Morse homology on noncompact manifolds

TL;DR

This paper explores the construction of Morse homology on noncompact manifolds by analyzing bidirect systems of chain complexes derived from Morse functions, focusing on the relationships between various limit processes.

Contribution

It investigates the different methods of taking limits in bidirect systems of Morse chain complexes on noncompact manifolds and clarifies their interrelations.

Findings

Different limit constructions can be applied to Morse homology on noncompact manifolds.

The relationships between these limit constructions are characterized and compared.

The study provides insights into the algebraic structures arising from Morse theory in noncompact settings.

Abstract

Given a Morse function on a manifold whose moduli spaces of gradient flow lines for each action window are compact up to breaking one gets a bidirect system of chain complexes. There are different possibilities to take limits of such a bidirect system. We discuss in this note the relation between these different limits.