On Descriptions of Products of Simplices

Abstract

The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology In addition, they give a shorter proof of a well known criterion on this subject.

Keywords

Convex polytope / Product of simplices / Moment-angle complex

References

[1] Ardila F, Billey S. Flag arrangements and triangulations of products of simplices. Adv. Math., 2007, 214(2): 495-524

[2] Bahri A, Bendersky M, Cohen F, Gitler S. Operations on polyhedral products and a new topological construction of infinite families of toric manifolds. Homology Homotopy Appl., 2015, 17: 137-160

[3] Buchstaber V M, Panov T E. Torus actions and their applications in topology and combinatorics, 2002, Providence, RI: Amer. Math. Soc.

[4] Buchstaber V M, Panov T E. Toric Topology, 2015, Providence, RI: Amer. Math. Soc.

[5] Burago D, Burago Y, Ivanov S. A course in metric geometry, Graduate Studies in Mathematics, 2001, Providence, RI: Amer. Math. Soc. 33

[6] Choi S, Kim J S. Combinatorial rigidity of 3-dimensional simplicial polytopes. Int. Math. Res. Not., 2011, 2011(8): 1935-1951

[7] Choi S, Masuda M, Suh D Y. Quasitoric manifolds over a product of simplices. Osaka J. Math., 2010, 47(1): 109-129

[8] Choi S, Masuda M, Suh D Y. Rigidity problems in toric topology: a survey. Proc. Steklov Inst. Math., 2011, 275: 177-190

[9] Choi S, Panov T, Suh D Y. Toric cohomological rigidity of simple convex polytopes. J. Lond. Math. Soc., 2010, 82(2): 343-360

[10] Coxeter H S M. Discrete groups generated by reflections. Annals of Math., 1934, 35(3): 588-621

[11] Davis M W, Januszkiewicz T. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 1991, 62(2): 417-451

[12] de Loera J A. Nonregular triangulations of products of simplices. Discrete Comput. Geom., 1996, 15(3): 253-264

[13] Francisco S. The Cayley trick and triangulations of products of simplices, Integer points in polyhedrageometry, number theory, algebra, optimization, 2005, Providence, RI: Amer. Math. Soc. 151-177

[14] Hattori A, Masuda M. Theory of multi-fans. Osaka J. Math., 2003, 40(1): 1-68

[15] Kobayashi S. Transformation Groups in Differential Geometry, 1972, Berlin: Springer-Verlag

[16] Kuroki S, Masuda M, Yu L. Small covers, infra-solvmanifolds and curvature. Forum Math., 2015, 27(5): 2981-3004

[17] Ustinovsky Y M. Doubling operation for polytopes and torus actions. Russian Math. Surveys, 2009, 64(5): 181-182

[18] Ustinovsky Y M. Toral rank conjecture for moment-angle complexes. Math. Notes, 2011, 90(2): 279-283

[19] Wiemeler M. Torus manifolds and non-negative curvature. J. Lond. Math. Soc., II. Ser, 2015, 91(3): 667-692

[20] Ziegler G M. Lectures on polytopes, Graduate Texts in Mathematics, 1998, New York: Springer-Verlag 152