#### Algebraic and Geometric Topology 3 (2003),
paper no. 27, pages 791-856.

## Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

### Hirotaka Tamanoi

**Abstract**.
Let G be a finite group and let M be a G-manifold. We introduce the
concept of generalized orbifold invariants of M/G associated to an
arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary
covering space of a connected manifold Sigma whose fundamental group
is Gamma. 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 Sigma 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 Sigma. 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 Gamma-equivariant G-principal bundles over
finite Gamma-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.
**Keywords**.

**AMS subject classification**.
Primary: 55N20, 55N91.
Secondary: 57S17, 57D15, 20E22, 37F20, 05A15.

**DOI:** 10.2140/agt.2003.3.791

**E-print:** `arXiv:math.GR/0309133`

Submitted: 11 February 2002.
(Revised: 31 July 2003.)
Accepted: 20 August 2003.
Published: 31 August 2003.

Notes on file formats
Hirotaka Tamanoi

Department of Mathematics, University of California

Santa Cruz, CA 95064, USA

Email: tamanoi@math.ucsc.edu

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