Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields

TL;DR

This paper develops a Floer homology framework applicable to both smooth Morse-Smale dynamical systems and combinatorial vector fields, enabling direct computation of homology from flow lines.

Contribution

It introduces a unified Floer homology approach for smooth and discrete systems, connecting flow line counts to homology in both contexts.

Findings

Floer boundary operator constructed for generalized Morse-Smale systems

Homology computed directly from flow lines in smooth and discrete cases

Applicable to compact smooth manifolds and CW complexes

Abstract

We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $\mathbb{Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.