Construction of the discrete Morse complex in the non-compact case

TL;DR

This paper extends discrete Morse theory to non-compact spaces by establishing an infinite analogue of the main theorem, under conditions on infinite paths, and provides a homological version.

Contribution

It introduces an infinite analogue of discrete Morse theory's main theorem applicable to non-compact cases, with conditions on infinite paths and a homological extension.

Findings

Proves an infinite analogue of the main theorem of discrete Morse theory.

Establishes conditions on infinite paths for the theorem to hold.

Provides a homological version of the discrete Morse theorem.

Abstract

We prove an infinite analogue of the main theorem of discrete Morse theory formulated in terms of discrete Morse matchings. Our theorem holds under the assumption that the given Morse matching induces finitely many equivalence classes of infinite directed simple paths. A homological version of the theorem is also given.