On cohomological dimension of homomorphisms

Abstract:

The (co)homological dimension of a homomorphism $\phi :G\to H$ is the maximal number $k$ such that the induced homomorphism in $k$-th (co)homology groups is nonzero for coefficients in some $H$-module. It is known that for geometrically finite groups $G, cd(G)=hd(G)$ and $cd(G\times G)=2cd(G)$. We prove analogous theorems for homomorphisms of geometrically finite groups.

The analogy stops working on the Eilenberg-Ganea equality $cd(G)=gd(G)$ where $cd(G)>2$ and $gd(G)$ is the geometric dimension of $G$. We show that for every $k>2$ there is a group homomorphism $\phi _k:\pi _k\to \mathbb {Z}^k$ with $cd(\phi _k)<k$ and $gd(\phi _k)=k$ where $\pi _k$ is the fundamental group of a closed aspherical $(k+1)$-dimensional manifold.