The Hausmann-Weinberger 4-manifold invariant of right-angled Artin groups.

摘要

There are standard constructions for building 4-manifolds with fundamental group isomorphic to a particular finitely presented group, G. This dissertation focuses on tools for constructing 4-manifolds that not only have fundamental group G but that are also minimal, in the sense that they minimize b2(M), the dimension of H2(M; Q ).;Let M (G) denote the class of closed, oriented topological 4-manifolds with fundamental group isomorphic to a fixed group G. For a finitely presented group G, define h(G) = min{b2( M)|M ∈ M (G)}. Calculations of h are known for free groups and free abelian groups, but little more. The underlying goal of the research represented in this dissertation is to generalize these calculations to right-angled Artin groups, of which free and free abelian groups are special cases. In particular, a right-angled Artin group has a presentation with a finite number of generators and the relations consist of commutators between generators. A nice thing about right-angled Artin groups is that their presentations can uniquely be represented by graphs, where each vertex represents a generator and each edge between vertices represents a commutator relation between those generators.;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 then see results calculating h for many infinite families of right-angled Artin groups as well as discuss the shortcomings with developing an inductive way to calculate h for all right-angled Artin groups.