On the LS-category of homomorphism of almost nilpotent groups

TL;DR

This paper establishes that for certain classes of almost nilpotent and virtually nilpotent groups, the LS-category of a homomorphism equals its cohomological dimension, providing a precise algebraic-topological relationship.

Contribution

It proves the equality of LS-category and cohomological dimension for homomorphisms between specific classes of nilpotent groups, extending known results.

Findings

$ ext{cat}() = ext{cd}()$ for epimorphisms between torsion-free, finitely generated almost nilpotent groups

$ ext{cat}() = ext{cd}()$ for homomorphisms between torsion-free, finitely generated virtually nilpotent groups

The results unify algebraic and topological invariants for these classes of groups

Abstract

We prove the equality $\cat(\phi)=\cd(\phi)$ for epimorphisms $\phi:\Gamma\to \Lambda$ between torsion-free, finitely generated almost nilpotent groups $\Gamma$ and $\Lambda$. In addition, we prove the equality $\cat(\phi)=\cd(\phi)$ for homomorphisms $\phi:\Gamma\to \Lambda$ between torsion-free, finitely generated virtually nilpotent groups.