2018/03/01 by Fredric D. Ancel, Ancel, Fredric, Pete Sparks +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1803.00644
A generalized dunce hat is a 2-dimensional polyhedron created by attaching the boundary of a disk Δ to a circle J via a map f:∂ Δ→ J with the property that there is a point v ∈ J such that f-1(\v\) is a finite set containing at least 3 points and f maps each component of ∂ Δ- f-1(\v\) homeomorphically onto J - \v\. 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 splittable in the sense of Gabai (i.e., \intr(M) = U ∪ V where U, V and U ∩ V are each homeomorphic to Euclidean 4-space).