A new proof of the Morse-Bott inequalities
ICTP (Main Building Seminar Room)
Professor Augustin Banyaga
(The Pennsylvania State University, University Park, USA)
Abstract: Let f be a Morse-Bott function on a compact oriented finite dimensional manifold M. The polynomial Morse inequalities and a perturbation technique approximating any Morse-Bott function by a Morse function show immediately that if MB_t(f) is the Morse-Bott polynomial of f , and P_t(M) is the Poincare polynomial of M , there exists a polynomial R(t) such that MB_t(f) = P_t(M) + (1+t)R(t). We prove that R(t) is a polynomial with non-negative integer coefficients, using the Morse Homology Theorem, and by studying the ranks of the kernels of the differentials of the Morse-Smale-Witten complexes of the Morse functions involved in the approximation scheme.
Our method works when all the critical submanifolds are oriented or when Z_2 coefficients are used. Our proof is much more direct than all previous proofs.
This is a joint work with David E. Hurtubise.
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