About the transverse fixed point formula for foliations (Q2743896)

The purpose of this article is to generalize the Atiyah-Segal localization theorem to the space of leaves. Let \((V,{\mathcal F})\) be a compact foliated manifold. Let \(H\) be a topologically cyclic compact Lie group of \({\mathcal F}\)-preserving diffeomorphisms of \(V.\) Denote by \(C^*(V,{\mathcal F})\) the maximal \(C^*\)-algebra associated with the foliation. The main theorem of the paper states that the localized \(H\)-equivariant K-theory of \(C^*(V,{\mathcal F})\) with respect to the ideal associated with any generator \(f\) of \(H\) only depends on the reduced holonomy groupoid of the globally \(H\)-invariant leaves.