Equivariant K-theory, groupoids and proper actions

Abstract

In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.Comment: 26 pages, v3 Correction to some lemmas and definitions in section 4. Added a page at the end of section 3. Other minor correction

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

“…Now we will prove that b(X, P) is an isomorphism when X is a finite G-CW-complex using a similar argument to the one used for untwisted G-equivariant K -theory [5]. …”

“…Proof If U is a G-cell, then G U is weakly equivalent to G M for some compact Lie group G and a finite G-CW-complex M. In section 3 of [5] it is proved that…”

“…Definition 5. 5 The extendable homotopy classes of sections of this bundle will be denoted by P K −n G (X ). The groups P K −n G (X ) are functorially associated to the pair (X, P) and so an isomorphism P → P of G-stable projective bundles on X induces an isomorphism P K −n G (X ) → P K −n G (X ) for all n ≥ 0.…”

“…Complex equivariant K -theory for actions for Lie groupoids was defined in [5] by using a special class of vector bundles, called extendable vector bundles. This defines a cohomology theory on the category of G-spaces.…”

“…In Sect. 3 we list some definitions and important results in equivariant K -theory for groupoid actions from [5]. We reproduce these two sections here for the sake of completeness, but they can skipped by a reader who is familiar with the results in that paper.…”