One-point reductions of finite spaces, h-regular CW-complexes and collapsibility

TL;DR

This paper explores one-point reduction techniques for finite topological spaces and introduces h-regular CW-complexes, demonstrating their homotopy equivalence with associated finite spaces, thus advancing combinatorial homotopy theory.

Contribution

It introduces h-regular CW-complexes, generalizes McCord's classical result, and links finite models to homotopy theory of cell complexes.

Findings

h-regular CW-complexes are modeled by finite spaces up to homotopy

One-point reduction methods simplify the study of homotopy in finite models

Generalization of McCord's theorem to h-regular CW-complexes

Abstract

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.