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Unformatted text preview: . di ﬀ geom.tex; April 12, 2006; 17:59; p. 201 204 CHAPTER 7. HOMOLOGY THEORY • Theorem 7.6.4 Morse Theorem. Let M be a closed manifold and f ; M → R be a smooth function. Suppose that the function f has only nondegenerate critical points. Let M p be the Morse type numbers, B p be the Betti numbers, M ( t ) be the Morse polynomial and P ( t ) be the Poincar´e polynomial. Then there is a polynomial Q ( t ) = n1 X p = q p t p with nonnegative coe ﬃ cients, q p ≥ , such that M ( t )P ( t ) = (1 + t ) Q ( t ) , that is, M pB p = q p + q p1 . Proof : 1. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 202...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Critical Point

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