# Group-twisted Alexander-Whitney and Eilenberg-Zilber maps

## TL;DR

This paper introduces group-twisted Alexander-Whitney and Eilenberg-Zilber maps for bimodule resolutions of skew group algebras, facilitating the transfer of homological information and simplifying deformation theory computations.

## Contribution

It develops new chain maps for skew group algebras that enable easier translation between resolutions, improving methods in homology and deformation classification.

## Key findings

- Defined group-twisted chain maps for bimodule resolutions
- Enabled transfer of homological information between resolutions
- Simplified classification of PBW deformations

## Abstract

We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to some known results classifying PBW deformations.