Functoriality and duality in Morse-Conley-Floer homology

TL;DR

This paper explores the functoriality and duality properties of Morse-Conley-Floer homology, providing direct proofs and extending results to local homology theories, thereby deepening understanding of the algebraic structure of dynamical systems.

Contribution

It offers direct proofs of functoriality in Morse homology and extends these results to Morse-Conley-Floer homology, including duality statements, without relying on isomorphisms to other theories.

Findings

Established functoriality for local Morse homology.

Proved Poincaré type duality for Morse-Conley-Floer homology.

Generalized results using isolating and flow maps.

Abstract

In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known~\cite{abbondandoloschwarz, aizenbudzapolski,audindamian, kronheimermrowka, schwarz}, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolating map and flow map allows the results to generalize to local Morse homology and Morse-Conley-Floer homology. We prove Poincar\'e type duality statements for local Morse homology and Morse-Conley-Floer homology.