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

Abstract

This paper focuses on tools for constructing 4–manifolds which have fundamental group $G$ isomorphic to a right-angled Artin group, and which are also minimal in the sense that they minimize $b_2\left(M\right)=dim{H}_2\left(M;\mathbb{Q}\right)$. For a finitely presented group $G$, define

$$h\left(G\right)=min\left\{b_2\left(M\right)\mid M\text{ a closed, oriented 4–manifold with }{\pi }_1\left(M\right)=G\right\}.$$

In this paper, we explore the ways in which we can bound $h\left(G\right)$ 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.