An Introduction to Morse Theory
About this Title
Yukio Matsumoto, University of Tokyo, Tokyo, Japan. Translated by Masahico Saito and Kiki Hudson
Publication: Translations of Mathematical Monographs
Publication Year: 2002; Volume 208
ISBNs: 978-0-8218-1022-4 (print); 978-1-4704-4633-8 (online)
MathSciNet review: MR1873233
MSC: Primary 57R70; Secondary 57-01, 57M25
In a very broad sense, ‘“spaces” are the primary objects of study in geometry, and “functions” are the objects of study in analysis. There are, however, deep relations between functions defined on a space and the shape of the space, and the study of these relations is the main theme of Morse theory. In particular, Morse's original insight was to examine the critical points of a function and to derive information about the shape of the space from the information about the critical points.
This book describes finite-dimensional Morse theory, which is an indispensable tool in the topological study of manifolds. That is, one can decompose manifolds into fundamental blocks such as cells and handles by Morse theory, and thereby compute a variety of topological invariants and discuss the shapes of manifolds. These aspects of Morse theory date from its origins and continue to be important in geometry and mathematical physics.
This textbook provides an introduction to Morse theory suitable for advanced undergraduates and graduate students.
Advanced undergraduates, graduate students, and research mathematicians interested in manifolds and cell complexes.
Table of Contents
- Morse theory on surfaces
- Extension to general dimensions
- Homology of manifolds
- Low-dimensional manifolds
- A view from current mathematics
- Answers to exercises