The Morse-Bott inequalities via dynamical systems

TL;DR

This paper establishes Morse-Bott inequalities using dynamical systems by relating gradient flow lines of a perturbed Morse-Bott function to those of Morse functions on critical submanifolds, proving a polynomial with nonnegative coefficients.

Contribution

It provides a dynamical systems proof of Morse-Bott inequalities, explicitly relating flow lines of perturbations to those of Morse functions on critical submanifolds, with results valid over oriented manifolds or with  coefficients.

Findings

Proves that R(t) has nonnegative integer coefficients.

Shows the number of flow lines of the perturbation matches that of the Morse functions.

Establishes a relationship between kernels of boundary operators in Morse homology.

Abstract

Let $f:M \to \mathbb{R}$ 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 $f_j$ 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 Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincar\'e polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by showing that the number of gradient flow lines of the perturbation of $f$ between two critical points $p,q \in C_j$ 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 Morse-Smale-Witten boundary operators associated to the Morse functions $f_j$ and the perturbation of $f$. This method works when $M$ and all the critical submanifolds are oriented or when $\mathbb{Z}_2$ coefficients are used.