634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
- 作者: Gaifullin A.1,2,3,4
-
隶属关系:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Skolkovo Institute of Science and Technology
- Lomonosov Moscow State University
- Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
- 期: 卷 88, 编号 3 (2024)
- 页面: 12-60
- 栏目: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/257715
- DOI: https://doi.org/10.4213/im9489
- ID: 257715
如何引用文章
详细
In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. There exist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\mathrm{C}_3$, $\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.
关键词
minimal triangulation, octonionic projective plane, manifold like a projective plane, Kühnel triangulation, Brehm–Kühnel triangulations, vertex-transitive triangulation, combinatorial manifold, minimal triangulation, octonionic projective plane, manifold like a projective plane, Kühnel triangulation, Brehm–Kühnel triangulations, vertex-transitive triangulation, combinatorial manifold
作者简介
Alexander Gaifullin
Steklov Mathematical Institute of Russian Academy of Sciences; Skolkovo Institute of Science and Technology; Lomonosov Moscow State University; Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
Email: agaif@mi-ras.ru
Scopus 作者 ID: 6602366976
Researcher ID: N-9247-2016
Doctor of physico-mathematical sciences, no status
参考
- А. В. Алексеевский, “О жордановых конечных коммутативных подгруппах простых комплексных групп Ли”, Функц. анализ и его прил., 8:4 (1974), 1–4
- P. Arnoux, A. Marin, “The Kühnel triangulation of the complex projective plane from the view point of complex crystallography. II”, Mem. Fac. Sci. Kyushu Univ. Ser. A, 45:2 (1991), 167–244
- J. C. Baez, “The octonions”, Bull. Amer. Math. Soc. (N.S.), 39:2 (2002), 145–205
- B. Bagchi, B. Datta, “On Kühnel's 9-vertex complex projective plane”, Geom. Dedicata, 50:1 (1994), 1–13
- B. Bagchi, B. Datta, “A short proof of the uniqueness of Kühnel's 9-vertex complex projective plane”, Adv. Geom., 1:2 (2001), 157–163
- B. Bagchi, B. Datta, “On $k$-stellated and $k$-stacked spheres”, Discrete Math., 313:20 (2013), 2318–2329
- B. Benedetti, F. H. Lutz, “Random discrete Morse theory and a new library of triangulations”, Exp. Math., 23:1 (2014), 66–94
- A. Björner, F. H. Lutz, “Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincare homology 3-sphere”, Exp. Math., 9:2 (2000), 275–289
- A. Borel, “Le plan projectif des octaves et les spheres comme espaces homogènes”, C. R. Acad. Sci. Paris, 230 (1950), 1378–1380
- U. Brehm, W. Kühnel, “Combinatorial manifolds with few vertices”, Topology, 26:4 (1987), 465–473
- U. Brehm, W. Kühnel, “15-vertex triangulations of an 8-manifold”, Math. Ann., 294:1 (1992), 167–193
- H. Bruggesser, P. Mani, “Shellable decompositions of cells and spheres”, Math. Scand., 29:2 (1971), 197–205
- F. Chapoton, L. Manivel, “Triangulations and Severi varieties”, Exp. Math., 22:1 (2013), 60–73
- J. H. Conway, D. A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A. K. Peters, Ltd./CRC Press, Natick, MA, 2003, xii+159 pp.
