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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 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.

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
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Publication
Received: 11 February 2002
Revised: 31 July 2003
Accepted: 20 August 2003
Published: 31 August 2003
Authors
Hirotaka Tamanoi
Department of Mathematics
University of California
Santa Cruz, CA 95064
USA