Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REAL POLYHEDRAL PRODUCTS, MOORE'S CONJECTURE, AND SIMPLICIAL ACTIONS ON REAL TORIC SPACES

  • Kim, Jin Hong (Department of Mathematics Education Chosun University)
  • Received : 2017.06.06
  • Accepted : 2017.10.26
  • Published : 2018.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore's conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all k-simplices with $k{\geq}2$. Moreover, we also show a structural theorem for a finite group G acting simplicially on the real toric space. In other words, we show that G always contains an element of order 2, and so the order of G should be even.

Keywords

Acknowledgement

Supported by : Chosun University

References

  1. A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), no. 3, 1634-1668. https://doi.org/10.1016/j.aim.2010.03.026
  2. A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, Homology Homotopy Appl. 17 (2015), no. 2, 137-160. https://doi.org/10.4310/HHA.2015.v17.n2.a8
  3. V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, 24, American Mathematical Society, Providence, RI, 2002.
  4. V. M. Buchstaber and T. E. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, American Mathematical Society, Providence, RI, 2015.
  5. S. Cho, S. Choi, and S. Kaji, Geometric representations of finite groups on real toric spaces, preprint (2017); arXiv:1704.08591v1.
  6. S. Choi, S. Kaji, and S. Theriault, Homotopy decomposition of a suspended real toric space, Bol. Soc. Mat. Mex. (3) 23 (2017), no. 1, 153-161. https://doi.org/10.1007/s40590-016-0090-1
  7. F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no. 1, 121-168. https://doi.org/10.2307/1971269
  8. M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417-451. https://doi.org/10.1215/S0012-7094-91-06217-4
  9. G. Denham and A. I. Suciu, Moment-angle complexes, monomial ideals and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson. Part 3, 25-60. https://doi.org/10.4310/PAMQ.2007.v3.n1.a2
  10. J. Grbic and S. Theriault, The homotopy type of the polyhedral product for shifted complexes, Adv. Math. 245 (2013), 690-715. https://doi.org/10.1016/j.aim.2013.05.002
  11. Y. Hao, Q. Sun, and S. Theriault, Moore's conjecture for polyhedral products, preprint (2017); arXiv:1701.07720v1.
  12. J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990.
  13. H. Ishida, Y. Fukukawa, and M. Masuda, Topological toric manifolds, Mosc. Math. J. 13 (2013), no. 1, 57-98, 189-190.
  14. J. A. Neisendorfer and P. S. Selick, Some examples of spaces with or without exponents, in Current trends in algebraic topology, Part 1 (London, Ont., 1981), 343-357, CMS Conf. Proc., 2, Amer. Math. Soc., Providence, RI, 1982.
  15. P. Selick, Moore conjectures, in Algebraic topology-rational homotopy (Louvain-laNeuve, 1986), 219-227, Lecture Notes in Math., 1318, Springer, Berlin, 1988..