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.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in manifolds and cell complexes.