Most unexposed taut one-relator presentation 2-complexes are finitely unsplittable

TL;DR

This paper shows that most unexposed taut one-relator presentation 2-complexes, including generalized dunce hats, are finitely unsplittable, impacting strategies for understanding the structure of certain 4-manifolds.

Contribution

It proves that almost all such complexes are finitely unsplittable, including generalized dunce hats, challenging previous approaches to 4-manifold decomposition.

Findings

Most unexposed taut one-relator presentation 2-complexes are finitely unsplittable.

Generalized dunce hats are included among these complexes.

This result undermines strategies for splitting the interior of certain 4-manifolds.

Abstract

The main result of this article is that among the family of one-relator presentation 2-complexes that might be expected to be finitely unsplittable (not the union of two proper subpolyhedra with finite first homology groups) almost all have this property. Included among these one-relator presentation 2-complexes are all generalized dunce hats. A generalized dunce hat is a 2-dimensional polyhedron created by attaching the boundary of a disk $\Delta$ to a circle $J$ via a map $f : \partial\Delta \rightarrow J$ with the property that there is a point $v$ in $J$ such that $f^{-1}(\{v\})$ is a finite set containing at least 3 points and $f$ maps each component of $\partial\Delta - f^{-1}(\{v\})$ homeomorphically onto $J - \{v\}$. The fact that generalized dunce hats are finitely unsplittable undermines a strategy for proving that the interior of the Mazur compact contractible 4-manifold $M$ is splittable in the sense of Gabai (i.e., $\text{int}(M) = U \cup V$ where $U$, $V$ and $U \cap V$ are each homeomorphic to Euclidean 4-space).