Existence of an epimorphism between finite aspherical spaces with cdp(φ)=1 and cat(φ)=2

Establish whether there exists a group epimorphism φ between finite aspherical spaces such that the cohomological dimension of the homomorphism equals one, cdp(φ)=1, while the Lusternik–Schnirelmann category of the induced classifying map Bφ: BΓ → BΛ equals two, cat(φ)=2.

Background

The paper studies invariants of group homomorphisms, including the Lusternik–Schnirelmann category cat(φ) and the cohomological dimension cdp(φ), defined via the classifying map Bφ: BΓ → BΛ. For surface groups, the authors show cat(φ)=cdp(φ)=1 exactly when φ factors through free groups, and otherwise cat(φ)=cdp(φ)=2.

For finite aspherical spaces, the authors point out an unresolved existence problem asking for an epimorphism with cdp(φ)=1 but cat(φ)=2, which would exhibit a gap between these two invariants. This relates to broader questions on the possible differences between cat(φ) and cdp(φ).