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

Marzieh Eidi, Jürgen Jost

TL;DR

This work develops a Floer-type boundary operator for generalized Morse-Smale dynamics on smooth manifolds and for Forman's combinatorial vector fields on finite complexes, enabling $\mathbb{Z}_2$-homology to be recovered directly from flow lines or $V$-paths. The approach replaces or augments orbits by pairs of rest points via Franks-type perturbations, constructs a Morse-Floer chain complex with generators derived from rest points and orbits, and proves $\partial^2=0$ under a simple or twist-free framework by establishing a chain-map equivalence to a gradient-like system. In the combinatorial setting, the construction translates moduli-space counts into equivalence classes of $V$-paths, yielding a discrete Floer-type complex that recovers the $\mathbb{Z}_2$-homology of the underlying complex, with explicit treatment of twisted cases via orbit-to-rest-point substitutions. Overall, the paper unifies smooth Conley/Morse-Floer ideas with Forman’s discrete Morse theory, providing a practical framework to compute homology from dynamical data in both continuous and combinatorial contexts.

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.

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

TL;DR

This work develops a Floer-type boundary operator for generalized Morse-Smale dynamics on smooth manifolds and for Forman's combinatorial vector fields on finite complexes, enabling -homology to be recovered directly from flow lines or -paths. The approach replaces or augments orbits by pairs of rest points via Franks-type perturbations, constructs a Morse-Floer chain complex with generators derived from rest points and orbits, and proves under a simple or twist-free framework by establishing a chain-map equivalence to a gradient-like system. In the combinatorial setting, the construction translates moduli-space counts into equivalence classes of -paths, yielding a discrete Floer-type complex that recovers the -homology of the underlying complex, with explicit treatment of twisted cases via orbit-to-rest-point substitutions. Overall, the paper unifies smooth Conley/Morse-Floer ideas with Forman’s discrete Morse theory, providing a practical framework to compute homology from dynamical data in both continuous and combinatorial contexts.

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 homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

Paper Structure

This paper contains 10 sections, 4 theorems, 25 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Generalized smooth Morse-Smale vector fields
  3. Preliminaries
  4. The chain complex for generalized Morse-Smale vector fields
  5. Computing Homology Groups of Smooth Manifolds
  6. Combinatorial Vector Fields
  7. Preliminaries
  8. The chain complex of combinatorial vector fields
  9. Computing Homology Groups of Simplicial Complexes
  10. Acknowledgments

Key Result

Lemma 2.5

Suppose $\phi_t$ is a generalized Morse-Smale flow on an orientable manifold with a periodic or homoclinic orbit of index $k$. Then for any neighborhood $U$ of that orbit there exists a new generalized Morse-Smale flow $\phi'_t$ whose vector field agrees with that of $\phi_t$ outside $U$ and which h

Figures (9)

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  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 17 more