Let f be a Morse-Bott function on a compact
finite dimensional manifold M. The polynomial Morse inequalities and an
explicit perturbation of f defined using Morse functions on the critical
submanifolds of f show immediately that MB(f,t)= P(M,t) + (1+t)R(t),
where MB(f,t) is the Morse-Bott polynomial of f and P(M,t) 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 M and all the
critical submanifolds are oriented or when Z2-coefficients are
used. |