Equivariant K-theory, groupoids and proper actions (Q2905334)

\textit{W. Lück} and \textit{B. Oliver} [Topology 40, No. 3, 585--616 (2001; Zbl 0981.55002)] constructed a complex equivariant \(K\)-theory for proper actions of a discrete group and showed that an analogue of the Atiyah-Segal completion theorem holds for this theory. In this paper the author defines a version of complex equivariant \(K\)-theory \(K_{\mathcal{G}}^*(-)\) for actions of a Lie groupoid \(\mathcal{G}=(\mathcal{G}_0, \mathcal{G}_1)\).NEWLINENEWLINEA \(\mathcal{G}\)-space is a manifold \(X\) equipped with maps \(\pi : X \to \mathcal{G}_0\) and \(\mu\) defined on pairs \((x, g)\) in \(X\times \mathcal{G}_1\) with \(\pi(x)=t(g)\) such that \(\mu(x, g)\) lies in \(X\) and \(\pi\mu(x, g)=s(g)\) holds, where \(s(g)\) and \(t(g)\) denote the source and target of \(g\). A \(\mathcal{G}\)-vector bundle is a vector bundle \(p : V \to X\) such that \(V\) is a \(\mathcal{G}\)-space with fiberwise linear action and \(p\) is a \(\mathcal{G}\)-equivariant map. Further, this \(p : V \to X\) is said to be \(\mathcal{G}\)-\textit{extendable} if there is a \(\mathcal{G}\)-vector bundle \(W \to \mathcal{G}_0\) such that \(V\) is a direct summand of \(\pi^*W\). Then \(K_{\mathcal{G}}(X)\) can be defined as the Grothendieck group of the monoid of these extendable \(\mathcal{G}\)-vector bundles over \(X\).NEWLINENEWLINEBy \textit{Bredon-compatible} it is meant that \(\mathcal{G}\)-vector bundles on \(U\) are extendable if given any \(\mathcal{G}\)-cell \(U\) (whose definition is omitted). The main result of this paper can be stated as follows: Assume that \(\mathcal{G}\) is Bredon-compatible. Then \(K_{\mathcal{G}}(-)\) defines a periodic cohomology theory on the category of fnite \(\mathcal{G}\)-\(CW\)-pairs, which admits a completion theorem of Atiyah and Segal when \({\mathcal{G}}\) is finite. Finally, given are some applications of this result to proper actions of a Lie group.