Completing the PL classification of the census of closed orientable 4-manifold triangulations with up to six pentachora: the residual candidate PL classes for $S^4$, $\mathbb{C}P^2$, and $QS^4(2)$

The census of Burke, Burton, and Spreer (Discrete & Computational Geometry, 2026) enumerates all 441,287 generalized triangulations of closed orientable 4-manifolds with up to six pentachora and determines their topological types, and it classifies their piecewise-linear (PL) types completely except in three blocks: among the 405,188 six-pentachoron triangulations homeomorphic to $S^4$ it leaves at most 4 candidate PL classes, among the 4,423 homeomorphic to $\mathbb{C}P^2$ at most 3, and among the 84 homeomorphic to the rational homology sphere $QS^4(2)$ at most 2; the four-pentachoron sphere $Q$ likewise resists connection to the standard sphere. The authors conjecture every residual class is the standard PL type, but this is unproven, and a non-standard residual class on $S^4$ would be an exotic 4-sphere, refuting the smooth 4-dimensional Poincaré conjecture. The open problem is to determine the exact number of distinct PL-homeomorphism types in each residual block and to decide, for each type, whether it is the standard PL structure — including the authors' Conjecture 2 that the four-ball subcomplex $B_C^4$ cut open from $Q$ is PL-homeomorphic to the standard 4-ball.

Background

A pentachoron is a 4-simplex, and a (generalized) triangulation of a closed 4-manifold is a finite collection of pentachora whose facets are glued in pairs so that the link of every vertex is a 3-sphere; unlike a simplicial complex, a generalized triangulation may identify two facets of the same simplex. Cairns and Whitehead showed every smooth 4-manifold admits such a triangulation, and for manifolds of dimension at most 6 every PL structure carries a unique compatible smooth structure, so PL and smooth structures on a 4-manifold are equivalent. A smooth (equivalently PL) structure on a topological 4-manifold is called exotic if it is homeomorphic but not diffeomorphic (equivalently not PL-homeomorphic) to the standard one; the smooth 4-dimensional Poincaré conjecture (S4PC) asserts that no exotic 4-sphere exists. By Pachner's theorem, two triangulations of the same PL 4-manifold are connected by a finite sequence of Pachner moves (bistellar moves replacing one set of pentachora by another spanning the same boundary), so PL-homeomorphism can in principle be certified by an explicit move sequence — but the homeomorphism problem for 4-manifolds is algorithmically undecidable, so only instance-specific heuristics and certificates exist. An isomorphism signature is a canonical string encoding a triangulation up to combinatorial isomorphism, used to pin down concrete triangulations.

Censuses — exhaustive lists of triangulations up to a given size — are the standard reference data of computational topology: in dimension 3 they extend to 11 tetrahedra, while in dimension 4 the closed orientable census previously stopped at 2, 4, and 6 pentachora (8, 784, and 440,495 triangulations), generated by Budney and Burton. Burke, Burton, and Spreer combined a biased random-walk search through the Pachner graph (their Up-Side-Down-Simplify heuristic) with reference triangulations built from Kirby diagrams (diagrams encoding handle decompositions of PL 4-manifolds) to classify the PL types of the whole census. They succeeded for every topological type except three: for six-pentachoron triangulations homeomorphic to $S^4$ (405,188 triangulations, at most 4 candidate PL classes), to $\mathbb{C}P^2$ (4,423 triangulations, at most 3), and to $QS^4(2)$ (84 triangulations, at most 2), improving the previous bounds of 36, 5, and unresolved type respectively obtained by polymake-based sphere-recognition heuristics applied to barycentric subdivisions. Here $QS^4(n)$ denotes the double of a rational homology ball $R(n)$ given by an explicit Kirby diagram; it is a rational homology 4-sphere (a closed 4-manifold whose homology with rational coefficients equals that of $S^4$) with fundamental group $\mathbb{Z}_n$.

The triangulations resisting classification have remarkable structure. The unique unconnected four-pentachoron 4-sphere, denoted $Q$ (isomorphism signature eAMPcaabcddd+aoa+aAa8aQara), contains a two-pentachoron subcomplex $C$ (signature cHIbbb0bRbpb) whose boundary closes up uniquely to the two-pentachoron ideal triangulation of a Cappell–Shaneson 2-knot complement, studied by Budney–Burton–Hillman. A 2-knot is an embedding of a 2-sphere in $S^4$, and Cappell–Shaneson 2-knots arise from matrices in $GL(3,\mathbb{Z})$; the Gluck construction — removing a tubular neighbourhood of a knotted 2-sphere in $S^4$ and regluing it by the once-rotating diffeomorphism of $S^2 \times S^1$ — always returns a topological 4-sphere that is standard for many knot classes (including twist-spun knots by Zeeman's theorem and many Cappell–Shaneson knots by Akbulut, Gompf, and Iwaki), motivating the expectation that census pathologies are also standard. Ungluing one facet pairing of $Q$ yields a topological 4-ball $B_C^4$ (signature eGzMkabcdddcaGa8aAa0awa) whose fundamental group is trivial (verified by coset enumeration) but whose presentation resists simplification; the authors conjecture $B_C^4$ is PL-homeomorphic to the standard 4-ball (their Conjecture 2). An exhaustive enumeration of all move sequences from $Q$ visiting triangulations of up to 10 pentachora found no connection to any other 4-sphere triangulation (their Proposition 4; Proposition 9 in the 2024 preprint version), so any connecting Pachner sequence must pass through at least 12 pentachora, and the residual classes are likewise certified pairwise disconnected within excess height 6. The subcomplex $C$, all of whose five triangles are dunce hats (the classical contractible but non-collapsible complex), appears across many resistant triangulations and is proposed as a combinatorial obstruction to simplification.

