How many simplices are needed to triangulate a Grassmannian?
Keywords
Minimal triangulation, Grassmann manifold, cup-length, lower bound theorem, manifold $g$-theoremAbstract
In this paper we use recently developed methods to compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $\G_k(\mathbb{R}^n)$. We first estimate the number of vertices that are needed for such a triangulation by giving a general lower bound and some more precise bounds for $k=2,3,4$. By applying the Lower Bound Theorem (LBT) of Barnette for triangulated manifolds, we then obtain estimates for the number of simplices in all dimensions. For higher-dimensional simplices these estimates can be considerably improved by using the recent progress on the Generalized Lower Bound Theorem for triangulated manifolds, which states that the $h''$-numbers of triangulated manifolds are unimodal, together with the computation of the Poincaré polynomial. For example, we are able to prove that the number of top-dimensional simplices in a triangulation of $\G_k(\mathbb{R}^n)$ grows exponentially with $n$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.References
K. Adiprasito, Combinatorial Lefschetz theorems beyond positivity, arXiv:math.CO/1812.10454v4 (2018), 76 pp.
K. Adiprasito, S. Awakumov and R. Karasev, A subexponential size RPn , https://arxiv.org/abs/2009.02703.
D. Barnette, The minimum number of vertices of a simple polytope, Israel J. Math. 10 (1971), 121–125.
D. Barnette, A proof of the lower bound conjecture for convex polytopes, Pacific J. Math. 46 (1973), 349–354.
D. Barnette, Graph theorems for manifolds, Israel J. Math. 16 (1973), 62–72.
A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207.
U. Brehm and W. Kühnel, 15-Vertex triangulations of 8-manifolds, Math. Ann. 294 (1992), 167–193.
L. Casian and Y. Kodama, On the cohomology of real Grassmann manifolds, arXiv:math. AG/1309.5520 (2013), 13 pp.
A.A. Gaifullin, Computation of characteristic classes of a manifold from its triangulation, Uspekhi Mat. Nauk 60 (2005), no. 4 (364), 37–66 (Russian); English transl.: Russian Math. Surveys 60 (2005), no. 4, 615–644.
A.A. Gaifullin, Configuration spaces, bistellar moves, and combinatorial formulas for the first Pontryagin class., Proc. Steklov Inst. Math. 268 (2010), 70–86.
I.M. Gel’fand and R.D. MacPherson, A combinatorial formula for the Pontrjagin classes, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 304–309.
D. Gorodkov, A 15-vertex triangulation of the quaternionic projective plane, Discrete Comput. Geom. 62 (2019), 348–373.
D. Govc, W. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 (2019), 31–48.
M. Gromov, Partial Differential Relations, Springer, Berlin, Heidelberg, New York, 1986.
A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
C. He, Torsions of integral homology and cohomology of real Grassmannians, arXiv:math.AT/1709.05623 (2017), 6 pp.
G. Kalai, Rigidity and the lower bound theorem 1, Invent. Math. 88 (1987), 125–151.
M. Karoubi and C. Weibel On the covering type of a space, L’Enseignement Math. 62 (2016), 457–474.
S. Klee and I. Novik, Face enumeration on simplicial complexes, Recent Trends in Combinatorics, Springer, Cham, 2016, pp. 653–686.
W. Kühnel, Higher-dimensional analogues of Caászár’s torus, Result. Math. 9 (1986), 95–106.
F.H. Lutz, Triangulated manifolds with few vertices: combinatorial manifolds, arXiv:math.CO/0506372v1 (2005), 37 pp.
P. McMullen and D. Walkup, A generalized lower-bound conjecture for simplicial polytopes, Mathematika 18 (1971), 264–273.
L. Milin, A combinatorial computation of the first Pontryagin class of the complex projective plane, Geom. Dedicata 49 (1994), 253–291.
S. Murai and E. Nevo, On the generalized lower bound conjecture for polytopes and spheres, Acta Math. 210 (2013), 185–202.
I. Novik, Upper Bound Theorems for homology manifolds, Israel J. Math. 108 (1998), 45–82.
I. Novik and E. Swartz, Applications of Klee’s Dehn–Sommerville relations, Discrete Comput. Geom. 42 (2009), 261–276.
I. Novik and E. Swartz, Gorenstein rings through face rings of manifolds, Compos. Math. 145 (2009), 993–1000.
I. Novik and E. Swartz, Socles of Buchsbaum modules, complexes and posets, Adv. Math. 222 (2009), 2059–2084.
P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), 125–142.
R. Stanley, The number of faces of a simplicial convex polytope, Adv. Math. 35 (1980), 236–238.
R. Stong, Cup products in Grassmannians, Top. Appl. 13 (1982), 103–113.
E. Swartz, Thirty-five years and counting, arXiv:math.CO/1411.0987 (2014), 29 pp.
L. Venturello and H. Zheng, A new family of triangulation of RP d , arXiv:math.CO/1910.07433 (2019), 13 pp.
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