An invitation to Morse theory / Liviu Nicolaescu.

Format:

Book

Author/Creator:

Nicolaescu, Liviu I.

Series:

Universitext

Language:

English

Subjects (All):

Morse theory.

Physical Description:

xiv, 241 pages : illustrations ; 24 cm.

Place of Publication:

New York ; London : Springer, 2007.

Summary:

This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory (over the reals). The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.k.a. Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds.

Contents:

Notations and conventions XI

1 Morse Functions 1

1.1 The Local Structure of Morse Functions 1

1.2 Existence of Morse Functions 17

2 The Topology of Morse Functions 23

2.1 Surgery, Handle Attachment, and Cobordisms 23

2.2 The Topology of Sublevel Sets 34

2.3 Morse Inequalities 46

2.4 Morse-Smale Dynamics 54

2.5 Morse-Floer Homology 64

2.6 Morse-Bott Functions 70

2.7 Min-Max Theory 74

3 Applications 87

3.1 The Cohomology of Complex Grassmannians 87

3.2 Lefschetz Hyperplane Theorem 92

3.3 Symplectic Manifolds and Hamiltonian Flows 99

3.4 Morse Theory of Moment Maps 117

3.5 S[superscript 1]-Equivariant Localization 135

4 Basics of Complex Morse Theory 151

4.1 Some Fundamental Constructions 152

4.2 Topological Applications of Lefschetz Pencils 156

4.3 The Hard Lefschetz Theorem 166

4.4 Vanishing Cycles and Local Monodromy 172

4.5 Proof of the Picard-Lefschetz formula 182

4.6 Global Picard-Lefschetz Formulae 187.

Notes:

Includes bibliographical references (pages [233]-235) and index.

ISBN:

0387495096

9780387495095

OCLC:

76851150