Introductory Lectures on Equivariant Cohomology

Name: Introductory Lectures on Equivariant Cohomology | (AMS-204)
Brand: Princeton University Press
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(AMS-204)

Loring W. Tu(Autor*in)

Princeton University Press

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Erschienen am 3. März 2020

200 Seiten

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Inhalt

Cover
Title
Copyright
Dedication
Contents
List of Figures
Preface
Acknowledgments
I Equivariant Cohomology in the Continuous Category
1 Overview
1.1 Actions of a Group
1.2 Orbits, Stabilizers, and Fixed Point Sets
1.3 Homogeneous Spaces
1.4 Equivariant Cohomology
Problems
2 Homotopy Groups and CW Complexes
2.1 Homotopy Groups
2.2 Fiber Bundles
2.3 Homotopy Exact Sequence of a Fiber Bundle
2.4 Attaching Cells
2.5 CW Complexes
2.6 Manifolds and CW Complexes
2.7 The Infinite Sphere
Problems
3 Principal Bundles
3.1 Principal Bundles
3.2 The Pullback of a Fiber Bundle
Problems
4 Homotopy Quotients and Equivariant Cohomology
4.1 A First Candidate for Equivariant Cohomology
4.2 Homotopy Quotients
4.3 Cartan's Mixing Space and Cartan's Mixing Diagram
4.4 Equivariant Cohomology Is Well-Defined
4.5 Algebraic Structure of Equivariant Cohomology
Problems
5 Universal Bundles and Classifying Spaces
5.1 Universal Bundles
5.2 Uniqueness of a CW Classifying Space
5.3 Milnor's Construction
5.4 Equivariant Cohomology of a Point
6 Spectral Sequences
6.1 Leray's Theorem
6.2 Leray's Theorem on the Product Structure
6.3 Example: The Cohomology of CP^2
Problems
7 Equivariant Cohomology of S^2 Under Rotation
7.1 Homotopy Quotient as a Fiber Bundle
7.2 Equivariant Cohomology of S^2 Under Rotation
Problems
8 A Universal Bundle for a Compact Lie Group
8.1 The Stiefel Variety
8.2 A Principal O(k)-Bundle
8.3 Homotopy Groups of a Stiefel Variety
8.4 Closed Subgroups of a Lie Group
8.5 Universal Bundle for a Compact Lie Group
8.6 Universal Bundle for a Direct Product
8.7 Infinite-Dimensional Manifolds
Problems
9 General Properties of Equivariant Cohomology
9.1 Functorial Properties
9.2 Free Actions
9.3 Coefficient Ring of Equivariant Cohomology
9.4 Equivariant Cohomology of a Disjoint Union
Problems
II Differential Geometry of a Principal Bundle
10 The Lie Derivative and Interior Multiplication
10.1 The Lie Derivative of a Vector Field
10.2 The Lie Derivative of a Differential Form
10.3 Interior Multiplication
Problems
11 Fundamental Vector Fields
11.1 Fundamental Vector Fields
11.2 Zeros of a Fundamental Vector Field
11.3 Vertical Vectors on a Principal Bundle
11.4 Translate of a Fundamental Vector Field
11.5 The Lie Bracket of Fundamental Vector Fields
Problems
12 Basic Forms
12.1 Basic Forms on R^2
12.2 Invariant Forms
12.3 Horizontal Forms
12.4 Basic Forms
Problems
13 Integration on a Compact Connected Lie Group
13.1 Bi-Invariant Forms on a Compact Connected Lie Group
13.2 Integration over a Compact Connected Lie Group
13.3 Invariance of the Integral
13.4 The Pullback of an Integral
13.5 Differentiation Under the Integral Sign
13.6 Cohomology Does Not Commute with Invariants
Problems
14 Vector-Valued Forms
14.1 Vector-Valued Forms
14.2 Lie Algebra Valued Forms
14.3 Matrix-Valued Forms
Problems
15 The Maurer-Cartan Form
15.1 The Lie Algebra g of a Lie Group and Its Dual g^?
15.2 Maurer-Cartan Equation with Respect to a Basis
15.3 The Maurer-Cartan Form
16 Connections on a Principal Bundle
16.1 Maps of Vector Bundles
16.2 Vertical and Horizontal Subbundles
16.3 Connections on a Principal Bundle
16.4 The Maurer-Cartan Form Is a Connection
16.5 Existence of a Connection on a Principal Bundle
17 Curvature on a Principal Bundle
17.1 Curvature
17.2 Properties of Curvature
III The Cartan Model
18 Differential Graded Algebras
18.1 Differential Graded Algebras
18.2 Tensor Product of Differential Graded Algebras
18.3 The Basic Subcomplex of a g-Differential Graded Algebra
