The Theory of Transformation Groups

Preface
Contents
1. Topological transformation groups
	1.1 Definitions and fundamental properties of transformation groups
	1.2 Topological groups
	1.3 Examples of topological groups and semi-direct products
	1.4 Topological transformation groups
	1.5 Fixed point sets
	1.6 Orbits and orbit spaces
	1.7 Homogeneous spaces and equivariant maps between them
	1.8 Orbit types and isotropy types
	1.9 Induced transformation groups
	Exercises
2. Fibre bundles
	2.1 Fibre bundles
	2.2 G_1-G_2 bispaces and twisted products
	2.3 Principal bundles and associated bundles
	2.4 Fibre products and induced bundles
	2.5 G-vector bundles
	2.6 The classification of G-vector bundles over G/H
	2.7 Invariant integration on a compact topological group
	2.8 G-invariant metrics on G-vector bundles and orthogonal complements
	Exercises
3. Manifolds and Lie groups
	3.1 Manifolds and examples
	3.2 Lie groups and their actions
	3.3 Tangent spaces
	3.4 Tangent G-vector bundles
	3.5 Vector fields and 1-parameter groups
	3.6 Left invariant vector fields on Lie groups and exponential map
	3.7 A closed subgroup of a Lie group is a Lie subgroup
	3.8 Differential structures on homogeneous spaces G/H and local cross-sections
	3.9 Semi-direct products of Lie groups and examples of homogeneous spaces
	Exercises
4. Structure of G-manifolds
	4.1 G-embeddings and orbits
	4.2 Geodesics and the exponential map
	4.3 G-invariant tubular neighbourhoods
	4.4 The slice theorem
	4.5 Structure theorems for G-manifolds
	4.6 Decomposition and blowing up of G-manifolds
	4.7 Finiteness of number of orbit types
	4.8 Principal orbits
	4.9 Dimension of compact Lie groups which act effectively on manifolds
	Exercises
5. Algebraization
	5.1 G-complexes
	5.2 Formulae for Euler characteristics and Möbius inversion
	5.3 The transfer
	5.4 Z_p-actions and the Smith homology group
	5.5 The Smith theory
	5.6 Applications to G-manifolds
	5.7 The Lefschetz fixed point theorem
	5.8 Free and semi-free transformation groups on spheres
	Exercises
6. Localization and Riemann-Roch type theorems
	6.1 Equivariant Gysin homomorphism
	6.2 Localization theorems
	6.3 The G-signature theorem
	6.4 Equivariant Riemann-Roch theorems
	6.5 The vanishing theorem of the \hat{A}-genus
	6.6 Vanishing theorems of exp(c/2)· \hat{A}-genera
	Exercises
Answers to Exercises
	Chapter 1
	Chapter 2
	Chapter 3・4
	Chapter 5・6
Bibliography
	1-22
	23-47
	48-69
	70-89
	90-113
	114-137
	138-140
Index