Volume 3, issue 2 (2003)

Download this article
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

Hirotaka Tamanoi

Algebraic & Geometric Topology 3 (2003) 791–856

arXiv: math.GR/0309133


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.

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