Singular homology of roots of unity

TL;DR

This paper extends singular homology theory to cech closure spaces, proving key theorems and computing homology of roots of unity with applications to graphs and complex spaces.

Contribution

It introduces singular homology for cech closure spaces and proves foundational theorems like excision, Mayer-Vietoris, and Hurewicz in this setting.

Findings

Established singular homology for cech closure spaces

Proved analogues of classical theorems in this new setting

Computed homology groups of roots of unity with closure structures

Abstract

We extend some basic results from the singular homology theory of topological spaces to the setting of \v{C}ech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces.