Morse理论入门

preface

notations and conventions

1 morse functions

1.1 the local structure of morse functions

1.2 existence of morse functions

2 the topology of morse functions

2.1 surgery, handle attachment, and cobordisms.

2.2 the topology of sublevel sets

2.3 morse inequalities

2.4 morse-smale dynamics

2.5 morse-floer homology

2.6 morse-bott functions

2.7 min-max theory

3　applications

3.1 the cohomology of complex grassmannians

3.2 lefschetz hyperplane theorem

3.3 symplectic manifolds and hamiltonian flows

3.4 morse theory of moment maps

3.5 s1-equivariant localization

4 basics of complex morse theory

as the the title suggests, the goal of this book is to give the reader a taste of the "unreasonable effectiveness" of morse theory. the main idea behind thistechnique can be easily visualized.　　suppose m is a smooth, compact manifold, which for simplicity we assume is embedded in a euclidean space e. we would like to understand basictopological invariants of m such as its homology, and we attempt a "slicing" technique.