Equivariant simplicial reconstruction

TL;DR

This paper presents parallel algorithms for compressing and reconstructing finite simplicial complexes with symmetries, leveraging a functorial approach based on complexes of groups to recover the original structure up to equivariant isomorphism.

Contribution

It introduces a novel functor-based method for reconstructing simplicial complexes with automorphisms, extending Bass-Serre theory to higher-dimensional complexes.

Findings

Algorithms are parallelizable and efficient.

Successfully reconstructs complexes up to equivariant isomorphism.

Generalizes classical group action reconstruction theories.

Abstract

We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the poset of simplices in) the orbit space to the 2-category of groups, whose higher structure is prescribed by isomorphisms arising from conjugation. Using this functor, we show how to algorithmically recover the original complex up to equivariant simplicial isomorphism. Our algorithms are derived from generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others) of the classical Bass-Serre theory for reconstructing group actions on trees.