equivariant Maps, the General S1-Action

194 Pages·1993·13.683 MB·English

by  Jorge Ize, Ivar Massabo, Alfonso Vignoli

Degree Theory
for Equivariant Maps,
1
the General S -Action
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(See the AMS catalog for earlier titles)
MEMOIRS
-LIT A f the
0
American Mathematical Society
Number 481
Degree Theory
for Equivariant Maps,
1
the General S -Action
Jorge Ize
Ivar Massabo
Alfonso Vignoli
November 1992 • Volume 100 • Number 481 (end of volume) • ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary 58B05; Secondary 34C25, 47H15, 54F45, 55Q91, 58E09.
Library of Congress Cataloging-in-Publication Data
Ize, Jorge, 1946-
Degree theory for equivariant maps, the general S ^action/Jorge Ize, Ivar Massabo, Alfonso
Vignoli.
p. cm. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 481)
Includes bibliographical references.
ISBN 0-8218-2542-9
1. Topological degree. 2. Mappings (Mathematics) 3. Homotopy groups. 4. Sphere. I. Mass
abo, Ivar, 1947- . II. Vignoli, Alfonso, 1940- . III. Title. IV. Series.
QA3.A57 no. 481
[QA612]
510s-dc20 92-28573
[514'.2] CIP
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TABLE OF CONTENTS
INTRODUCTION 1
CHAPTER ONE: PRELIMINARIES 13
1.1. S -actions 13
1.2. Almost semi-free action 14
1.3. Equivariant homotopy 15
1.4. The extension degree 16
1.5. Equivariant homotopy groups of spheres 18
1.6. Equivariant degree in the almost semi-free case 19
CHAPTER TWO: EXTENSIONS OF 5!-MAPS 21
2.1. The fundamental cell lemma 21
2.2. The Extension Theorem 24
2.3. The Extension degree 35
2.4. Properties of the Extension degree 39
VI TABLE OF CONTENTS
CHAPTER THREE: HOMOTOPY GROUPS OF S^MAPS 41
3.1. Trivial invariant part, the case p > 1 42
3.2. Nontrivial invariant part, the case p = 0 46
3.3. Behavior under suspension 52
3.4. Relationship with the set of Ji-degrees 61
3.5. Symmetry Breaking 63
CHAPTER FOUR: DEGREE OF S^-MAPS 69
4.1. Range of degi (/; ft) 70
s
4.2. Infinite dimensional degree 74
4.3. Computation of the S -degree 76
4.4. Global Continuation 82
4.5. Global Bifurcation 85
CHAPTER FIVE: S^INDEX OF AN ISOLATED NON-STATIONARY
ORBIT AND APPLICATIONS 101
5.1. The case p > 1 102
5.2. The case p = 0 103
5.3. p = 0, hyperbolic orbits 106
5.4. Autonomous differential equations 114
5.5. Gradient maps 116
5.6. Differential equations with fixed period 116
5.7. Differential equations with first integrals 118
5.8 Symmetry breaking for differential equations 120
TABLE OF CONTENTS vn
CHAPTER SIX: INDEX OF AN ISOLATED ORBIT OF STATIONARY
SOLUTIONS AND APPLICATIONS 125
6.1. Computation of the 5 -Index 125
6.2. Application to bifurcation 129
6.3. Hopf bifurcation for autonomous differential equations 132
6.4. Hopf bifurcation for systems with first integrals 135
6.5. Hopf bifurcation and symmetry breaking 154
CHAPTER SEVEN: VIRTUAL PERIODS AND ORBIT INDEX 159
7.1. Virtual periods 159
7.2. The Orbit Index 165
APPENDIX: ADDITIVITY UP TO ONE SUSPENSION 173
REFERENCES 177
ABSTRACT
In the first paper of this series we constructed a general degree theory for maps which
are equi variant with respect to a linear action of an arbitrary compact Lie group. This
degree takes values in certain equi variant homotopy groups of spheres. These groups were
computed for the particular case when the circle group S acts almost semi-freely. In the
present paper we consider general S -actions, which may differ on the domain and on the
range, with isotropy subspaces with one dimension more on the domain. In the special
case of self-maps the S1-degree is given by the usual degree of the invariant part, while for
one parameter S -maps one has an integer for each isotropy subgroup different from S . In
particular we recover all the S -degrees introduced in special cases by other authors and
we are also able to interpret period doubling results on the basis of our S -degree. The
applications concern essentially periodic solutions of ordinary differential equations.
Key words and phrases .
Equivariant topological degree, S -homotopy group of spheres.
ACKNOWLEDGMENTS
Part of the research for this paper was done during visits of the first author to the
universities of Cosenza and Roma II.
The support of these universities, of the Consiglio Nazionale delle Ricerche and of the
Centro Matematico Yito Volterra, of the University of Roma II, was essential for these visits.
The research for the problems with first integrals (sections 5.7 and 6.4) was conducted during
a visit of the first author to the University of New England, Australia. Long conversations
with E.N. Dancer were quite stimulating. Special thanks are due to Alma Rosa Rodriguez
who has beautifully typed and processed the manuscript.