Computing homology of $\mathbb{Z}_k$-complexes from their quotients

TL;DR

This paper presents an algebraic method to recover the homology of a simplicial complex with a finite group action from its quotient, using matrix representations and Smith normal form for the case of cyclic groups.

Contribution

It introduces a novel algebraic framework combining complexes of groups and matrix representations to compute homology from quotient spaces, especially for cyclic groups.

Findings

Homology of $ ext{Z}_k$-complexes can be recovered from quotient data.

Matrix representations over group rings admit Smith normal form for cyclic groups.

The approach complements existing geometric algorithms for equivariant complexes.

Abstract

In this paper, we investigate the question of how one can recover the homology of a simplicial complex $X$ equipped with a regular action of a finite group $G$ from the structure of its quotient space $X/G.$ Specifically, we describe a process for enriching the structure of the chain complex $C_\ast(X/G; \mathbb{F})$ using the data of a complex of groups, a framework developed by Bridson and Corsen for encoding the local structure of a group action. We interpret this data through the lens of matrix representations of the acting group, and combine this structure with the standard simplicial boundary matrices for $X/G$ to construct a surrogate chain complex. In the case $G = \mathbb{Z}_k,$ the group ring $\mathbb{F}G$ is commutative and matrices over $\mathbb{F}G$ admit a Smith normal form, allowing us to recover the homology of $G$ from this surrogate complex. This algebraic approach complements the geometric compression algorithm for equivariant simplicial complexes described by Carbone, Nanda, and Naqvi.