Problem Statement

Determine the complete PL classification of the census of closed orientable triangulated 4-manifolds with up to six pentachora. Concretely, for each residual block of triangulations already known to be topologically homeomorphic to a fixed 4-manifold — (a) the 405,188 six-pentachoron triangulations homeomorphic to $S^4$, currently partitioned into at most 4 candidate PL classes; (b) the 4,423 six-pentachoron triangulations homeomorphic to $\mathbb{C}P^2$, at most 3 candidate PL classes; (c) the 84 six-pentachoron triangulations homeomorphic to the rational homology sphere $QS^4(2)$ (the double of the rational ball $R(2)$ with $\pi_1 \cong \mathbb{Z}_2$), at most 2 candidate PL classes; and (d) the four-pentachoron triangulations homeomorphic to $S^4$, at most 2 candidate PL classes, represented by the exceptional sphere $Q$ with isomorphism signature eAMPcaabcddd+aoa+aAa8aQara — determine the exact number of distinct PL-homeomorphism types into which they fall, and decide for each type whether it is PL-homeomorphic to the standard PL structure of that manifold (the canonical structure defined by the reference Kirby diagrams) or represents a genuinely distinct PL, hence smooth, structure. In particular, decide Conjecture 2 of Burke–Burton–Spreer: the four-pentachoron topological 4-ball $B_C^4$ (isomorphism signature eGzMkabcdddcaGa8aAa0awa), obtained from $Q$ by ungluing one facet pairing and containing the Cappell–Shaneson subcomplex $C$, is PL-homeomorphic to the standard 4-ball. Note the stakes: every residual triangulation is already known to be a topological $S^4$, $\mathbb{C}P^2$, or $QS^4(2)$, so a non-standard residual class for $S^4$ would be an exotic 4-sphere refuting the smooth 4-dimensional Poincaré conjecture, and a non-standard class for $\mathbb{C}P^2$ would be the first exotic smooth structure on $\mathbb{C}P^2$; exhaustive search shows any Pachner sequence connecting $Q$ to the standard sphere must pass through at least 12 pentachora.

References

1. Rhuaidi Antonio Burke, Benjamin A. Burton, Jonathan Spreer: Small Triangulations of 4-Manifolds and the 4-Manifold Census, Discrete & Computational Geometry (2026), DOI 10.1007/s00454-026-00818-w; preprint arXiv:2412.04768, https://arxiv.org/abs/2412.04768
2. Rhuaidi Antonio Burke, Benjamin A. Burton, Arunima Ray, Jonathan Spreer, Finn Thompson, Orion Zymaris: Triangulating Spun 2-Knot Complements, arXiv:2607.05923 (2026), https://arxiv.org/abs/2607.05923
3. Jonathan Spreer, Lucy Tobin: Small triangulations of simply connected 4-manifolds, Journal of Applied and Computational Topology (2025), DOI 10.1007/s41468-025-00221-z; preprint arXiv:2501.18933, https://arxiv.org/abs/2501.18933
4. Rafael Torres: A novel construction of homotopy 4-spheres and real projective 4-spaces à la Cappell-Shaneson, arXiv:2607.15062 (2026), https://arxiv.org/abs/2607.15062
5. Selman Akbulut: Cappell-Shaneson homotopy spheres are standard, Annals of Mathematics (2) 171(3) (2010), 2171–2175, DOI 10.4007/annals.2010.171.2171, https://doi.org/10.4007/annals.2010.171.2171
6. Robert E. Gompf: More Cappell–Shaneson spheres are standard, Algebr. Geom. Topol. 10(3) (2010), 1665–1681, DOI 10.2140/agt.2010.10.1665, https://doi.org/10.2140/agt.2010.10.1665
7. Kazunori Iwaki: Infinite families of standard Cappell–Shaneson homotopy 4-spheres, Topology and its Applications 366 (2025), 109293, DOI 10.1016/j.topol.2025.109293, https://doi.org/10.1016/j.topol.2025.109293
8. Ryan Budney, Benjamin A. Burton, Jonathan Hillman: Triangulating a Cappell–Shaneson knot complement, Mathematical Research Letters 19(5) (2012), 1117–1126, DOI 10.4310/MRL.2012.v19.n5.a12, https://doi.org/10.4310/MRL.2012.v19.n5.a12
9. Hermann Gluck: The embedding of two-spheres in the four-sphere, Bulletin of the American Mathematical Society 67 (1961), 586–589, DOI 10.1090/S0002-9904-1961-10703-9, https://doi.org/10.1090/S0002-9904-1961-10703-9
10. E. C. Zeeman: Twisting spun knots, Transactions of the American Mathematical Society 115 (1965), 471–495, DOI 10.2307/1994281, https://doi.org/10.2307/1994281
11. U. Pachner: Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 57 (1987), 69–86, DOI 10.1007/BF02941601, https://doi.org/10.1007/BF02941601
12. Maria Rita Casali, Paola Cristofori: Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problems, arXiv:2502.01757 (2025), https://arxiv.org/abs/2502.01757
13. Rhuaidi Antonio Burke: Practical software for triangulating and simplifying 4-manifolds, Journal of Computational Geometry 16(2) (2025), 109–144, DOI 10.20382/jocg.v16i2a4, https://doi.org/10.20382/jocg.v16i2a4; preprint arXiv:2402.15087, https://arxiv.org/abs/2402.15087