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.


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.

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Hirotaka Tamanoi
Department of Mathematics, University of California
Santa Cruz, CA 95064, USA

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