Translations of Mathematical Monographs Iwanami Series in Modern Mathematics 2002; 219 pp; softcover Volume: 208 ISBN10: 0821810227 ISBN13: 9780821810224 List Price: US$53 Member Price: US$42.40 Order Code: MMONO/208
 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 finitedimensional and infinitedimensional spaces. In particular, it is believed that Morse theory on infinitedimensional spaces will become more and more important in the future as mathematics advances. This book describes Morse theory for finite dimensions. Finitedimensional Morse theory has an advantage in that it is easier to present fundamental ideas than in infinitedimensional Morse theory, which is theoretically more involved. Therefore, finitedimensional Morse theory is more suitable for beginners to study. On the other hand, finitedimensional 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 "The first twothirds 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 higherdimensional 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  Morse theory on surfaces
 Extension to general dimensions
 Handlebodies
 Homology of manifolds
 Low dimensional manifolds
 A view from current mathematics
 Answers to exercises
 Bibliography
 Recommended reading
 Index
