Lectures on morse homology莫尔斯同源演讲 pdf 在线 2025 epub 免费 书籍 下载

This book presents in great detail all the results one needs to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. Most of these results can be found scattered throughout the literature dating from the mid to late 1900's in some form or other, but often the results are proved in different contexts with a multitude of different notations and different goals. This book collects all these results together into a single reference with complete and detailed proofs.

The core material in this book includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory. More advanced topics include Morse theory on Grassmann manifolds and Lie groups, and an overview of Floer homology theories. With the stress on completeness and by its elementary approach to Morse homology, this book is suitable as a textbook for a graduate level course, or as a reference for working mathematicians and physicists.

Preface

1. Introduction

　1.1 Overview

　1.2 Algebraic topology

　1.3 Basic Morse theory

　1.4 Stable and unstable manifolds

　1.5 Basic differential topology

　1.6 Morse-Smale functions

　1.7 The Morse Homology Theorem

　1.8 Morse theory on Grassmann manifolds

　1.9 Floer homology theories

　1.10 Guide to the book

2. The CW-Homology Theorem

　2.1 Singular homology

　2.2 Singular cohomology

　2.3 CW-complexes

　2.4 CW-homology

　2.5 Some homotopy theory

3. Basic Morse Theory

　3.1 Morse functions

　3.2 The gradient flow of a Morse function

　3.3 The CW-complex associated to a Morse function

　3.4 The Morse Inequalities

　3.5 Morse-Bott functions

4. The Stable/Unstable Manifold Theorem

　4.1 The Stable/Unstable Manifold Theorem for a Morse function

　4.2 The Local Stable Manifold Theorem

　4.3 The Global Stable/Unstable Manifold Theorem

　4.4 Examples of stable/unstable manifolds

5. Basic Differential Topology

　5.1 Immersions and submersions

　5.2 Transversality

　5.3 Stability

　5.4 General position

　5.5 Stability and density for Morse functions

　5.6 Orientations and intersection numbers

　5.7 The Lefschetz Fixed Point Theorem

6. Morse-Smale Functions

　6.1 The Morse-Smale transversality condition

　6.2 The A-Lemma

　6.3 Consequences of the A-Lemma

　6.4 The CW-complex associated to a Morse-Smale function

7. The Morse Homology Theorem

　7.1 The Morse-Smale-Witten boundary operator

　7.2 Examples using the Morse Homology Theorem

　7.3 The Conley index

　7.4 Proof of the Morse Homology Theorem

　7.5 Independence of the choice of the index pairs

8. Morse Theory On Grassmann Manifolds

　8.1 Morse theory on the adjoint orbit of a Lie group

　8.2 A Morse function on an adjoint orbit of the unitary group

　8.3 An almost complex structure on the adjoint orbit

　8.4 The critical points and indices of fA : U(n+k)·xo→R

　8.5 A Morse function on the complex Grassmann manifold

　8.6 The gradient flow lines of fA : Gn,n,+k(C)→R

　8.7 The homology of Gn, n+k(C)

　8.8 Further generalizations and applications

9. An Overview of Floer Homology Theories

　9.1 Introduction to Floer homology theories

　9.2 Symplectic Floerhomology

　9.3 Floer homology for Lagrangian intersections

　9.4 Instanton Floer homology

　9.5 A symplectic flavor of the instanton homology

Hints and References for Selected Problems

Bibliography

Symbol Index

Index

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