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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ardila, F. and Billey, S., Flag arrangements and triangulations of products of simplices, Adv. Math., 214(2), 2007, 495–524.
Bahri, A., Bendersky, M., Cohen, F. and Gitler, S., Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, Homology Homotopy Appl., 17, 2015, 137–160.
Buchstaber, V. M. and Panov, T. E., Torus actions and their applications in topology and combinatorics, University Lecture Series, 24, Amer. Math. Soc., Providence, RI, 2002.
Buchstaber, V. M. and Panov, T. E., Toric Topology, Mathematical Surveys and Monographs, 204, Amer. Math. Soc., Providence, RI, 2015.
Burago, D., Burago, Y. and Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc., Providence, RI, 2001.
Choi, S. and Kim, J. S., Combinatorial rigidity of 3-dimensional simplicial polytopes, Int. Math. Res. Not., 2011(8), 2011, 1935–1951.
Choi, S., Masuda, M. and Suh, D. Y., Quasitoric manifolds over a product of simplices, Osaka J. Math., 47(1), 2010, 109–129.
Choi, S., Masuda, M. and Suh, D. Y., Rigidity problems in toric topology: a survey, Proc. Steklov Inst. Math., 275, 2011, 177–190.
Choi, S., Panov, T. and Suh, D. Y., Toric cohomological rigidity of simple convex polytopes, J. Lond. Math. Soc., 82(2), 2010, 343–360.
Coxeter, H. S. M., Discrete groups generated by reflections, Annals of Math., 35(3), 1934, 588–621.
Davis, M. W. and Januszkiewicz, T., Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62(2), 1991, 417–451.
de Loera, J. A., Nonregular triangulations of products of simplices, Discrete Comput. Geom., 15(3), 1996, 253–264.
Francisco, S., The Cayley trick and triangulations of products of simplices, Integer points in polyhedrageometry, number theory, algebra, optimization, Contemp. Math., 374, Amer. Math. Soc., Providence, RI, 2005, 151–177.
Hattori, A. and Masuda, M., Theory of multi-fans, Osaka J. Math., 40(1), 2003, 1–68.
Kobayashi, S., Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, 1972.
Kuroki, S., Masuda, M. and Yu, L., Small covers, infra-solvmanifolds and curvature, Forum Math., 27(5), 2015, 2981–3004.
Ustinovsky, Y. M., Doubling operation for polytopes and torus actions, Russian Math. Surveys, 64(5), 2009, 181–182.
Ustinovsky, Y. M., Toral rank conjecture for moment-angle complexes, Math. Notes, 90(2), 2011, 279–283.
Wiemeler, M., Torus manifolds and non-negative curvature, J. Lond. Math. Soc., II. Ser, 91(3), 2015, 667–692.
Ziegler, G. M., Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1998.
Acknowledgements
The authors want to thank Hanchul Park and Suyoung Choi for some helpful comments and thank Shicheng Xu and Jiaqiang Mei for some valuable discussions on the geometry of Alexandrov spaces.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This work was supported by the National Natural Science Foundation of China (No. 11871266) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Rights and permissions
About this article
Cite this article
Yu, L., Masuda, M. On Descriptions of Products of Simplices. Chin. Ann. Math. Ser. B 42, 777–790 (2021). https://doi.org/10.1007/s11401-021-0290-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-021-0290-5