By Yukio Matsumoto
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