Cover
Progress in Mathematics 111
Title Page
Copyright Page
Contents
List of symbols
Chapter 1. Introduction
1.1 Background
1.1.1 Classical Morse Theory
1.1.2 Relative Morse Theory
1.1.3 The Continuation Principle
1.2 Overview
1.2.1 The Construction of the Morse Homology
1.2.2 The Axiomatic Approach
1.3 Remarks on the Methods
1.4 Table of Contents
1.5 Acknowledgments
Chapter 2. The Trajectory Spaces
2.1 The Construction of the Trajectory Spaces
2.2 Fredholm Theory
2.2.1 The Fredholm Operator on the Trivial Bundle
2.2.2 The Fredholm Operator on Non-Trivial Bundles
2.2.3 Generalization to Fredholm maps
2.3 Transversality
2.3.1 The Regularity Conditions
2.3.2 The Regularity Results
2.4 Compactness
2.4.1 The Space of Unparametrized Trajectories
2.4.2 The Compactness Result for Unparametrized Trajectories
2.4.3 The Compactness Result for Homotopy Trajectories
2.5 Gluing
2.5.1 Gluing for the Time-Independent Trajectory Spaces
2.5.2 Gluing of Trajectories of the Time-Dependent Gradient Flow
Chapter 3. Orientation
3.1 Orientation and Gluing in the Trivial Case
3.1.1 The Determinant Bundle
3.1.2 Gluing and Orientation for Fredholm Operators
3.2 Coherent Orientation
3.2.1 Orientation and Gluing on the Manifold M
Chapter 4. Morse Homology Theory
4.1 The Main Theorems of Morse Homology
4.1.1 Canonical Orientations
4.1.2 The Morse Complex
4.1.3 The Canonical Isomorphism
4.1.4 Topology and Coherent Orientation
4.2 The EilenbergβSteenrod Axioms
4.2.1 Extension of Morse Functions and Induced Morse Functions on Vector Bundles
4.2.2 The Homology Functor and Homotopy Invariance
4.2.3 Relative Morse Homology
4.2.4 Summary
4.3 The Uniqueness Result
Chapter 5. Extensions
5.1 Morse Cohomology
5.2 PoincarΓ© Duality
5.3 Products
Appendix A. Curve Spaces and Banach Bundles
A.1 The Manifold of Maps P[sup(1,2)]sub(x,y)
A.2 Banach Bundles on P[sup(1,2)]sub(x,y)
Appendix B. The Geometric Boundary Operator
Bibliography
Index
