PL standardness of the four-pentachoron census sphere Q

Source 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.