An Introduction to Morse Theory

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American Mathematical Soc., 2002 - Mathematics - 219 pages
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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
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