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

Alyson Hildum

Algebraic & Geometric Topology 15 (2015) 1599–1641

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 b2(M) = dimH2(M; ). For a finitely presented group G, define

h(G) = min{b2(M) M a closed, oriented 4–manifold with π1(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.

Hausmann–Weinberger invariant, right-angled Artin group, RAAG
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F36
Received: 7 March 2014
Revised: 3 September 2014
Accepted: 4 September 2014
Published: 19 June 2015
Alyson Hildum
Deptartment of Mathematics & Statistics
McMaster University
1280 Main Street West
Hamilton ON L8S 4K1