3 - Orbifold K-theory

Summary

Introduction

Orbifold K-theory is the K-theory associated to orbifold vector bundles. This can be developed in the full generality of groupoids, but as we have seen in Chapter 1, any effective orbifold can be expressed as the quotient of a smooth manifold by an almost free action of a compact Lie group. Therefore, we can use methods from equivariant topology to study the K-theory of effective orbifolds. In particular, using an appropriate equivariant Chern character, we obtain a decomposition theorem for orbifold K-theory as a ring. A byproduct of our orbifold K-theory is a natural notion of orbifold Euler number for a general effective orbifold. What we lose in generality is gained in simplicity and clarity of exposition. Given that all known interesting examples of orbifolds do indeed arise as quotients, we feel that our presentation is fairly broad and will allow the reader to connect orbifold invariants with classical tools from algebraic topology. In order to compute orbifold K-theory, we make use of equiariant Bredon cohomology with coefficients in the representation ring functor. This equivariant theory is the natural target for equivariant Chern characters, and seems to be an important technical device for the study of orbifolds.

A key physical concept in orbifold string theory is twisting by discrete torsion.