A nerve lemma for gluing together incoherent discrete Morse functions

TL;DR

This paper introduces a new theorem that combines the nerve lemma and discrete Morse theory, enabling complex decompositions into non-contractible parts under certain conditions, with broad applications in topological combinatorics.

Contribution

It presents a novel theorem that interpolates between the nerve lemma and discrete Morse theory, expanding their combined applicability.

Findings

The theorem allows decompositions into non-contractible pieces.

Proof is based on diagrams of spaces but does not require their theory.

Applications in topological combinatorics are demonstrated.

Abstract

Two of the most useful tools in topological combinatorics are the nerve lemma and discrete Morse theory. In this note we introduce a theorem that interpolates between them and allows decompositions of complexes into non-contractible pieces as long as discrete Morse theory ensures that they behave well enough. The proof is based on diagrams of spaces, but that theory is not needed for the formulation or applications of the theorem.