  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).
