Twisted K-theory for actions of Lie groupoids and its completion theorem

Abstract

In this paper we define twisted equivariant K -theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, we show that this defines a periodic cohomology theory on the category of finite G-CW-complexes with G-stable projective bundles by comparing with a suitable representable cohomology theory. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.

Proper actions and decompositions in equivariant K-theory

“…Lie group G, we extend the definition of [LO01] of equivariant K-theory for proper actions to twisted equivariant K-theory for the twisting relevant to our decompositions. See also [Can11].…”

“…We remark that if every central extension G of G by S 1 satisfies the property (AN ), it implies the property (K) defined in Section 2.3. The property (AN ) is closely related to the property of being Bredon-compatible defined by J. Cantarero in [Can12] and [Can11] for representable grupoids. We use the property (AN ) here, because it happens to be easier to verify for some nice cases.…”