## Algebraic & Geometric Topology

### 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 $Γ$, 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

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

Dates
Revised: 31 July 2003
Accepted: 20 August 2003
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882418

Digital Object Identifier
doi:10.2140/agt.2003.3.791

Mathematical Reviews number (MathSciNet)
MR1997338

Zentralblatt MATH identifier
1037.57022

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882418

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