An introduction to Morse theory. Transl. from the Japanese by Kiki Hudson and Masahico Saito (Q2778313)

This excellent monograph deals with an important topic of modern mathematics: Morse-theory of finite-dimensional manifolds and some of its applications. It represents the English translation of the Japanese edition published in 1997.NEWLINENEWLINENEWLINEThe book contains five chapters and I will briefly present each of them. Chapter 1 is ``Morse Theory on Surfaces'' and it presents basic notions and results concerning critical points of functions, Hessian, the Morse lemma, Morse functions on surfaces, and handle decompositions. Chapter 2 is entitled ``Extension to General Dimension'' and it contains four sections related to the following aspects: manifolds of dimension \(m\), Morse functions, gradient-like vector fields, raising and lowering critical points. In Chapter 3, ``Handlebodies'', handle decompositions associated to Morse functions in general dimensions are presented. Some Morse functions on classical manifolds such as projective spaces and Lie groups are constructed, and their numbers of critical points as well as the associated indices are computed. In each case the handle decomposition is very well explained. Chapter 4 has the title ``Homology of Manifolds'' and the main problems studied are the following: homology groups, Morse inequalities, Poincaré duality, intersection forms. Chapter 5, ``Low-dimensional Manifolds'', is devoted to the study of low-dimensional (dimension 4 and less) manifolds. In these dimensions, handlebodies can be explicitly visualized by Heegaard diagrams and Kirby diagrams. The author also mentions the connections to knot theory and the following aspects: fundamental groups, closed surfaces and 3-dimensional manifolds, 4-dimensional manifolds (Heegaard diagrams and the Kirby calculus).NEWLINENEWLINENEWLINEEach chapter ends with a summary and with some proposed exercises. The book also contains a ``View from Current Mathematics'', a short part representing a brief history of Morse theory. The book ends with answers and solutions to all proposed exercises and problems, with a suggestive bibliography containing 23 titles and with a useful index.NEWLINENEWLINENEWLINEThe book is written in a very clear and rigorous manner and it is recommended for researchers and graduate students working in differential topology, differential geometry and nonlinear analysis.