Multiplicative structure in equivariant cohomology

TL;DR

This paper develops a new algebraic framework for understanding the multiplicative structure in equivariant cohomology, providing explicit models for specific group actions to facilitate computations.

Contribution

It introduces strongly homotopy-comultiplicative resolutions and applies them to enrich Moore's theorem, enabling better models for equivariant cohomology calculations.

Findings

Enriched Moore's theorem with comultiplicative structure

Constructed explicit cochain algebra models for circle actions

Provided small models for group actions on simplicial sets

Abstract

We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore's theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology. In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small, explicit cochain algebra models that we describe in detail.