2019/07/15 by Fredric D. Ancel, Ancel, Fredric D., Pete Sparks +1
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary: 57N13 #Secondary: 57N15
paper · pdf · doi:10.48550/arxiv.1907.06742
openalex publication_date 2019/07/15 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
The main result of this article is that among the family of one-relator\npresentation 2-complexes that might be expected to be finitely unsplittable\n(not the union of two proper subpolyhedra with finite first homology groups)\nalmost all have this property. Included among these one-relator presentation\n2-complexes are all generalized dunce hats. A generalized dunce hat is a\n2-dimensional polyhedron created by attaching the boundary of a disk \Δ\nto a circle J via a map f : \∂\Δ \→ J with the property\nthat there is a point v in J such that f-1( v ) is a finite set\ncontaining at least 3 points and f maps each component of \∂\Δ -\nf-1( v ) homeomorphically onto J - v . The fact that generalized\ndunce hats are finitely unsplittable undermines a strategy for proving that the\ninterior of the Mazur compact contractible 4-manifold M is splittable in the\nsense of Gabai (i.e., ∫(M) = U \∪ V where U, V and U \∩ V\nare each homeomorphic to Euclidean 4-space).\n