On the LS-category of homomorphisms

We prove the equality $\cat(ϕ)=\cd(ϕ)$ for homomorphisms $ϕ:Γ\to Λ$ of a torsion free finitely generated nilpotent groups $Γ$ to an arbitrary group $Λ$. We construct an epimorphism $ψ:G\to H$ between geometrically finite groups with $\cat(ψ)> \cd(ψ)$.

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