# An Introduction to Morse Theory

American Mathematical Soc., 2002 - Mathematics - 219 pages
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.

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### Contents

 Morse Theory on Surfaces 1 12 Hessian 3 13 The Morse lemma 8 14 Morse functions on surfaces 14 15 Handle decomposition 22 Summary 31 Extension to General Dimensions 33 22 Morse functions 41
 34 Canceling handles 120 Summary 130 Exercises 131 Homology of Manifolds 133 42 Morse inequality 141 43 Poincaré duality 148 44 Intersection forms 158 Summary 164

 23 Gradientlike vector fields 56 24 Raising and lowering critical points 69 Summary 71 Handlebodies 73 32 Examples 83 33 Sliding handles 105
 Lowdimensional Manifolds 167 52 Closed surfaces and 3dimensional manifolds 173 53 4dimensional manifolds 186 Summary 197 Copyright