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On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

Published 25 Oct 2023 in math.CO, math.GR, and math.GT | (2310.16679v2)

Abstract: 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 $\mathbb{RP}2$, a $9$-vertex triangulation of the complex projective plane $\mathbb{CP}2$, and $15$-vertex triangulations of the quaternionic projective plane $\mathbb{HP}2$. Recently, the author has constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane $\mathbb{OP}2$. The four most symmetrical have symmetry group $\mathrm{C}33\rtimes \mathrm{C}{13}$ 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 $\mathbb{OP}2$ 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 $\mathrm{S}{27}$ 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 $\mathrm{C}_33\rtimes \mathrm{C}{13}$ 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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References (31)
  1. J. F. Adams, “On the non-existence of elements of Hopf invariant one”, Ann. Math., 72:1 (1960), 20–104.
  2. A. V. Alekseevskii, “Finite commutative Jordan subgroups of complex simple Lie groups”, Funkt. Analiz i Prilozh., 8:4 (1974), 1–4 (in Russian); English transl.: Funct. Anal. Appl., 8:4 (1974), 277–279.
  3. P. Alexandroff, “On local properties of closed sets”, Ann. Math., 36:1 (1935), 1–35.
  4. P. Arnoux, A. Marin, “The Kühnel triangulation of the complex projective plane from the view point of complex crystallography (part II)”, Mem. Fac. Sci. Kyushu Univ. Ser. A, 45:2 (1991), 167–244.
  5. B. Bagchi, B. Datta, “On Kühnel’s 9-vertex complex projective plane”, Geom. Dedicata, 50 (1994), 1–13.
  6. B. Bagchi, B. Datta, “Non-existence of 6666-dimensional pseudomanifolds with complementarity”, Adv. Geom., 4 (2004), 537–550; arXiv:math/0404225.
  7. H. U. Besche, B. Eick, E. A. O’Brien, “The groups of order at most 2000200020002000”, Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 1–4.
  8. G. E. Bredon, “Orientation in generalized manifolds and applications to the theory of transformation groups”, Michigan Math. J., 7:1 (1960), 35–64.
  9. G. E. Bredon, “The cohomology ring structure of a fixed point set”, Ann. Math., 80:3 (1964), 524–537.
  10. G. E. Bredon, “Cohomological aspects of transformation groups”, in Proceedings of the conference on transformation groups (ed. P. S. Mostert), Springer-Verlag, 1968, 245–280.
  11. U. Brehm, W. Kühnel, “Combinatorial manifolds with few vertices”, Topology, 26:4 (1987), 465–473.
  12. U. Brehm, W. Kühnel, “15-vertex triangulations of an 8-manifold”, Math. Ann., 294 (1992), 167–193.
  13. E. Čech, “Sur les nombres de Betti locaux”, Ann. Math., 35:3 (1934), 678–701.
  14. F. Chapoton, L. Manivel, “Triangulations and Severi varieties”, Exp. Math., 22:1, 60–73; arXiv:1109.6490.
  15. A. M. Cohen, D. B. Wales, “Finite subgroups of F4⁢(ℂ)subscript𝐹4ℂF_{4}(\mathbb{C})italic_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_C ) and E6⁢(ℂ)subscript𝐸6ℂE_{6}(\mathbb{C})italic_E start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( blackboard_C )”, Proc. London Math. Soc., 74:1 (1997), 105–150.
  16. B. Datta, “Pseudomanifolds with complementarity”, Geom. Dedicata, 73 (1998), 143–155.
  17. T. Dokchitser, https:/​/people.maths.bris.ac.uk/∼similar-to\sim∼matyd/GroupNames/ (under construction).
  18. J. Eells, N. H. Kuiper, “Manifolds which are like projective planes”, Publ. Math. Inst. Hautes Etud. Sci., 14 (1962), 181–222.
  19. A. A. Gaifullin, “634634634634 vertex-transitive and more than 10103superscript1010310^{103}10 start_POSTSUPERSCRIPT 103 end_POSTSUPERSCRIPT non-vertex-transitive 27272727-vertex triangulations of manifolds like the octonionic projective plane”, Izv. RAN. Ser. Matem., 88:3 (to appear); arXiv:2207.08507.
  20. A. A. Gaifullin, “New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane”, arXiv:2311.11309.
  21. D. A. Gorodkov, “A minimal triangulation of the quaternionic projective plane”, Uspekhi Matem. Nauk 71:6(432) (2016), 159–160 (in Russian); English transl.: Russian Math. Surveys, 71:6 (2016), 1140–1142.
  22. D. Gorodkov, “A 15-vertex triangulation of the quaternionic projective plane”, Discrete Comput. Geom., 62:2 (2019), 348–373; arXiv:1603.05541.
  23. M. Hall, Jr., “Simple groups of order less than one million”, J. Algebra, 20:1 (1972), 98–102.
  24. G. Higman, “Finite groups in which every element has prime power order”, J. London Math. Soc., 32:3 (1957), 335–342.
  25. V. Klee, “A combinatorial analogue of Poincaré’s duality theorem”, Canad. J. Math., 16 (1964), 517–531.
  26. L. Kramer, “Projective planes and their look-alikes”, J. Differential Geometry, 64 (2003), 1–55; arXiv:math/0206145.
  27. W. Kühnel, T. F. Banchoff, “The 9-vertex complex projective plane”, Math. Intelligencer, 5:3 (1983), 11–22.
  28. W. Kühnel, G. Lassmann, “The unique 3-neighborly 4-manifold with few vertices”, J. Comb. Theory, Ser. A, 35 (1983), 173–184.
  29. I. Novik, “Upper bound theorems for homology manifolds”, Isr. J. Math., 108:1 (1998), 45–82.
  30. P. A. Smith, “Transformations of finite period. II”, Ann. Math., 40:3 (1939), 690–711.
  31. M. Suzuki, “On a class of doubly transitive groups”, Ann. Math., 75:1 (1962), 105–145.
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