Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2018: 0.29

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 69, Issue 2

Issues

Example of C-rigid polytopes which are not B-rigid

Suyoung Choi / Kyoungsuk Park
Published Online: 2019-03-19 | DOI: https://doi.org/10.1515/ms-2017-0236

Abstract

A simple polytope P is said to be B-rigid if its combinatorial structure is characterized by its Tor-algebra, and is said to be C-rigid if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over P. It is known that a B-rigid simple polytope is C-rigid. In this paper, we show that the B-rigidity is not equivalent to the C-rigidity.

MSC 2010: 52B35; 14M25; 05E40; 55NXX

Keywords: cohomologically rigid; B-rigid; quasitoric manifold; simple polytope; Peterson graph

The first named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(NRF-2016R1D1A1A09917654).

References

  • [1]

    Bosio, F.: Combinatorially rigid simple polytopes with d + 3 facets, arXiv:1511.09039 (2015).Google Scholar

  • [2]

    Buchstaber, V. M.: Lectures on toric topology. In: Proceedings of Toric Topology Workshop KAIST 2008, Vol. 10, Trends in Math. New Series, no. 1, Information Center for Mathematical Sciences, KAIST, 2008, pp. 1–64.Google Scholar

  • [3]

    Buchstaber, V. M.—Erokhovets, N. Y.: Fullerenes, polytopes and toric topology. In: Combinatorial and Toric Homotopy. Introducatory lectures, pp. 67–178, Vol. 35, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., River Edge, NJ, 2017.Google Scholar

  • [4]

    Buchstaber, V. M.—Erokhovets, N.—Masuda, M.—Panov, T. E.—Park, S.: Cohomological rigidity of manifolds defined by right-angled 3-dimensional polytopes, Uspekhi Mat. Nauk 72(2) (2017), 3–66 (in Russian); Russian Math. Surveys 72(2) (2017), 199–256 (English translation).Google Scholar

  • [5]

    Buchstaber, V. M.—Panov, T. E.: Toric Topology, Math. Surveys Monogr. 204, Amer. Math. Soc., Providence, RI, 2015.Google Scholar

  • [6]

    Choi, S.: Different moment-angle manifolds arising from two polytopes having the same bigraded Betti numbers, Algebr. Geom. Topol. 13(6) (2013), 3639–3649.CrossrefWeb of ScienceGoogle Scholar

  • [7]

    Choi, S.—Kim, J. S.: Combinatorial rigidity of 3-dimensional simplicial polytopes, Int. Math. Res. Not. IMRN 8 (2011), 1935–1951.Google Scholar

  • [8]

    Choi, S.—Panov, T. E.—Suh, D. Y.: Toric cohomological rigidity of simple convex polytopes, J. Lond. Math. Soc. 82(2) (2010), 343–360.CrossrefGoogle Scholar

  • [9]

    Davis, M. W.—Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62(2) (1991), 417–451.CrossrefGoogle Scholar

  • [10]

    Erokhovets, N. Yu.: Moment-angle manifolds of simple n-dimensional polytopes with n + 3 facets, Uspekhi Mat. Nauk 66(5) (2011), 187–188 (in Russian); Russian Math. Surveys 66(5) (2011), 1006–1008 (English translation).Google Scholar

  • [11]

    Erokhovets, N. Yu.: Buchstaber invariant theory of simplicial complexes and convex polytopes, Trudy MIAN 286(1) (2014), 144–206 (in Russian); Proc. Steklov Inst. Math. 286(1) (2014), 128–187 (English translation).Google Scholar

  • [12]

    Garrison, A.—Scott, R: Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc. 131(3) (2003), 963–971.CrossrefGoogle Scholar

  • [13]

    Gretenkort, J.—Kleinschmidt, P.—Sturmfels, B.: On the existence of certain smooth toric varieties, Discrete Comput. Geom. 5(3) (1990), 255–262.CrossrefGoogle Scholar

  • [14]

    Grünbaum, B.: Convex Polytopes. Grad. Texts in Math. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler.Google Scholar

  • [15]

    de Medrano, S. L.: Topology of the intersection of quadrics in Rn. In: Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math. 1370, Springer, Berlin, 1989, pp. 280–292.Google Scholar

  • [16]

    Masuda, M.—Suh, D. Y.: Classification problems of toric manifolds via topology. Toric topology, Contemp. Math. 460, Amer. Math. Soc., Providence, RI, 2008, pp. 273–286.Google Scholar

  • [17]

    Park, K.: Combinatorics of coxeter groups with permutation tableaux and cohomological rigidity of simple polytopes, Ph.D. Thesis, Ajou university, 2015, pp. 1–110.Google Scholar

About the article

Received: 2017-07-02

Accepted: 2018-04-11

Published Online: 2019-03-19

Published in Print: 2019-04-24


(Communicated by David Buhagiar)


Citation Information: Mathematica Slovaca, Volume 69, Issue 2, Pages 437–448, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0236.

Export Citation

© 2019 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in