Alyson Hildum | Searcharxiv

arXiv2018

Topological 4-manifolds with right-angled Artin fundamental groups

We classify closed, topological spin$^+$ 4-manifolds with fundamental group $π$ of cohomological dimension $\leq 3$ (up to s-cobordism), after stabilization by connected sum with at most $b_3(π)$ copies of $S^2\times S^2$. In general we must also assume that $π$ 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.

math.GT↗

arXiv2014

The minimum $b_2$ problem for right-angled Artin groups

This paper focuses on tools for constructing 4-manifolds that have fundamental group $G$ isomorphic to a right-angled Artin group and that are also minimal, in the sense that they minimize $b_2(M)$, the dimension of $H_2(M;\mathbb{Q})$. For a finitely presented group $G$, define $h(G) = \min\{ b_2(M) | M \in \mathcal M(G) \}$. In this paper, we explore the ways in which we can bound $h(G)$ from below using group cohomology and the tools necessary to build 4-manifolds that realize these lower bounds. We give solutions for right-angled Artin groups, or RAAGs, when the graph associated to $G$ has no 4-cliques, and further we reduce this problem to the case when the graph is connected and contains only 4-cliques. We then give solutions for many infinite families of RAAGs and provide a conjecture to the solution for all RAAGs.

math.GT↗