Geometric invariants of locally compact groups, homological version

In this talk we present the homological version of $\Sigma$-theory for locally compact Hausdorff groups, the corresponding talk for the homotopical version will be given by José in the same parallel session. 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.
We provide criteria for type $\mathrm{CP}_m$ and homological locally compact $\Sigma^m$. In the setting of an exact sequence of locally compact Hausdorff groups, we study in which way compactness properties of the kernel/extension/quotient can be derived from the other two groups in the sequence. This project is joint work with Kai-Uwe Bux and José Pedro Quintanilha.