Homology of Simplicial G-complexes and Group Rings

Abstract: There has been a growing trend to use the homology of simplicial complexes to study complex data structures because of its resilience to deformation and noise. In this talk, 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*(X/G; 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 = Zk, the group ring FG is commutative and matrices over FG 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.

algebraic topologycategory theory

Audience: researchers in the topic

Bilkent Topology Seminar

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