634 vertex-transitive and more than \(10^{103}\) non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane (Q6568707)

A combinatorial \(d\)-manifold is a simplicial complex with the property that the link of each vertex is a \((d-1)\)-dimensional combinatorial sphere. The symmetry group \(\operatorname{Sym}(K)\) of a simplicial complex \(K\) is the group of all permutations of the vertex set of \(K\) which send simplices to simplices and non-simplices to non-simplices. A combinatorial manifold is called vertex-transitive, if its symmetry group acts transitively on its set of vertices.\N\N\textit{U. Brehm} and \textit{W. Kühnel} [Topology 26, 465--473 (1987; Zbl 0681.57009)] showed that a combinatorial \(d\)-manifold with less than \(3\lceil{}d/2\rceil{}+3\) vertices is PL homeomorphic to a sphere. Moreover, they proved that a combinatorial \(d\)-manifold with exactly \(3d/2+3\) vertices is PL homeomorphic to either a sphere or a so-called \textit{manifold like a projective plane} (compare [\textit{J. Eells jun.} and \textit{N. H. Kuiper}, Publ. Math., Inst. Hautes Étud. Sci. 14, 181--222 (1962; Zbl 0109.15701)]), where the latter possibility may occur only for \(d\in\{2,4,8,16\}\).\N\NThe case \(d=2\) corresponds to a unique \(6\)-vertex triangulation of \(\mathbb{RP}^2\), and the case \(d=4\) to a unique \(9\)-vertex triangulation of \(\mathbb{CP}^2\) [\textit{W. Kühnel} and \textit{T. F. Banchoff}, Math. Intell. 5, No. 3, 11--22 (1983; Zbl 0534.51009); \textit{W. Kühnel} and \textit{G. Lassmann}, J. Comb. Theory, Ser. A 35, 173--184 (1983; Zbl 0526.52008)]. Both of these triangulations are vertex-transitive.\N\NRegarding the case \(d=8\), several \(15\)-vertex triangulations like a projective plane have been identified and studied [\textit{U. Brehm} and \textit{W. Kühnel}, Math. Ann. 294, No. 1, 167--193 (1992; Zbl 0734.57017); \textit{F. H. Lutz}, ``Triangulated Manifolds with Few Vertices: Combinatorial Manifolds'', Preprint, \url{arXiv:math/0506372}; \textit{A. A. Gaifullin}, ``New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane'', Preprint, \url{arXiv:2311.11309}], see also [\textit{D. A. Gorodkov}, Russ. Math. Surv. 71, No. 6, 1140--1142 (2016; Zbl 1369.57027); translation from Usp. Mat. Nauk 71, No. 6, 159--160 (2016); \textit{D. Gorodkov}, Discrete Comput. Geom. 62, No. 2, 348--373 (2019; Zbl 1425.57016)]. In particular, there is a vertex-transitive \(15\)-vertex triangulation of \(\mathbb {HP}^2\) on which the alternating group \(A_5\) acts as a symmetry group.\N\NThe article is concerned with the case \(d=16\). The author constructs multiple combinatorial \(27\)-vertex \(16\)-manifolds like a projective plane. More precisely, he proves the following theorem:\N\NTheorem 1.1. There are at least\N\N\[\frac{1}{351}\cdot (2^{351}+13\cdot 2^{118} + 81\cdot 2^{29}) + 2\]\N\Ncombinatorially distinct \(27\)-vertex combinatorial triangulations of \(16\)-manifolds like the octonionic projective plane. The symmetry groups of these triangulations are \(\mathbb{Z}_3^3\rtimes \mathbb{Z}_{13}\) (in \(4\) cases), \(\mathbb{Z}_3^3\) (in \(630\) cases), and \(\mathbb{Z}_{13}\), \(\mathbb{Z}_3^2\), \(\mathbb{Z}_3\), or the trivial group (in the remaining cases). The \(634\) triangulations with symmetry groups \(\mathbb{Z}_3^3\rtimes \mathbb{Z}_{13}\) and \(\mathbb{Z}_3^3\) are vertex-transitive, and all other triangulations are not. In addition, the four triangulations \(K_1, K_2, K_3, K_4\) with the symmetry group \(\mathbb{Z}_3^3\rtimes \mathbb{Z}_{13}\) have the following properties:\N\begin{enumerate}\N\item \(\operatorname{Sym}(K_i)\) acts transitively and freely on the 351 undirected edges of \(K_i\),\N\item \(\operatorname{Sym}(K_i)\) acts freely on the set of \(16\)-simplices of \(K_i\), so \(K_i\) contains exactly \(286\) orbits of \(16\)-dimensional simplices, each consisting of \(351\) simplices.\N\end{enumerate}\NMoreover, \(K_1\), \(K_2\), \(K_3\), and \(K_4\) are (up to isomorphism) the only \(27\)-vertex combinatorial triangulations of \(16\)-manifolds like a projective plane whose symmetry groups contain a subgroup isomorphic to \(\mathbb{Z}_3^3\rtimes \mathbb{Z}_{13}\). Two of the four combinatorial manifolds \(K_1\), \(K_2\), \(K_3\), and \(K_4\), and all above-mentioned combinatorial manifolds with smaller symmetry groups are PL homeomorphic to each other.\N\NThe question whether the constructed simplicial complexes are homeomorphic to \(\mathbb{OP}^2\) remains open.\N\NThe manuscript [\textit{A. A. Gaifullin}, Sb. Math. 215, No. 7, 869--910 (2024; Zbl 07945700); translation from Mat. Sb. 215, No. 7, 3--51 (2024)] is a continuation of the present article.