On cohomological dimension of homomorphisms (Q6621276)

For a group homomorphism \(\phi:G\rightarrow H\), the cohomological dimension \(\mathrm{cd}(\phi)\) is defined to be the largest \(n\) such that there is a non-zero homomorphism \(\phi^*:H^n(H,A) \rightarrow H^n(G,A)\) for some \(\mathbb{Z}H\)-module \(A\), and the homological dimension \(\mathrm{hd}(\phi)\) is similarly defined. These concepts are analogous to the concepts of cohomological and homological dimensions, \(\mathrm{cd}(G)\) and \(\mathrm{hd}(G)\), for groups \(G\). This article considers properties of \(\mathrm{cd}(G)\) and \(\mathrm{hd}(G)\) and explores if analogous results hold for \(\mathrm{cd}(\phi)\) and \(\mathrm{hd}(\phi)\).\N\NFor geometrically finite groups (that is, groups for which there is a finite classifying CW-complex) we have \(\mathrm{hd}(G)=\mathrm{cd}(G)\); Theorem 1.1 of the current article states that, if \(G\) is a geometrically finite group and \(\phi:G\rightarrow H\) is a homomorphism, then \(\mathrm{hd}(\phi)=\mathrm{cd}(\phi)\). For a geometrically finite group \(G\) it has been shown in [\textit{A. Dranishnikov}, Algebra Discrete Math. 28, No. 2, 203--212 (2019; Zbl 1448.20043)] that \(\mathrm{cd}(G\times G)=2\mathrm{cd}(G)\); Theorem 1.2 states that, if \(G,H\) are geometrically finite groups and \(\phi:G\rightarrow H\) is a homomorphism, then \(\mathrm{cd}(\phi\times \phi)=2\mathrm{cd}(\phi)\). The main tools involved in proving these theorems are Davis's trick with aspherical manifolds, combined with a characterisation of cohomological dimension of a group homomorphism in terms of projective resolutions, obtained here.\N\NThe Eilenberg-Ganea theorem states that, for a group \(G\), \(\mathrm{cd}(G)=\mathrm{gd}(G)\) whenever \(\mathrm{cd}(G)\geq 3\). The Eilenberg-Ganea conjecture is that the same equality holds when \(\mathrm{cd}(G)=2\); therefore a counterexample to it would require \(\mathrm{cd}(G)=2\) and \(\mathrm{gd}(G)=3\). In [\textit{A. Dranishnikov} and \textit{N. Kuanyshov}, Math. Z. 305, No. 1, Paper No. 14, 12 p. (2023; Zbl 1527.55003)] it was shown that there is an example of a group homomorphism \(\phi:G\rightarrow H\) such that \(\mathrm{cd}(\phi)=2\) and \(\mathrm{gd}(\phi)=3\). It is shown here that the coincidence of the appearance of the numbers 2,3 in that result and in the Eilenberg-Ganea conjecture is a coincidence. Specifically, by Corollary 7.2, for every \(k>2\) there is a group homomorphism \(\phi_k:G_k\rightarrow \mathbb{Z}^k\) with \(\mathrm{cd}(\phi_k)<k\) and \(\mathrm{gd}(\phi_k)=k\), where \(G_k\) is the fundamental group of a closed, aspherical, \((k+1)\)-dimensional manifold.