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2003 Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces
Hirotaka Tamanoi
Algebr. Geom. Topol. 3(2): 791-856 (2003). DOI: 10.2140/agt.2003.3.791

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.

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Hirotaka Tamanoi. "Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces." Algebr. Geom. Topol. 3 (2) 791 - 856, 2003. https://doi.org/10.2140/agt.2003.3.791

Information

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

zbMATH: 1037.57022
MathSciNet: MR1997338
Digital Object Identifier: 10.2140/agt.2003.3.791

Subjects:
Primary: 55N20, 55N91
Secondary: 05A15, 20E22, 37F20, 57D15, 57S17

Rights: Copyright © 2003 Mathematical Sciences Publishers

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