Iwanami Series in Modern Mathematics
2002; 219 pp; Softcover
MSC: Primary 57;
Print ISBN: 978-0-8218-1022-4
Product Code: MMONO/208
List Price: $56.00
AMS Member Price: $44.80
MAA Member Price: $50.40
Electronic ISBN: 978-1-4704-4633-8
Product Code: MMONO/208.E
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An Introduction to Morse Theory
Share this pageYukio Matsumoto
In a very broad sense, “spaces” are objects of
study in geometry, and “functions” are 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, its
feature is to look at the critical points of a function, and to derive
information on the shape of the space from the information about the
critical points.
Morse theory deals with both finite-dimensional and infinite-dimensional
spaces. In particular, it is believed that Morse theory on infinite-dimensional
spaces will become more and more important in the future as mathematics
advances.
This book describes Morse theory for finite dimensions. Finite-dimensional
Morse theory has an advantage in that it is easier to present fundamental ideas
than in infinite-dimensional Morse theory, which is theoretically more
involved. Therefore, finite-dimensional Morse theory is more suitable for
beginners to study.
On the other hand, finite-dimensional Morse theory has its own significance,
not just as a bridge to infinite dimensions. It 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 will continue to be a treasure in
geometry for years to come.
This textbook aims at introducing Morse theory to advanced undergraduates and
graduate students. It is the English translation of a book originally published
in Japanese.
Readership
Advanced undergraduates, graduate students, and research mathematicians interested in manifolds and cell complexes.
Reviews & Endorsements
The first two-thirds of the book is accessible to anyone with knowledge of calculus in \(\mathbf{R}^n\) and elementary topology. The book begins with the basic ideas of Morse theory … on surfaces. This avoids some of the technical problems of the higher-dimensional case … and allows a very pictorial introduction. The text, which was translated in part by Kiki Hudson, and in part by Masahico Saito, is very readable.
-- Mathematical Reviews
Table of Contents
Table of Contents
An Introduction to Morse Theory
- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Preface to English translation xi12
- Objectives xiii14
- Morse theory on surfaces 116
- Extension to general dimensions 3348
- Handlebodies 7388
- Homology of manifolds 133148
- Low-dimensional manifolds 167182
- A view from current mathematics 199214
- Answers to exercises 203218
- Bibliography 213228
- Recommended reading 215230
- Index 217232
- Back Cover Back Cover1238