On the LS-category of homomorphisms

TL;DR

This paper establishes the equality of LS-category and cohomological dimension for homomorphisms between certain nilpotent groups, and provides a counterexample where these invariants differ.

Contribution

It proves the equality of LS-category and cohomological dimension for homomorphisms of torsion-free finitely generated nilpotent groups, and constructs a counterexample for general groups.

Findings

$ ext{cat}() = ext{cd}()$ for specific group homomorphisms

Counterexample with $ ext{cat}() > ext{cd}()$ for geometrically finite groups

Provides new insights into the relationship between LS-category and cohomological dimension

Abstract

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