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.
Similar content being viewed by others
References
Allday C, Puppe V. Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge: Cambridge University Press, 1993
Arnold V I. The cohomology ring of the colored braid group (in Russian). Mat Zametki, 1969, 5: 227–231
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: 12241–12244
Beben P, Grbić J. Configuration spaces and polyhedral products. Adv Math, 2017, 314: 378–425
Berrick A J, Cohen F R, Wong Y L, et al. Configurations, braids and homotopy groups. J Amer Math Soc, 2006, 19: 265–326
Birman J S. Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton: Princeton University Press, 1975
Bott R. Configuration spaces and embedding invariants. Turkish J Math, 1996, 20: 1–17
Buchstaber V M, Panov T E. Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series, vol. 24. Providence: Amer Math Soc, 2002
Cao X Y, Lü Z. Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture. J Algebraic Combin, 2012, 35: 121–140
Choi S Y, Masuda M, Suh D Y. Rigidity problems in toric topology: A survey. Proc Steklov Inst Math, 2011, 275: 177–190
Choi S Y, Panov T E, Suh D Y. Toric cohomological rigidity of simple convex polytopes. J Lond Math Soc (2), 2010, 82: 343–360
Cohen D C. Monodromy of fiber-type arrangements and orbit configuration spaces. Forum Math, 2001, 13: 505–530
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, 207–351
Cohen F R. On configuration spaces, their homology, and Lie algebras. J Pure Appl Algebra, 1995, 100: 19–42
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: 2289–2300
Cohen F R, Xicoténcatl M A. On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups. Topology Appl, 2002, 118: 17–29
Davis M, Januszkiewicz T. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math J, 1991, 61: 417–451
Fadell E, Neuwirth L. Configuration spaces. Math Scand, 1962, 10: 111–118
Farber M. Invitation to Topological Robotics. Zuürich Lectures in Advanced Mathematics. Zuürich: Eur Math Soc, 2008
Feichtner E M, Ziegler G M. On orbit configuration spaces of spheres. Topology Appl, 2002, 118: 85–102
Félix Y, Thomas J C. Rational Betti numbers of configuration spaces. Topology Appl, 2000, 102: 139–149
Ghrist R. Configuration spaces and braid groups on graphs in robotics. AMS/IP Stud Adv Math, 2001, 24: 29–40
Ishida H, Fukukawa Y, Masuda M. Topological toric manifolds. Mosc Math J, 2013, 13: 57–98
Lü Z, Tan Q B. Equivariant Chern numbers and the number of fixed points for unitary torus manifolds. Math Res Lett, 2011, 18: 1319–1325
Luü Z, Yu L. Topological types of 3-dimensional small covers. Forum Math, 2011, 23: 245–284
Masuda M. Equivariant cohomology distinguishes toric manifolds. Adv Math, 2008, 218: 2005–2012
Masuda M, Suh D Y. Classification problems of toric manifolds via topology. Contemp Math, 2008, 460: 273–286
Munkres J R. Elements of Algebraic Topology. Menlo Park: Addison-Wesley Publishing Company, 1984
Nambu Y. Second configuration space and third quantization. Progr Theoret Phys, 1949, 4: 96–98
Stafa M. The Mayer-Vietoris spectral sequence. http://www.math.tulane.edu/~mstafa/papers/mayer.vietoris.ss.pdf, 2015
Straten V, Petronia M. The topology of the configurations of Desargues and Pappus. Rep Math Colloquium, 1948, 8: 3–17
Totaro B. Configuration spaces of algebraic varieties. Topology, 1996, 35: 1057–1067
Ustinovsky Y M. Toral rank conjecture for moment-angle complexes. Math Notes, 2011, 90: 279
Vassiliev V A. Complements of Discriminants of Smooth Maps: Topology and Applications. Translations of Mathematical Monographs, vol. 98. Providence: Amer Math Soc, 1992
Xicoténcatl M A. Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces. PhD Thesis. Rochester: University of Rochester, 1997
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, 233–249
Yamashita Y, Nishi H, Kojima S. Configuration spaces of points on the circle and hyperbolic Dehn fillings, II. Geom Dedicata, 2002, 89: 143–157
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, J., Lü, Z. & Wu, J. Orbit configuration spaces of small covers and quasi-toric manifolds. Sci. China Math. 64, 167–196 (2021). https://doi.org/10.1007/s11425-018-9526-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-018-9526-6
Keywords
- orbit configuration space
- small cover
- quasi-toric manifold
- (real) moment-angle manifold
- Euler characteristic
- homotopy type