Generalized dunce hats are not splittable

A \emph{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 $f:\partial \Delta \to J$withthepropertythatthereisapoint$v \in J$suchthat$f^{-1}({v})$isafinitesetcontainingatleast3pointsand$f$ maps each component of $\mathrm{\partial }\mathrm{\Delta }-f^{-1}(\{v\})$ homeomorphically onto $J - {v}.$ $\mathbf{\text{Theorem:}}$ 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., $\text{intr}(M)=U\cup V$ where $U,$ $V$ and $U\cap V$ are each homeomorphic to Euclidean 4-space).