Degree Theory for Equivariant Maps, the General S1-Action.
Ize, Jorge.
Degree Theory for Equivariant Maps, the General S1-Action. - 1st ed. - 1 online resource (194 pages) - Memoirs of the American Mathematical Society ; v.100 . - Memoirs of the American Mathematical Society .
Intro -- TABLE OF CONTENTS -- INTRODUCTION -- CHAPTER ONE: PRELIMINARIES -- 1.1. S[sup(1)]-actions -- 1.2. Almost semi-free action -- 1.3. Equivariant homotopy -- 1.4. The extension degree -- 1.5. Equivariant homotopy groups of spheres -- 1.6. Equivariant degree in the almost semi-free case -- CHAPTER TWO: EXTENSIONS OF 5[sup(1)]-MAPS -- 2.1. The fundamental cell lemma -- 2.2. The Extension Theorem -- 2.3. The Extension degree -- 2.4. Properties of the Extension degree -- CHAPTER THREE: HOMOTOPY GROUPS OF S[sup(1)]-MAPS -- 3.1. Trivial invariant part, the case p ≥ 1 -- 3.2. Nontrivial invariant part, the case p = 0 -- 3.3. Behavior under suspension -- 3.4. Relationship with the set of K-degrees -- 3.5. Symmetry Breaking -- CHAPTER FOUR: DEGREE OF S[sup(1)]-MAPS -- 4.1. Range of deg[sub(S1)](f -- Ω) -- 4.2. Infinite dimensional degree -- 4.3. Computation of the S[sup(1)]-degree -- 4.4. Global Continuation -- 4.5. Global Bifurcation -- CHAPTER FIVE: S[sup(1)]-INDEX OF AN ISOLATED NON-STATIONARY ORBIT AND APPLICATIONS -- 5.1. The case p ≥ 1 -- 5.2. The case p = 0 -- 5.3. p = 0, hyperbolic orbits -- 5.4. Autonomous differential equations -- 5.5. Gradient maps -- 5.6. Differential equations with fixed period -- 5.7. Differential equations with first integrals -- 5.8 Symmetry breaking for differential equations -- CHAPTER SIX: INDEX OF AN ISOLATED ORBIT OF STATIONARY SOLUTIONS AND APPLICATIONS -- 6.1. Computation of the S[sup(1)-Index -- 6.2. Application to bifurcation -- 6.3. Hopf bifurcation for autonomous differential equations -- 6.4. Hopf bifurcation for systems with first integrals -- 6.5. Hopf bifurcation and symmetry breaking -- CHAPTER SEVEN: VIRTUAL PERIODS AND ORBIT INDEX -- 7.1. Virtual periods -- 7.2. The Orbit Index -- APPENDIX: ADDITIVITY UP TO ONE SUSPENSION -- REFERENCES.
9781470400583
Topological degree.
Mappings (Mathematics).
Homotopy groups.
Sphere.
Electronic books.
QA3 -- .I94 1992eb
514/.2
Degree Theory for Equivariant Maps, the General S1-Action. - 1st ed. - 1 online resource (194 pages) - Memoirs of the American Mathematical Society ; v.100 . - Memoirs of the American Mathematical Society .
Intro -- TABLE OF CONTENTS -- INTRODUCTION -- CHAPTER ONE: PRELIMINARIES -- 1.1. S[sup(1)]-actions -- 1.2. Almost semi-free action -- 1.3. Equivariant homotopy -- 1.4. The extension degree -- 1.5. Equivariant homotopy groups of spheres -- 1.6. Equivariant degree in the almost semi-free case -- CHAPTER TWO: EXTENSIONS OF 5[sup(1)]-MAPS -- 2.1. The fundamental cell lemma -- 2.2. The Extension Theorem -- 2.3. The Extension degree -- 2.4. Properties of the Extension degree -- CHAPTER THREE: HOMOTOPY GROUPS OF S[sup(1)]-MAPS -- 3.1. Trivial invariant part, the case p ≥ 1 -- 3.2. Nontrivial invariant part, the case p = 0 -- 3.3. Behavior under suspension -- 3.4. Relationship with the set of K-degrees -- 3.5. Symmetry Breaking -- CHAPTER FOUR: DEGREE OF S[sup(1)]-MAPS -- 4.1. Range of deg[sub(S1)](f -- Ω) -- 4.2. Infinite dimensional degree -- 4.3. Computation of the S[sup(1)]-degree -- 4.4. Global Continuation -- 4.5. Global Bifurcation -- CHAPTER FIVE: S[sup(1)]-INDEX OF AN ISOLATED NON-STATIONARY ORBIT AND APPLICATIONS -- 5.1. The case p ≥ 1 -- 5.2. The case p = 0 -- 5.3. p = 0, hyperbolic orbits -- 5.4. Autonomous differential equations -- 5.5. Gradient maps -- 5.6. Differential equations with fixed period -- 5.7. Differential equations with first integrals -- 5.8 Symmetry breaking for differential equations -- CHAPTER SIX: INDEX OF AN ISOLATED ORBIT OF STATIONARY SOLUTIONS AND APPLICATIONS -- 6.1. Computation of the S[sup(1)-Index -- 6.2. Application to bifurcation -- 6.3. Hopf bifurcation for autonomous differential equations -- 6.4. Hopf bifurcation for systems with first integrals -- 6.5. Hopf bifurcation and symmetry breaking -- CHAPTER SEVEN: VIRTUAL PERIODS AND ORBIT INDEX -- 7.1. Virtual periods -- 7.2. The Orbit Index -- APPENDIX: ADDITIVITY UP TO ONE SUSPENSION -- REFERENCES.
9781470400583
Topological degree.
Mappings (Mathematics).
Homotopy groups.
Sphere.
Electronic books.
QA3 -- .I94 1992eb
514/.2