An Introduction to Morse Theory (Translations of Mathematical Monographs, Vol. 208) (Translations of Mathematical Monographs)

Chapter 1. Morse Theory On Surfaces 1 -- 1.1. Critical Points Of Functions 1 -- 1.2. Hessian 3 -- 1.3. The Morse Lemma 8 -- 1.4. Morse Functions On Surfaces 14 -- 1.5. Handle Decomposition 22 -- A. The Case When The Index Of Po Is Zero 26 -- B. The Case When The Index Of Po Is One 26 -- C. The Case When The Index Of Po Is Two 29 -- D. Handle Decompositions 30 -- Chapter 2. Extension To General Dimensions 33 -- 2.1. Manifolds Of Dimension M 33 -- A. Functions On A Manifold And Maps Between Manifolds 33 -- B. Manifolds With Boundary 34 -- C. Functions And Maps On Manifolds With Boundary 38 -- 2.2. Morse Functions 41 -- A. Morse Functions On M-manifolds 41 -- B. The Morse Lemma For Dimension M 44 -- C. Existence Of Morse Functions 47 -- 2.3. Gradient-like Vector Fields 56 -- A. Tangent Vectors 56 -- B. Vector Fields 61 -- C. Gradient-like Vector Fields 63 -- 2.4. Raising And Lowering Critical Points 69 -- Chapter 3. Handlebodies 73 -- 3.1 Handle Decompositions Of Manifolds 73 -- 3.3. Sliding Handles 105 -- 3.4. Canceling Handles 120 -- Chapter 4. Homology Of Manifolds 133 -- 4.1. Homology Groups 133 -- 4.2. Morse Inequality 141 -- A. Handlebodies And Cell Complexes 141 -- B. Proof Of The Morse Inequality 147 -- C. Homology Groups Of Complex Projective Space Cp[superscript M] 147 -- 4.3. Poincare Duality 148 -- A. Cohomology Groups 148 -- B. Proof Of Poincare Duality 150 -- 4.4. Intersection Forms 158 -- A. Intersection Numbers Of Submanifolds 159 -- B. Intersection Forms 159 -- C. Intersection Numbers Of Submanifolds And Intersection Forms 163 -- Chapter 5. Low-dimensional Manifolds 167 -- 5.1. Fundamental Groups 167 -- 5.2. Closed Surfaces And 3-dimensional Manifolds 173 -- A. Closed Surfaces 173 -- B. 3-dimensional Manifolds 181 -- 5.3. 4-dimensional Manifolds 186 -- A. Heegaard Diagrams For 4-dimensional Manifolds 186 -- B. The Case N = D[superscript 4] 190 -- C. Kirby Calculus 194 -- A View From Current Mathematics 199. Yukio Matsumoto ; Translated By Kiki Hudson, Masahico Saito. Includes Bibliographical References (p. 213-214) And Index. 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. 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. This book introduces basic concepts related to finite dimensions, including critical points, the Hessian, and handle decompressions. It first uses surfaces to illustrate these ideas, and then generalizes them to apply to higher dimensions. This treatment then informs a discussion of handlebodies, homology, and low-dimensional manifold theory. Illustrations are provided throughout. Annotation c. Book News, Inc., Portland, OR (booknews.com)