Year: 2026
Type: preprint
Abstract: We prove that the four-pentachoron homotopy \(4\)-sphere \(Q\) singled out in the census of Burke, Burton and Spreer is PL homeomorphic to the standard \(4\)-sphere. The key construction identifies the compact exterior of an explicit locally flat \(2\)-sphere in an auxiliary 39-vertex model of \(Q\) with the exterior of Budney, Burton and Hillman, then certifies its fibration by a normal cut followed by a relative collapse. The exterior identification uses finite checked moves, an exact regular-neighbourhood construction and a PL lemma controlling the passage through an ideal triangulation. Cutting the compact exterior along the specified normal hypersurface gives a manifold that collapses onto one boundary copy of the punctured \(3\)-torus in 209,144 elementary pairs; a marked product lemma identifies the cut as a product. The resulting integral monodromy realizes \(Q\) as a Cappell--Shaneson filling whose two surgery framings are standard by Gompf's results. The fibration certificate is independent of Issa's earlier PL-compatible constructions. Combined with the published census, the result establishes standardness for every homotopy \(4\)-sphere triangulated with at most four pentachora.
Source: arXiv (Cornell University)
Authors: Faber, Vance, Murphy, Michael
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223115914