An Introduction to Morse TheoryAmerican Mathematical Soc., 2002 - 219 pagine 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 withboth 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 theoryhas 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 andhandles 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 graduatestudents. It is the English translation of a book originally published in Japanese. |
Sommario
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 |
Parole e frasi comuni
1-handle 3-manifold assume attaching map attaching sphere base point belt sphere C₁ called cell complex Chapter closed manifold closed surface co-core compute consider continuous map coordinate neighborhood coordinate system x1 COROLLARY corresponds critical point critical value definition degenerate critical point denoted diffeomorphism diffeomorphism type dimension disk Euler number EXAMPLE ƏD² Figure function f fundamental group gradient-like vector field h₁ h₂ handle decomposition handlebody Heegaard diagram Hessian homeomorphic homology groups homotopy integral curve intersection form intersection number isotopy Kirby Lemma loop manifolds with boundary map h mapping cylinder matrix Mc-e Morse function Morse theory non-degenerate critical point obtain perturbed Poincaré duality point of index points of f positive number projective space proof of Theorem prove real numbers respect smooth function SO(m standard form SU(m submanifolds tangent vector U₁ მე მყ