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On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
Alexander A. Gaifullin
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In 1987 Brehm and K\"uhnel showed that any triangulation of a d-manifold (without boundary) that is not homeomorphic to the sphere has at least 3d/2+3 vertices. Moreover, triangulations with exactly 3d/2+3 vertices may exist only for `manifolds like projective planes', which can have dimensions $2$, $4$, $8$, and $16$ only. There is a $6$-vertex triangulation of the real projective plane ℝℙ², a $9$-vertex triangulation of the complex projective plane ℂℙ², and $15$-vertex triangulations of the quaternionic projective plane ℍℙ². Recently, the author has constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane 𝕆ℙ². The four most symmetrical have symmetry group C₃³⋊C₁₃ of order $351$. These triangulations were constructed using a computer program after the symmetry group was guessed. However, it remained unclear why exactly this group is realized as the symmetry group and whether $27$-vertex triangulations of manifolds like 𝕆ℙ² exist with other (possibly larger) symmetry groups. In this paper we find strong restrictions on symmetry groups of such $27$-vertex triangulations. Namely, we present a list of $26$ subgroups of S₂₇ containing all possible symmetry groups of $27$-vertex triangulations of manifolds like the octonionic projective plane. (We do not know whether all these subgroups can be realized as symmetry groups.) The group C₃³⋊C₁₃ is the largest group in this list, and the orders of all other groups do not exceed $52$. A key role in our approach is played by the use of Smith and Bredon's results on the topology of fixed point sets of finite transformation groups.
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