Geometric invariants of locally compact groups: the homological perspective

Abstract

In this paper we develop the homological version of $Σ$-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 $Σ$-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 $Σ^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $Σ^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $Σ$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$.