Abstract:

Abstract: Let $f:M^n\rightarrow \mathbb{R}$ be a function with only nondegenerate critical points. Denote by $Crit_kf$ those critical points of f that have index k, let $c_k$ denote their total number. Consider the free abelian groups $C_k=\mathbb{Z}^{c_k}$, $C_k$ has one generator for each critical point of index k that f has. It is well known that that the strong Morse inequalities $c_k-c_{k-1}+\dots\pm c_0\geq b_k-b_{k-1}+\dots\pm b_0$ for $k=0,\dots,n-1$ and $c_n-c_{n-1}+\dots\pm c_0=b_n-b_{n-1}+\dots\pm b_0$, are equivalent to the existence of boundary homomorphisms $\partial_k:C_k\rightarrow C_{k-1}$ whose homology groups have rank, $b_k=Rank(H_k(M;\mathbb{Z}))$.There are several ways of getting to a boundary operator that will work. In this talk we will discuss one approach to constructing such a chain complex for a manifold M, given a metric g on M and a Morse function f on M. All approaches of which I am aware are based on the following observation. Associated to every Morse function f on M is a dynamical system given by the negative gradient flow of f. To define $\partial_k:C_k\rightarrow C_{k-1}$ we will investigate this dynamical system.