Introduction |
Equivariant Cohomology in Topology / 1: |
Equivariant Cohomology via Classifying Bundles / 1.1: |
Existence of Classifying Spaces / 1.2: |
Bibliographical Notes for Chapter 1 / 1.3: |
G* Modules / 2: |
Differential-Geometric Identities / 2.1: |
The Language of Superalgebra / 2.2: |
From Geometry to Algebra / 2.3: |
Cohomology / 2.3.1: |
Acyclicity / 2.3.2: |
Chain Homotopies / 2.3.3: |
Free Actions and the Condition (C) / 2.3.4: |
The Basic Subcomplex / 2.3.5: |
Equivariant Cohomology of G* Algebras / 2.4: |
The Equivariant de Rham Theorem / 2.5: |
Bibliographical Notes for Chapter 2 / 2.6: |
The Weil Algebra / 3: |
The Koszul Complex / 3.1: |
Classifying Maps / 3.2: |
W* Modules / 3.4: |
Bibliographical Notes for Chapter 3 / 3.5: |
The Weil Model and the Cartan Model / 4: |
The Mathai-Quillen Isomorphism / 4.1: |
The Cartan Model / 4.2: |
Equivariant Cohomology of W* Modules / 4.3: |
H ((A ⊗ E)bas) does not depend on E / 4.4: |
The Characteristic Homomorphism / 4.5: |
Commuting Actions / 4.6: |
The Equivariant Cohomology of Homogeneous Spaces / 4.7: |
Exact Sequences / 4.8: |
Bibliographical Notes for Chapter 4 / 4.9: |
Cartan's Formula / 5: |
The Cartan Model for W* Modules / 5.1: |
Bibliographical Notes for Chapter 5 / 5.2: |
Spectral Sequences / 6: |
Spectral Sequences of Double Complexes / 6.1: |
The First Term / 6.2: |
The Long Exact Sequence / 6.3: |
Useful Facts for Doing Computations / 6.4: |
Functorial Behavior / 6.4.1: |
Gaps / 6.4.2: |
Switching Rows and Columns / 6.4.3: |
The Cartan Model as a Double Complex |
HG(A) as an S(g*)G-Module |
Morphisms of G* Modules |
Restricting the Group |
Bibliographical Notes for Chapter 6 / 6.5: |
Fermionic Integration / 7: |
Definition and Elementary Properties / 7.1: |
Integration by Parts / 7.1.1: |
Change of Variables / 7.1.2: |
Gaussian Integrals / 7.1.3: |
Iterated Integrals / 7.1.4: |
The Fourier Transform / 7.1.5: |
The Mathai-Quillen Construction |
The Fourier Transform of the Koszul Complex / 7.2: |
Bibliographical Notes for Chapter 7 / 7.3: |
Characteristic Classes / 8: |
Vector Bundles / 8.1: |
The Invariants / 8.2: |
G = U(n) / 8.2.1: |
G = O(n) / 8.2.2: |
G = SO(2n) / 8.2.3: |
Relations Between the Invariants / 8.3: |
Restriction from U(n) to O(n) / 8.3.1: |
Restriction from SO(2n) to U(n) / 8.3.2: |
Restriction from U(n) to U(k) × U(ℓ) / 8.3.3: |
Symplectic Vector Bundles / 8.4: |
Consistent Complex Structures / 8.4.1: |
Characteristic Classes of Symplectic Vector Bundles / 8.4.2: |
Equivariant Characteristic Classes / 8.5: |
Equivariant Chern classes / 8.5.1: |
Equivariant Characteristic Classes of a Vector Bundle Over a Point / 8.5.2: |
Equivariant Characteristic Classes as Fixed Point Data / 8.5.3: |
The Splitting Principle in Topology |
Bibliographical Notes for Chapter 8 |
Equivariant Symplectic Forms / 9: |
Equivariantly Closed Two-Forms / 9.1: |
The Case M = G / 9.2: |
Equivariantly Closed Two-Forms on Homogeneous Spaces / 9.3: |
The Compact Case / 9.4: |
Minimal Coupling / 9.5: |
Symplectic Reduction / 9.6: |
The Duistermaat-Heckman Theorem / 9.7: |
The Cohomology Ring of Reduced Spaces / 9.8: |
Flag Manifolds / 9.8.1: |
Delzant Spaces / 9.8.2: |
Reduction: The Linear Case / 9.8.3: |
Equivariant Duistermaat-Heckman |
Group Valued Moment Maps |
The Canonical Equivariant Closed Three-Form on G / 9.10.1: |
The Exponential Map / 9.10.2: |
G-Valued Moment Maps on Hamiltonian G-Manifolds / 9.10.3: |
Conjugacy Classes / 9.10.4: |
Bibliographical Notes for Chapter 9 / 9.11: |
The Thom Class and Localization / 10: |
Fiber Integration of Equivariant Forms / 10.1: |
The Equivariant Normal Bundle / 10.2: |
Modifying ν / 10.3: |
Verifying that τ is a Thom Form / 10.4: |
The Thom Class and the Euler Class / 10.5: |
The Fiber Integral on Cohomology / 10.6: |
Push-Forward in General / 10.7: |
Localization / 10.8: |
The Localization for Torus Actions / 10.9: |
Bibliographical Notes for Chapter 10 / 10.10: |
The Abstract Localization Theorem / 11: |
Relative Equivariant de Rham Theory / 11.1: |
Mayer-Vietoris / 11.2: |
S(g*) Modules / 11.3: |
The Chang-Skjelbred Theorem / 11.4: |
Some Consequences of Equivariant Formality / 11.6: |
Two Dimensional G-Manifolds / 11.7: |
A Theorem of Goresky-Kottwitz-Mac Pherson / 11.8: |
Bibliographical Notes for Chapter 11 / 11.9: |
Appendix |
Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie / Henri Cartan |
La transgression dans un groupe de Lie et dans un espace fibré principal |
Bibliography |
Index |
Introduction |
Equivariant Cohomology in Topology / 1: |
Equivariant Cohomology via Classifying Bundles / 1.1: |