- G. Danaraj, V. Klee, “Which spheres are shellable?”, Algorithmic aspects of combinatorics, Ann. Discrete Math., 2, North-Holland Publ. Comp., 1978, 33–52
- R. Dougherty, V. Faber, M. Murphy, “Unflippable tetrahedral complexes”, Discrete Comput. Geom., 32:3 (2004), 309–315
- J. Eells, Jr., N. H. Kuiper, “Manifolds which are like projective planes”, Inst. Hautes Etudes Sci. Publ. Math., 14 (1962), 5–46
- A. Engström, “Discrete Morse functions from Fourier transforms”, Exp. Math., 18:1 (2009), 45–53
- А. А. Гайфуллин, “Локальные формулы для комбинаторных классов Понтрягина”, Изв. РАН. Сер. матем., 68:5 (2004), 13–66
- А. А. Гайфуллин, “Построение комбинаторных многообразий с заданными наборами линков вершин”, Изв. РАН. Сер. матем., 72:5 (2008), 3–62
- А. А. Гайфуллин, “Минимальная триангуляция комплексной проективной плоскости, допускающая шахматную раскраску четырехмерных симплексов”, Геометрия, топология и математическая физика. II, Сборник статей. К 70-летию со дня рождения академика Сергея Петровича Новикова, Труды МИАН, 266, МАИК «Наука/Интерпериодика», М., 2009, 33–53
- А. А. Гайфуллин, “Пространства конфигураций, бизвездные преобразования и комбинаторные формулы для первого класса Понтрягина”, Дифференциальные уравнения и топология. I, Сборник статей. К 100-летию со дня рождения академика Льва Семеновича Понтрягина, Труды МИАН, 268, МАИК «Наука/Интерпериодика», М., 2010, 76–93
- A. A. Gaifullin, Triangulations of the quaternionic projective plane and manifolds like the octonionic projective plane
- A. A. Gaifullin, On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
- A. A. Gaifullin, New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane
- А. А. Гайфуллин, Д. А. Городков, “Явный вид локальной комбинаторной формулы для первого класса Понтрягина”, УМН, 74:6(450) (2019), 161–162
- Д. А. Городков, “Минимальная триангуляция кватернионной проективной плоскости”, УМН, 71:6(432) (2016), 159–160
- D. Gorodkov, “A 15-vertex triangulation of the quaternionic projective plane”, Discrete Comput. Geom., 62:2 (2019), 348–373
- R. L. Griess, Jr., “Elementary abelian $p$-subgroups of algebraic groups”, Geom. Dedicata, 39:3 (1991), 253–305
- B. Grünbaum, V. P. Sreedharan, “An enumeration of simplicial $4$-polytopes with $8$ vertices”, J. Combinatorial Theory, 2:4 (1967), 437–465
- J. Kahn, M. Saks, D. Sturtevant, “A topological approach to evasiveness”, Combinatorica, 4:4 (1984), 297–306
- E. G. Köhler, F. H. Lutz, Triangulated manifolds with few vertices: vertex-transitive triangulations I
- L. Kramer, “Projective planes and their look-alikes”, J. Differential Geom., 64:1 (2003), 1–55
- W. Kühnel, Tight polyhedral submanifolds and tight triangulations, Lecture Notes in Math., 1612, Springer-Verlag, Berlin, 1995, vi+122 pp.
- W. Kühnel, T. F. Banchoff, “The 9-vertex complex projective plane”, Math. Intelligencer, 5:3 (1983), 11–22
- W. Kühnel, G. Lassmann, “The unique 3-neighborly 4-manifold with few vertices”, J. Combin. Theory Ser. A, 35:2 (1983), 173–184
- J. M. Landsberg, L. Manivel, “The projective geometry of Freudenthal's magic square”, J. Algebra, 239:2 (2001), 477–512
- R. Lazarsfeld, “An example of 16-dimensional projective variety with a 25-dimensional secant variety”, Math. Letters, 7 (1981), 1–4
- W. B. R. Lickorish, “Simplicial moves on complexes and manifolds”, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, 299–320
- F. H. Lutz, Triangulated manifolds with few vertices: combinatorial manifolds
- J. Milnor, “On manifolds homeomorphic to the 7-sphere”, Ann. of Math. (2), 64:2 (1956), 399–405
- B. Morin, M. Yoshida, “The Kühnel triangulation of the complex projective plane from the view point of complex crystallography. I”, Mem. Fac. Sci. Kyushu Univ. Ser. A, 45:1 (1991), 55–142
- A. Nabutovsky, “Geometry of the space of triangulations of a compact manifold”, Comm. Math. Phys., 181:2 (1996), 303–330
- U. Pachner, “Bistellare Äquivalenz kombinatorischer Mannigfaltigkeiten”, Arch. Math. (Basel), 30:1 (1978), 89–98
- U. von Pachner, “Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten”, Abh. Math. Sem. Univ. Hamburg, 57 (1987), 69–86
- К. Рурк, Б. Сандерсон, Введение в кусочно линейную топологию, Мир, М., 1974, 208 с.
- N. Shimada, “Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds”, Nagoya Math. J., 12 (1957), 59–69
- И. А. Володин, В. Е. Кузнецов, А. Т. Фоменко, “О проблеме алгоритмического распознавания стандартной трехмерной сферы”, УМН, 29:5(179) (1974), 71–168
- I. Yokota, “Realization of automorphisms $sigma$ of order $3$ and $G^{sigma}$ of compact exceptional Lie groups $G$. I. $G=G_2,F_4,E_6$”, J. Fac. Sci. Shinshu Univ., 20:2 (1985), 131–144
- Ф. Л. Зак, “Проекции алгебраических многообразий”, Матем. сб., 116(158):4(12) (1981), 593–602
- Ф. Л. Зак, “Многообразия Севери”, Матем. сб., 126(168):1 (1985), 115–132
- E. C. Zeeman, Seminar on combinatorial topology, Inst. Hautes Etudes Sci., Paris, 1963