[Paper Review] New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane

Alexander A. Gaifullin|arXiv (Cornell University)|Nov 19, 2023

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper presents a comprehensive classification of 15-vertex triangulations of the quaternionic projective plane ℍP², constructing 72 new examples with diverse symmetry groups—including C₇, C₆, C₅, C₆×C₂, and others—beyond the original three by Brehm and Kühnel. The key contribution is proving that, up to isomorphism, there are exactly 75 such triangulations with symmetry groups of order at least 4, significantly expanding the known landscape of minimal triangulations of non-spherical 8-manifolds.

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 $\mathrm{C}_6 imes\mathrm{C}_2$, 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.

Motivation & Objective

• To construct new 15-vertex combinatorial triangulations of the quaternionic projective plane ℍP² with symmetry groups not previously known to admit such structures.
• To classify all 15-vertex triangulations of ℍP² with symmetry groups of order at least 4, extending the known list beyond the original three Brehm–Kühnel examples.
• To investigate the distribution of combinatorial invariants such as the number of 8-simplices in three 8-simplices (m₈), distinguished subcomplexes (t), and 6-simplices in three 8-simplices (m₃) across the new triangulations.
• To explore the role of symmetry and combinatorial structure in distinguishing triangulations that are PL homeomorphic to ℍP² but not isomorphic as simplicial complexes.

Proposed method

• The authors employ computational algebraic topology techniques, including systematic enumeration and isomorphism rejection, to generate new 15-vertex triangulations of ℍP² via transformation group actions and combinatorial search.
• They utilize the fact that 15 vertices is the minimal number for a combinatorial 8-manifold not PL homeomorphic to S⁸, leveraging the Brehm–Kühnel construction as a starting point.
• The symmetry groups of the triangulations are analyzed by realizing them as subgroups of S₁₅, and isomorphism classes are determined by vertex labeling and group action on vertex orbits.
• Combinatorial invariants such as m₈ (number of 8-simplices in three 8-simplices), t (number of distinguished subcomplexes), and m₃ (number of 6-simplices in exactly three 8-simplices) are computed and compared across triangulations.
• A graph 𝒢₀ is constructed to model relationships between triangulations, revealing connected components and symmetries such as twin pairs and almost twins with identical invariants but different symmetry groups.
• Random walks in the graph space are used to sample additional triangulations and assess extremal values of invariants like m₈ and m₃.

Experimental results

Research questions

• RQ1 Are there 15-vertex triangulations of ℍP² with symmetry groups other than A₅, A₄, and S₃, as previously known?
• RQ2 What is the complete count of isomorphism classes of 15-vertex triangulations of ℍP² with symmetry groups of order at least 4?
• RQ3 What are the extremal values of combinatorial invariants such as m₈, m₃, and t across all 15-vertex triangulations of ℍP²?
• RQ4 Why do certain pairs of triangulations (twins or almost twins) have identical invariants but different symmetry groups?
• RQ5 Is there a 27-vertex 16-manifold like a projective plane that contains a 16-simplex in two distinct distinguished subcomplexes?

Key findings

• There are exactly 75 isomorphism classes of 15-vertex triangulations of ℍP² with symmetry groups of order at least 4, including the original three Brehm–Kühnel examples and 72 new ones.
• The paper constructs 19 triangulations with symmetry group C₇, 14 with C₆, 26 with C₅, one with C₆×C₂, and one new example with A₄ and 11 with S₃.
• The triangulation ℍP²₁₅(C₆,9) is extremal: it achieves the minimal m₃ = 1038 and nearly maximal m₈ = 21 among non-𝒢₀ triangulations.
• The largest value of m₈ among triangulations not in the connected component 𝒢₀ is 22, achieved at a triangulation with trivial symmetry group, while the largest m₈ in 𝒢₀ is 30.
• The maximum value of t (number of distinguished subcomplexes) is 28, achieved at four distinct triangulations with symmetry group C₇, while the minimum t = 4 is achieved at ℍP²₁₅(A₄,2).
• The triangulation ℍP²₁₅(C₆,9) has the smallest m₃ = 1038 and is the only one with m₃ < 1100, suggesting it is a key object for further study.