Problems
19 The Weil Algebra and the Weil Model
19.1 The Weil Algebra and the Weil Map
19.2 The Weil Map Relative to a Basis
19.3 The Weil Algebra as a g-DGA
19.4 The Cohomology of the Weil Algebra
19.5 An Algebraic Model for the Universal Bundle
19.6 An Algebraic Model for the Homotopy Quotient
19.7 Functoriality of the Weil Model
Problems
20 Circle Actions
20.1 The Weil Algebra for a Circle Action
20.2 The Weil Model for a Circle Action
20.3 The Cartan Model for a Circle Action
20.4 The Cartan Differential for a Circle Action
20.5 Example: The Action of a Circle on a Point
21 The Cartan Model in General
21.1 The Weil-Cartan Isomorphism
21.2 The Cartan Differential
21.3 Intrinsic Description of the Cartan Differential
21.4 Pullback of Equivariant Forms
21.5 The Equivariant de Rham Theorem
21.6 Equivariant Forms for a Torus Action
21.7 Example: The Action of a Torus on a Point
21.8 Equivariantly Closed Extensions
Problems
22 Outline of a Proof of the Equivariant de Rham Theorem
22.1 The Cohomology of the Base
22.2 Equivariant de Rham Theorem for a Free Action
22.3 Equivariant de Rham Theorem in General
22.4 Cohomology of a Classifying Space
IV Borel Localization
23 Localization in Algebra
23.1 Localization with Respect to a Variable
23.2 Localization and Torsion
23.3 Antiderivations Under Localization
23.4 Localization and Exactness
Problems
24 Free and Locally Free Actions
24.1 Equivariant Cohomology of a Free Action
24.2 Locally Free Actions
24.3 Equivariant Cohomology of a Locally Free Circle Action
Problems
25 The Topology of a Group Action
25.1 Smooth Actions of a Compact Lie Group
25.2 Equivariant Vector Bundles
25.3 The Normal Bundle of a Submanifold
25.4 Equivariant Tubular Neighborhoods
25.5 Equivariant Mayer-Vietoris Sequence
26 Borel Localization for a Circle Action
26.1 Borel Localization
26.2 Example: The Ring Structure of H^*S^1 (S^2)
Problems
V The Equivariant Localization Formula
27 A Crash Course in Representation Theory
27.1 Representations of a Group
27.2 Local Data at a Fixed Point
Problems
28 Integration of Equivariant Forms
28.1 Manifolds with Boundary
28.2 Integration of Equivariant Forms
28.3 Stokes's Theorem for Equivariant Integration
28.4 Integration of Equivariant Forms for a Circle Action
Problems
29 Rationale for a Localization Formula
29.1 Circle Actions with Finitely Many Fixed Points
29.2 The Spherical Blow-Up
Problems
30 Localization Formulas
30.1 Equivariant Localization Formula for a Circle Action
30.2 Application: The Area of a Sphere
30.3 Equivariant Characteristic Classes
30.4 Localization Formula for a Torus Action
Problems
31 Proof of the Localization Formula for a Circle Action
31.1 On the Spherical Blow-Up
31.2 Surface Area of a Sphere
32 Some Applications
32.1 Integration of Invariant Forms
32.2 Rational Cohomology of a Homogeneous Space
32.3 Topological Invariants of a Homogeneous Space
32.4 Symplectic Geometry and Classical Mechanics
32.5 Stationary Phase Approximation
32.6 Exponents at Fixed Points
32.7 Gysin Maps
Appendices
A. Proof of the Equivariant de Rham Theorem by Loring W. Tu and Alberto Arabia
A.1 The Weil Algebra
A.2 The Weil Map
A.3 Cohomology of Basic Subcomplexes
A.4 Cartan's Operator K
A.5 A Degree-Lowering Property
A.6 The Weil-Cartan Isomorphism
A.7 Equivariant de Rham Theorem for a Free Action
A.8 Approximation Theorems
A.9 Proof of the Equivariant de Rham Theorem in General
A.10 Approximations of EG
A.11 Approximations of the Homotopy Quotient MG
A.12 A Spectral Sequence for the Cartan Model
A.13 Ordinary Cohomology and the Cohomology of the Cartan Model
B. A Comparison Theorem for Spectral Sequences by Alberto Arabia
C. Commutativity of Cohomology with Invariants by Alberto Arabia
Hints and Solutions to Selected End-of-Section Problems
List of Notations
Bibliography
Index

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