Advertisement

Functional Analysis and Its Applications

, Volume 52, Issue 4, pp 297–307 | Cite as

The Universal Euler Characteristic of V-Manifolds

  • S. M. Gusein-ZadeEmail author
  • I. Luengo
  • A. Melle-Hernández
Article
  • 6 Downloads

Abstract

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.

Key words

finite group actions V-manifold orbifold additive topological invariant lambda-ring Macdonald identity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Atiyah and G. Segal, “On equivariant Euler characteristics,” J. Geom. Phys., 6:4 (1989), 671–677.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Bryan and J. Fulman, “Orbifold Euler characteristics and the number of commuting mtuples in the symmetric groups,” Ann. Comb., 2:1 (1998), 1–6.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    W. Chen and Y. Ruan, “Orbifold Gromov–Witten theory,” in: Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, 25–85.CrossRefGoogle Scholar
  4. [4]
    L. Dixon, J. Harvey, C. Vafa, and E. Witten, “Strings on orbifolds,” Nuclear Phys. B, 261:4 (1985), 678–686.MathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Dixon, J. Harvey, C. Vafa, and E. Witten, “Strings on orbifolds. II,” Nuclear Phys. B, 274:2 (1986), 285–314.MathSciNetCrossRefGoogle Scholar
  6. [6]
    M. R. Dixon and T. A. Fournelle, “The indecomposability of certain wreath products indexed by partially ordered sets,” Arch. Math., 43 (1984), 193–207.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. Farsi and Ch. Seaton, “Generalized orbifold Euler characteristics for general orbifolds and wreath products,” Algebr. Geom. Topol., 11:1 (2011), 523–551.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. M. Gusein-Zade, “Equivariant analogues of the Euler characteristic and Macdonald type formulas,” Uspekhi Mat. Nauk, 72:1(433) (2017), 3–36; English transl.: Russian Math. Surveys, 72:1 (2017), 1–32.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “A power structure over the Grothendieck ring of varieties,” Math. Res. Lett., 11:1 (2004), 49–57.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “On the power structure over the Grothendieck ring of varieties and its applications,” Trudy Mat. Inst. Steklov., 258 (2007), 58–69; English transl.: Proc. Steklov Inst. Math., 258:1 (2007), 53–64.MathSciNetzbMATHGoogle Scholar
  11. [11]
    S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “Equivariant versions of higher order orbifold Euler characteristics,” Mosc. Math. J., 16:4 (2016), 751–765.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “Grothendieck ring of varieties with finite groups actions,” Proc. Edinb. Math. Soc., Ser. 2 (to appear); https://arxiv.org/abs/1706.00918.Google Scholar
  13. [13]
    F. Hirzebruch and T. Höfer, “On the Euler number of an orbifold,” Math. Ann., 286:1–3 (1990), 255–260.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D. Knutson, λ-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Math., vol. 308, Springer-Verlag, Berlin–New York, 1973.Google Scholar
  15. [15]
    P. M. Neumann, “On the structure of standard wreath products of groups,” Math. Z., 84 (1964), 343–373.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    I. Satake, “On a generalization of the notion of manifold,” Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 359–363.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    I. Satake, “The Gauss–Bonnet theorem for V-manifolds,” J. Math. Soc. Japan, 9 (1957), 464–492.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. Tamanoi, “Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory,” Algebr. Geom. Topol., 1 (2001), 115–141.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. tom Dieck, Transformation Groups, De Gruyter Studies in Math., vol. 8, Walter de Gruyter, Berlin, 1987.CrossRefGoogle Scholar
  20. [20]
    C. E. Watts, “On the Euler characteristic of polyhedra,” Proc. Amer. Math. Soc., 13 (1962), 304–306.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • S. M. Gusein-Zade
    • 1
    Email author
  • I. Luengo
    • 2
    • 3
  • A. Melle-Hernández
    • 4
  1. 1.Moscow State University, Faculty of Mechanics and MathematicsMoscowRussia
  2. 2.ICMAT, Madrid, Spain Department of Algebra, Geometry, and TopologyComplutense University of MadridMadridSpain
  3. 3.ICMATMadridSpain
  4. 4.Institute of Interdisciplinary Mathematics, Department of Algebra, Geometry, and TopologyComplutense University of MadridMadridSpain

Personalised recommendations