Topological 4-manifolds with right-angled Artin fundamental groups

TL;DR

This paper classifies certain 4-manifolds with specific fundamental groups, using stabilization and algebraic conditions, expanding understanding of their topological and geometric structures.

Contribution

It provides a classification of topological spin$^+$ 4-manifolds with right-angled Artin fundamental groups under specific algebraic and topological conditions.

Findings

Classification up to s-cobordism after stabilization

Applicable to torsion-free 3-manifold groups and certain RAAGs

Conditions involve K-theory and assembly maps

Abstract

We classify closed, topological spin$^+$ 4-manifolds with fundamental group $\pi$ of cohomological dimension $\leq 3$ (up to s-cobordism), after stabilization by connected sum with at most $b_3(\pi)$ copies of $S^2\times S^2$. In general we must also assume that $\pi$ also satisfies certain K-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of 3-manifolds and all right-angled Artin groups whose defining graphs have no 4-cliques.