Generalized equivariant homology on simplicial complexes

TL;DR

This paper explores generalized equivariant homology theories on topological pairs with group actions, establishing their equivalence to known homology theories on equivariant simplicial complexes, thus unifying different approaches.

Contribution

It demonstrates the equivalence of equivariant simplicial homology, the second derived term of the Atiyah--Hirzebruch spectral sequence, and equivariant singular homology.

Findings

Equivalence of equivariant simplicial homology and singular homology.

Identification of the second derived term of the spectral sequence with known homology.

Unified framework for equivariant homology theories.

Abstract

A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of equivariant simplicial homology (also known as Bredon homology), the second derived term of the Atiyah--Hirzebruch spectral sequence, and equivariant singular homology is demonstrated.