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

TL;DR

This work addresses recovering the homology of a finite regular $\mathbb{Z}_k$-complex from its quotient by enriching the quotient chain complex with a complex-of-groups data to form a surrogate chain complex over $\mathbb{F}G$. The authors introduce isotropy transfer triples and $G$-boundary matrices, and establish a representation-theoretic link between the quotient boundary and the original boundary via a $\mathbb{Z}_k$-representation $\rho_\alpha$, yielding a Smith normal form-based rank formula. The main result shows that, given a Smith normal form $D$ of the quotient boundary, $\mathrm{rank}(\partial_d)$ equals the sum of ranks of the blocks $\rho_\alpha(D_{i,i})$, allowing the computation of $H_d(X;\mathbb{F})$ from $X/G$ data, independent of choices of lift and bases. This provides a memory/time-efficient algebraic pathway for equivariant homology computations and connects complex-of-groups-based compression with a representation-theoretic perspective, offering avenues for extension beyond cyclic actions.

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.

Theorems & Definitions (60)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Definition 2.7
• Definition 2.8
• Definition 2.9