#### Volume 3, issue 2 (2003)

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Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

### Hirotaka Tamanoi

Algebraic & Geometric Topology 3 (2003) 791–856
 arXiv: math.GR/0309133
##### 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
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
##### Mathematical Subject Classification 2000
Primary: 55N20, 55N91
Secondary: 57S17, 57D15, 20E22, 37F20, 05A15