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.
Keywords.
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.
Notes on file formats
Hirotaka Tamanoi
Department of Mathematics, University of California
Santa Cruz, CA 95064, USA
Email: tamanoi@math.ucsc.edu
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