Geometric invariants of locally compact groups: The homological perspective

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. The 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, the $\Sigma$-theory of the extension can tell whether the kernel is of type ${\mathrm{CP}}_m$.