Generalized dunce hats are not splittable

A \emph{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 $J - \{v\}.$ \textbf{Theorem:} No generalized dunce hat is the union of two proper subpolyhedra that each have finite first homology groups. This result undermines a strategy for proving that the interior of the Mazur compact contractible 4-manifold M is \emph{splittable in the sense of Gabai} (i.e., $\intr(M) = U \cup V$ where $U,$ $V$ and $U \cap V$ are each homeomorphic to Euclidean 4-space).