New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane

TL;DR

This work substantially expands the catalog of 15-vertex combinatorial triangulations of the quaternionic projective plane $\mathbb{HP}^2$, showing that, up to isomorphism, there are 75 such triangulations with symmetry groups of order at least 4, including many new examples with $ ext{C}_7$, $ ext{C}_6 imes ext{C}_2$, $ ext{C}_6$, $ ext{C}_5$, and $ ext{S}_3$. The authors deploy a blend of fixed-point theory for prime-order actions, explicit models like Kühnel’s $ ext{CP}^2_9$, and heavy computer-aided enumeration via the program exttt{find} to construct and classify these triangulations, organized into several symmetry-case subsections. A core result is that all constructed 15-vertex manifolds are PL homeomorphic to $ ext{$ ext{HP}^2$}$, confirmed through first Pontryagin-class computations performed by Gorodkov and leveraging Kramer’s framework. The paper further develops the triple flip graph $oldsymbol{\mathcal{G}}$ and its equivariant variants, demonstrating that all 75 triangulations with $| ext{Sym}(K)|>3$ can be reached from a small set of base examples; it also raises several open questions about the full landscape of 15-vertex triangulations and the structure of these flip-graphs, with implications for broader conjectures on homology manifolds in low dimensions.

Abstract

Brehm and Kühnel (1992) constructed three 15-vertex combinatorial 8-manifolds `like the quaternionic projective plane' with symmetry groups $\mathrm{A}_5$, $\mathrm{A}_4$, and $\mathrm{S}_3$, respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to $\mathbb{HP}^2$. Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to $S^8$. In the present paper we construct a lot of new 15-vertex triangulations of $\mathbb{HP}^2$. A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group $\mathrm{A}_5$. Namely, we find 19 triangulations with symmetry group $\mathrm{C}_7$, one triangulation with symmetry group $\mathrm{C}_6\times\mathrm{C}_2$, 14 triangulations with symmetry group $\mathrm{C}_6$, 26 triangulations with symmetry group $\mathrm{C}_5$, one new triangulation with symmetry group $\mathrm{A}_4$, and 11 new triangulations with symmetry group $\mathrm{S}_3$. Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of $\mathbb{HP}^2$ with 15 vertices and symmetry group of order at least 4: the three Brehm-Kühnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups $\mathrm{C}_3$ and $\mathrm{C}_2$, as well as the trivial symmetry group.

Figures (5)

• Figure 1: The connected component of $\mathbb{HP}^2_{15}(\mathrm{A}_5)$ in $\mathcal{G}$
• Figure 2: The graph $\mathcal{G}_{\mathrm{A}_4}$
• Figure 6: The graph $\mathcal{G}_{\mathrm{C}_5}$
• Figure 7: The connected components of $\mathbb{HP}^2_{15}(\mathrm{A}_5)$ in $\mathcal{G}_{\mathrm{C}_3}$
• Figure 8: The connected components of $\mathbb{HP}^2_{15}(\mathrm{A}_5)$ in $\mathcal{G}_{\mathrm{C}_2}$

Theorems & Definitions (79)

• Theorem 1.1: Brehm, Kühnel BrKu87
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5: Novik, Nov98
• Theorem 1.6
• Remark 1.7
• Theorem 2.1: Smith, Smi39, cf. Bor60
• Remark 2.2
• Remark 2.3