By Yukio Matsumoto

ISBN-10: 0821810227

ISBN-13: 9780821810224

Bankruptcy 1. Morse concept on Surfaces 1 -- 1.1. serious issues of services 1 -- 1.2. Hessian three -- 1.3. The Morse lemma eight -- 1.4. Morse services on surfaces 14 -- 1.5. deal with decomposition 22 -- a. The case while the index of po is 0 26 -- b. The case while the index of po is one 26 -- c. The case whilst the index of po is 2 29 -- d. deal with decompositions 30 -- bankruptcy 2. Extension to basic Dimensions 33 -- 2.1. Manifolds of size m 33 -- a. services on a manifold and maps among manifolds 33 -- b. Manifolds with boundary 34 -- c. services and maps on manifolds with boundary 38 -- 2.2. Morse services forty-one -- a. Morse capabilities on m-manifolds forty-one -- b. The Morse lemma for size m forty four -- c. lifestyles of Morse features forty seven -- 2.3. Gradient-like vector fields fifty six -- a. Tangent vectors fifty six -- b. Vector fields sixty one -- c. Gradient-like vector fields sixty three -- 2.4. elevating and decreasing severe issues sixty nine -- bankruptcy three. Handlebodies seventy three -- 3.1 deal with decompositions of manifolds seventy three -- 3.3. Sliding handles one zero five -- 3.4. Canceling handles one hundred twenty -- bankruptcy four. Homology of Manifolds 133 -- 4.1. Homology teams 133 -- 4.2. Morse inequality 141 -- a. Handlebodies and mobile complexes 141 -- b. facts of the Morse inequality 147 -- c. Homology teams of advanced projective area CP[superscript m] 147 -- 4.3. Poincare duality 148 -- a. Cohomology teams 148 -- b. facts of Poincare duality one hundred fifty -- 4.4. Intersection varieties 158 -- a. Intersection numbers of submanifolds 159 -- b. Intersection varieties 159 -- c. Intersection numbers of submanifolds and intersection varieties 163 -- bankruptcy five. Low-dimensional Manifolds 167 -- 5.1. basic teams 167 -- 5.2. Closed surfaces and three-d manifolds 173 -- a. Closed surfaces 173 -- b. third-dimensional manifolds 181 -- 5.3. four-dimensional manifolds 186 -- a. Heegaard diagrams for four-dimensional manifolds 186 -- b. The case N = D[superscript four] a hundred ninety -- c. Kirby calculus 194 -- A View from present arithmetic 199

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A31 a32 a33 . . a3i . . . ai1 ai2 ai3 . . aii . . . g. 14000 . . 13999 . . but otherwise the decimal expansion is unique. In order to have uniqueness, we only consider decimal expansions which do not end in an infinite sequence of 9’s. e. a real number z not in the range of f . b1 b2 b3 . . bi . . by “going down the diagonal” as follows: Select b1 = a11 , b2 = a22 , b3 = a33 , . . ,bi = aii ,. . We make sure that the decimal expansion for z does not end in an infinite sequence of 9’s by also restricting bi = 9 for each i; one explicit construction would be to set bn = ann + 1 mod 9.

E. a real number z not in the range of f . b1 b2 b3 . . bi . . by “going down the diagonal” as follows: Select b1 = a11 , b2 = a22 , b3 = a33 , . . ,bi = aii ,. . We make sure that the decimal expansion for z does not end in an infinite sequence of 9’s by also restricting bi = 9 for each i; one explicit construction would be to set bn = ann + 1 mod 9. 40)11 , since for each i it is clear that z differs from the i’th member of the sequence in the i’th place of z’s decimal expansion. But this implies that f is not onto.

If ρ ⊆ A × B is a relation with domain A, then there exists a function f : A → B with f ⊆ ρ. 3. If g : B → A is onto, then there exists f : A → B such that g ◦ f = identity on A. 4. For some of the most commonly used equivalent forms we need some further concepts. 2 A relation ≤ on a set X is a partial order on X if, for all x, y, z ∈ X, 1. (x ≤ y) ∧ (y ≤ x) ⇒ x = y (antisymmetry), and This says that a ball in R3 can be divided into five pieces which can be rearranged by rigid body motions to give two disjoint balls of the same radius as before!

### An introduction to Morse theory by Yukio Matsumoto

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