Foliation groupoids and their cyclic homology (Q5929055)

A groupoid \(G\) is a small category in which every arrow is invertible. The tangent spaces of \(s^{-1}(x)\) at \(1_x\), \(s\) being the source map relative to \(G\), constitute a bundle \({\mathfrak g}\) over the set of objects \(G_0\) of \(G\) and the differential of the target map relative to \(G\) induces a map of vector bundles \( \alpha: {\mathfrak g} \rightarrow TG_0\) called the anchor map of \(G\). NEWLINENEWLINENEWLINEThe paper is devoted to prove two results. First, one gives three equivalent statements for a smooth groupoid \(G\) which are: (1) \(G\) is Morita equivalent to a smooth étale groupoid. (2) The Lie algebroid \({\mathfrak g}\) of \(G\) has an injective anchor map. (3) All isotropy Lie groups of \(G\) are discrete. NEWLINENEWLINENEWLINENotice that since Lie algebroids with injective anchor map are exactly foliations, the above result asserts that a Lie groupoid is equivalent to an étale one exactly when it integrates a foliation. For this reason, the authors call these groupoids foliation groupoids. NEWLINENEWLINENEWLINEFinally, the second result of the paper states that equivalent foliation groupoids have isomorphic Hochschild, cyclic and periodic cyclic homology groups. The proof provides explicit isomorphisms.