Algebraic & Geometric Topology

Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

Hirotaka Tamanoi

Full-text: Open access


Let G be a finite group and let M be a G–manifold. We introduce the concept of generalized orbifold invariants of MG associated to an arbitrary group Γ, an arbitrary Γ–set, and an arbitrary covering space of a connected manifold Σ whose fundamental group is Γ. Our orbifold invariants have a natural and simple geometric origin in the context of locally constant G–equivariant maps from G–principal bundles over covering spaces of Σ to the G–manifold M. We calculate generating functions of orbifold Euler characteristic of symmetric products of orbifolds associated to arbitrary surface groups (orientable or non-orientable, compact or non-compact), in both an exponential form and in an infinite product form. Geometrically, each factor of this infinite product corresponds to an isomorphism class of a connected covering space of a manifold Σ. The essential ingredient for the calculation is a structure theorem of the centralizer of homomorphisms into wreath products described in terms of automorphism groups of Γ–equivariant G–principal bundles over finite Γ–sets. As corollaries, we obtain many identities in combinatorial group theory. As a byproduct, we prove a simple formula which calculates the number of conjugacy classes of subgroups of given index in any group. Our investigation is motivated by orbifold conformal field theory.

Article information

Algebr. Geom. Topol., Volume 3, Number 2 (2003), 791-856.

Received: 11 February 2002
Revised: 31 July 2003
Accepted: 20 August 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N20: Generalized (extraordinary) homology and cohomology theories 55N91: Equivariant homology and cohomology [See also 19L47]
Secondary: 57S17: Finite transformation groups 57D15 20E22: Extensions, wreath products, and other compositions [See also 20J05] 37F20: Combinatorics and topology 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

automorphism group centralizer combinatorial group theory covering space equivariant principal bundle free group $\Gamma$–sets generating function Klein bottle genus (non)orientable surface group orbifold Euler characteristic symmetric products twisted sector wreath product


Tamanoi, Hirotaka. Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces. Algebr. Geom. Topol. 3 (2003), no. 2, 791--856. doi:10.2140/agt.2003.3.791.

Export citation


  • P. Bantay, Symmetric products, permutation orbifolds, and discrete torsion .
  • T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Math. 98, Springer-Verlag, New York (1985)
  • K. S. Brown, Cohomology of Groups, Graduate Texts in Math. 87, Springer-Verlag, New York (1982)
  • J. Bryan and J. Fulman, Orbifold Euler characteristics and the number of commuting $m$-tuples in the symmetric groups , Annals of Combinatorics, 2 (1998), 1–6
  • R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde, Elliptic genera of symmetric products and second quantized strings , Comm. Math. Phys., 185 (1997), 197–209
  • L. Dixon, J. Harvey, C. Vafa and E. Witten, Strings on orbifolds , Nuclear Physics, B 261 (1985), 678–686
  • F. Hirzebruch and H. Höfer, On the Euler number of an orbifold , Math. Annalen, 286 (1990), 255–260
  • M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel, Generalized group characters and complex oriented cohomology theories , J. Amer. Math. Soc., 13 (2000), 553–594
  • A. Kerber and B. Wagner, Gleichungen in endlichen Gruppen , Arch. Math., 35 (1980), 252–262
  • I. G. Macdonald, Poincaré polynomials of symmetric products , Proc. Camb. Phil. Soc., 58 (1962), 123–175
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press (1995, second edition)
  • A. D. Mednykh, Hurwitz problem on the number of nonequivalent coverings of a compact Riemann surface , Siber. Math. J., 23 (1983), 415–420
  • A. D. Mednykh and G. G. Pozdnyakova, Number of nonequivalent coverings over a nonorientable compact surface , Siber. Math. J., 27 (1986), 99–106
  • P. Shanahan, Atiyah-Singer Index Theorem, Lecture Notes in Math. 638, Springer-Verlag, New York (1978)
  • R. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge (1999)
  • M. Suzuki, Group Theory \romI, Grundlehren der mathematischen Wissenschaften 247, Springer-Verlag, New York (1982)
  • H. Tamanoi, Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory , Algebraic and Geometric Topology, 1 (2001), 115–141
  • W. Wang, Equivariant K-theory, wreath products, and Heisenberg algebra , Duke Math. J., 103 (2000), 1–23 s