Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

Tamanoi, Hirotaka¹

¹Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 791-856

Résumé

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.

Détail

DOI : 10.2140/agt.2003.3.791

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

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