Ergodic Theory and Dynamical Systems

Research Article

The Morse–Bott inequalities via a dynamical systems approach


a1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email:

a2 Department of Mathematics and Statistics, Penn State Altoona, Altoona, PA 16601-3760, USA (email:


Let f:Mxs211D 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,qxs2208Cj 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 xs21242 coefficients are used.

(Received September 01 2007)

(Revised August 08 2008)