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Orbit configuration spaces of small covers and quasi-toric manifolds

  • Junda Chen
  • Zhi LüEmail author
  • Jie Wu
Articles
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Abstract

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces have non-free group actions, and they are all noncompact, but still built via simple convex polytopes. We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of the Euler characteristic for the classical configuration space, which generalizes the Félix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

Keywords

orbit configuration space small cover quasi-toric manifold (real) moment-angle manifold Euler characteristic homotopy type 

MSC(2010)

55R80 57S25 52B20 55P91 55N91 14M25 

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Notes

Acknowledgements

The first author and the second author were supported by National Natural Science Foundation of China (Grant Nos. 11371093, 11431009 and 11661131004). The third author was supported by National Natural Science Foundation of China (Grant No. 11028104).

References

  1. 1.
    Allday C, Puppe V. Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge: Cambridge University Press, 1993zbMATHGoogle Scholar
  2. 2.
    Arnold V I. The cohomology ring of the colored braid group (in Russian). Mat Zametki, 1969, 5: 227–231MathSciNetGoogle Scholar
  3. 3.
    Bahri A, Bendersky M, Cohen F R, et al. Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces. Proc Natl Acad Sci USA, 2009, 106: 12241–12244MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beben P, Grbić J. Configuration spaces and polyhedral products. Adv Math, 2017, 314: 378–425MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berrick A J, Cohen F R, Wong Y L, et al. Configurations, braids and homotopy groups. J Amer Math Soc, 2006, 19: 265–326MathSciNetCrossRefGoogle Scholar
  6. 6.
    Birman J S. Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton: Princeton University Press, 1975Google Scholar
  7. 7.
    Bott R. Configuration spaces and embedding invariants. Turkish J Math, 1996, 20: 1–17MathSciNetzbMATHGoogle Scholar
  8. 8.
    Buchstaber V M, Panov T E. Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series, vol. 24. Providence: Amer Math Soc, 2002CrossRefGoogle Scholar
  9. 9.
    Cao X Y, Lü Z. Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture. J Algebraic Combin, 2012, 35: 121–140MathSciNetCrossRefGoogle Scholar
  10. 10.
    Choi S Y, Masuda M, Suh D Y. Rigidity problems in toric topology: A survey. Proc Steklov Inst Math, 2011, 275: 177–190MathSciNetCrossRefGoogle Scholar
  11. 11.
    Choi S Y, Panov T E, Suh D Y. Toric cohomological rigidity of simple convex polytopes. J Lond Math Soc (2), 2010, 82: 343–360MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cohen D C. Monodromy of fiber-type arrangements and orbit configuration spaces. Forum Math, 2001, 13: 505–530MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cohen F R. The homology of Cn+1 spaces, n ⩾ 0. In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol. 533. Berlin-Heidelberg: Springer, 1976, 207–351CrossRefGoogle Scholar
  14. 14.
    Cohen F R. On configuration spaces, their homology, and Lie algebras. J Pure Appl Algebra, 1995, 100: 19–42MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cohen F R, Kohno T, Xicoténcatl M A. Orbit configuration spaces associated to discrete subgroups of PSL(2, ℝ). J Pure Appl Algebra, 2009, 213: 2289–2300MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cohen F R, Xicoténcatl M A. On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups. Topology Appl, 2002, 118: 17–29MathSciNetCrossRefGoogle Scholar
  17. 17.
    Davis M, Januszkiewicz T. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math J, 1991, 61: 417–451MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fadell E, Neuwirth L. Configuration spaces. Math Scand, 1962, 10: 111–118MathSciNetCrossRefGoogle Scholar
  19. 19.
    Farber M. Invitation to Topological Robotics. Zuürich Lectures in Advanced Mathematics. Zuürich: Eur Math Soc, 2008Google Scholar
  20. 20.
    Feichtner E M, Ziegler G M. On orbit configuration spaces of spheres. Topology Appl, 2002, 118: 85–102MathSciNetCrossRefGoogle Scholar
  21. 21.
    Félix Y, Thomas J C. Rational Betti numbers of configuration spaces. Topology Appl, 2000, 102: 139–149MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ghrist R. Configuration spaces and braid groups on graphs in robotics. AMS/IP Stud Adv Math, 2001, 24: 29–40MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ishida H, Fukukawa Y, Masuda M. Topological toric manifolds. Mosc Math J, 2013, 13: 57–98MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lü Z, Tan Q B. Equivariant Chern numbers and the number of fixed points for unitary torus manifolds. Math Res Lett, 2011, 18: 1319–1325MathSciNetCrossRefGoogle Scholar
  25. 25.
    Luü Z, Yu L. Topological types of 3-dimensional small covers. Forum Math, 2011, 23: 245–284MathSciNetzbMATHGoogle Scholar
  26. 26.
    Masuda M. Equivariant cohomology distinguishes toric manifolds. Adv Math, 2008, 218: 2005–2012MathSciNetCrossRefGoogle Scholar
  27. 27.
    Masuda M, Suh D Y. Classification problems of toric manifolds via topology. Contemp Math, 2008, 460: 273–286MathSciNetCrossRefGoogle Scholar
  28. 28.
    Munkres J R. Elements of Algebraic Topology. Menlo Park: Addison-Wesley Publishing Company, 1984zbMATHGoogle Scholar
  29. 29.
    Nambu Y. Second configuration space and third quantization. Progr Theoret Phys, 1949, 4: 96–98MathSciNetCrossRefGoogle Scholar
  30. 30.
    Stafa M. The Mayer-Vietoris spectral sequence. http://www.math.tulane.edu/~mstafa/papers/mayer.vietoris.ss.pdf, 2015
  31. 31.
    Straten V, Petronia M. The topology of the configurations of Desargues and Pappus. Rep Math Colloquium, 1948, 8: 3–17MathSciNetzbMATHGoogle Scholar
  32. 32.
    Totaro B. Configuration spaces of algebraic varieties. Topology, 1996, 35: 1057–1067MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ustinovsky Y M. Toral rank conjecture for moment-angle complexes. Math Notes, 2011, 90: 279MathSciNetCrossRefGoogle Scholar
  34. 34.
    Vassiliev V A. Complements of Discriminants of Smooth Maps: Topology and Applications. Translations of Mathematical Monographs, vol. 98. Providence: Amer Math Soc, 1992Google Scholar
  35. 35.
    Xicoténcatl M A. Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces. PhD Thesis. Rochester: University of Rochester, 1997Google Scholar
  36. 36.
    Xicoténcatl M A. On orbit configuration spaces and the rational cohomology of F(ℝPn, k). In: Une Dégustation Topologique: Homotopy Theory in the Swiss Alps. Contemporary Mathematics, vol. 265. Providence: Amer Math Soc, 2000, 233–249CrossRefGoogle Scholar
  37. 37.
    Yamashita Y, Nishi H, Kojima S. Configuration spaces of points on the circle and hyperbolic Dehn fillings, II. Geom Dedicata, 2002, 89: 143–157MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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