The Morse–Bott inequalities via a dynamical systems approach

Abstract

AbstractLet f:M→ℝ be a Morse–Bott 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 Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré 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 fj. This leads to a relationship between the kernels of the Morse–Smale–Witten boundary operators associated to the Morse functions fj and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients are used.