The Morse-Bott inequalities via a dynamical systems approach

Let f:M be a MorseBott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds C_j of f show immediately that MB_t(f)=P_t(M)+(1+t)R(t), where MB_t(f) is the MorseBott polynomial of f and P_t(M) is the Poincar polynomial of M. We prove that R(t) is a polynomial with non-negative integer coefficients by showing that the number of gradient flow lines of the perturbation of f between two critical points p,q∈Cj of relative index one coincides with the number of gradient flow lines between p and q of the Morse function f_j. This leads to a relationship between the kernels of the MorseSmaleWitten boundary operators associated to the Morse functions f _jand the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients areused.