differential geometry w notes from teacher_Part_102

# differential geometry w notes from teacher_Part_102 - di...

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7.6. RELATIVE HOMOLOGY AND MORSE THEORY 203 The number of critical points of index p is called the p -th Morse type num- ber and denoted by M p . Let t be a formal variable. The polynomial M ( t ) = n X p = 0 M p t p is called the Morse polynomial . For each real number a R we deﬁne M a = { x M | f ( x ) a } and M - a = { x M | f ( x ) < a } . A real number a is called a homotopically critical value of the function f if some relative homology group H p ( M a , M - a ; G ) is non-zero. It turns out that for non-degenerate critical points a value is homotopically critical if and only if it is critical. That is, for non-degenerate critical points the critical values of f are pre- cisely the values at which new relative cycles appear. Let B p ( M ) = dim H p ( M ) be the Betti numbers. The polynomial P ( t ) = n X p = 0 B p t p is called the Poincar´e polynomial

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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 non-degenerate 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 ) = n-1 X p = q p t p with non-negative coe ﬃ cients, q p ≥ , such that M ( t )-P ( t ) = (1 + t ) Q ( t ) , that is, M p-B p = q p + q p-1 . 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.

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