Abstract
In this paper we study the cubical stuctures and fundamental groups of real toric spaces. We give an explicit presentation of the fundamental group of the real toric space over a simple polytope. Then using this presentation, we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.
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References
Brøndsted, A.: An Introduction to Convex Polytopes, Springer-Verlag, New York, 1983
Bjørner, A., Brenti, F.: Combinatorics of Coxeter Groups. Grad. Texts in Math., Vol. 231, Springer, Berlin-Heidelberg, 2005
Buchstaber, V. M., Panov, T. E.: Algebraic topology of manifolds defined by simple polytopes (Russian). Uspekhi Mat. Nauk, 53(3), 195–196 (1998); (Translated in English) Russian Math. Surveys, 53(3), 623–625 (1998)
Buchstaber, V. M., Panov, T. E.: Torus actions, combinatorial topology, and homological algebra. Russian Math. Surveys, 55, 825–921 (2000)
Buchstaber, V. M., Panov, T. E.: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series, Vol. 24, Amer. Math. Soc., Providence, Rhode Island, 2002
Buchstaber, V. M., Panov, T. E.: Toric topology, Amer. Math. Soc., Providence, Rhode Island, 2015
Choi, S., Kaji, S., Theriault, S.: Homotopy decomposition of a suspended real toric space. Bol. Soc. Mat. Mexicana (3), 23(1), 153–161 (2017)
Danilov, V. I.: The geometry of toric varieties. Uspekhi Mat. Nauk, 33(2), 85–134 (1978)
Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup., 3(4), 507–588 (1970)
Davis, M. W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 62, 417–451 (1991)
Erokhovets, N.: Buchstaber invariant of simple polytopes. Russian Math. Surveys, 63(5), 962–964 (2008)
Fukukawa, Y., Masuda, M: Buchstaber invariants of skeleta of a simplex. Osaka J. Math., 48(2), 549–582 (2011)
Jurkiewicz, J.: Chow ring of projective nonsingular torus embedding. Colloq. Math., 43(2), 261–270 (1981)
Lü, Z., Ma, J. and Sun, Y.: Elementary symmetric polynomials in Stanley-Reisner face ring. arXiv:1602.08837v1
Lü, Z., Panov, T.: Moment-angle complexes from simplicial posets. Cent. Eur. J. Math., 9(4), 715–730 (2011)
Porter, G. J.: The homotopy groups of wedges of suspensions. Amer. J. Math., 88(3), 655–663 (1966)
Shen, Q. F.: Lower bound for Buchstaber invariants of real universal complexes. To appear in Osaka J. Math.
Sun, Y.: Buchstaber invariants of universal complexes. Chinese Ann. Math. Ser. B, 38(6), 1335–1344 (2017)
Wu, L., Yu, L.: Fundamental groups of small covers revisited. Int. Math. Res. Not., 10, 7262–7298 (2021)
Acknowledgements
We appreciate Dr. Wu Lisu for giving the elementary idea of this explicit presentation of fundamental groups. In addition, we thank the referees for their time, and the comments which are constructive and forward-looking.
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Dedicated to Professor Banghe Li on His 80th Birthday
Partially supported by the NSFC (Grant No. 11971112), and the China Scholarship Council (Grant No. 202106100095)
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Lü, Z., Zhang, S. Fundamental Groups of Real Toric Spaces over Simple Polytopes. Acta. Math. Sin.-English Ser. 38, 1887–1900 (2022). https://doi.org/10.1007/s10114-022-2301-1
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DOI: https://doi.org/10.1007/s10114-022-2301-1