Chern character for twisted K-theory of orbifolds

Abstract

For an orbifold X and α ∈ H 3 ( X , Z ) , we introduce the twisted cohomology H c ∗ ( X , α ) and prove that the non-commutative Chern character of Connes–Karoubi establishes an isomorphism between the twisted K-groups K α ∗ ( X ) ⊗ C and the twisted cohomology H c ∗ ( X , α ) . This theorem, on the one hand, generalizes a classical result of Baum–Connes, Brylinski–Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C , and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem–Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai–Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.