Let $f$ be a Morse-Bott function on a compact oriented finite dimensional manifold $M$. The polynomial Morse inequalities and a perturbation of $f$ into a Morse-Smale functions ( by adding to $f$ Morse functions on the critical submanifolds) show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincare polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by studying the kernels of the Morse-Smale-Witten boundary operators associated to the Morse functions on the critical submanifolds of $f$. Our method works when all the critical submanifolds are oriented or when $Z_2$ coefficients are used. Our method is much more direct than all previous methods. This is a joint work with David Hurtubise. |