On the treewidth of triangulated 3-manifolds

TL;DR

This paper investigates the relationship between the topological complexity of 3-manifolds and the combinatorial width parameters of their triangulations, showing that not all manifolds admit bounded treewidth triangulations and establishing bounds relating treewidth to Heegaard genus.

Contribution

It proves that some 3-manifolds cannot have triangulations with bounded treewidth, and links width parameters to topological invariants like Heegaard genus.

Findings

Existence of infinite families of 3-manifolds without bounded treewidth triangulations.

Bound on Heegaard genus in terms of triangulation treewidth.

Explicit connections between topology and graph width parameters.

Abstract

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1)).