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

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 $f:\partial \Delta \to J$ 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 $\partial \Delta -f^{-1}(\{v\})$ 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).

The main theorem of this article generalizes the main theorem of "Generalized dunce hats are not splittable" (arXiv:1803.00644)

References in corpus (1)

• Contractible n-Manifolds and the Double n-Space Property