Geometric invariants of locally compact groups: the homological perspective

TL;DR

This paper develops a homological framework for $ ext{Sigma}$-theory in locally compact groups, establishing criteria and connections to classical theory, with implications for understanding group extensions and invariants.

Contribution

It introduces the homological version of $ ext{Sigma}$-theory for locally compact groups, linking it to the homotopical version and classical $ ext{Sigma}$-theory, with criteria for types $ ext{CP}_m$ and $ ext{C}_m$.

Findings

Provides criteria for type $ ext{CP}_m$ and $ ext{C}_m$ in locally compact groups.

Establishes a Hurewicz-like theorem connecting homological and homotopical versions.

Analyzes extensions and how $ ext{Sigma}$-theory properties are preserved or derived.

Abstract

In this paper we develop the homological version of $\Sigma$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $\Sigma$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $\Sigma^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $\Sigma^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $\Sigma$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$.