PL standardness of the four-pentachoron census sphere Q
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.
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BibTeX
@misc{faber2026standardness,
title = {{PL standardness of the four-pentachoron census sphere Q}},
author = {Vance Faber and Michael Murphy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27283},
eprint = {2610.27283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27283},
note = {CC0},
}
RIS
TY - UNPB
TI - PL standardness of the four-pentachoron census sphere Q
AU - Vance Faber
AU - Michael Murphy
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27283
UR - https://arxiv.org/abs/2610.27283
AB - 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.
ER -