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

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 $\mathrm{\Delta }$ to a circle $J$ via a map $f:\mathrm{\partial }\mathrm{\Delta }\to J$ 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 $\partial\Delta - f^{-1}({v})$homeomorphicallyonto$J - {v}$. 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).