Independent resolutions for totally disconnected dynamical systems I: Algebraic case

TL;DR

This paper introduces an algebraic framework for independent resolutions in totally disconnected dynamical systems and explores their applications to group homology and cohomology, laying groundwork for future K-theory applications.

Contribution

It presents the first algebraic approach to independent resolutions and demonstrates their use in group (co)homology, preparing for K-theory applications in subsequent work.

Findings

Algebraic formulation of independent resolutions

Applications to group homology and cohomology

Foundation for K-theory computations in future research

Abstract

This is the first out of two papers on independent resolutions for totally disconnected dynamical systems. In the present paper, we discuss independent resolutions from an algebraic point of view. We also present applications to group homology and cohomology. This first paper sets the stage for our second paper, where we explain how to use independent resolutions in K-theory computations for crossed products attached to totally disconnected dynamical systems.