Circle Actions and Z/k-manifolds

We establish a S1-equivariant index theorem for Dirac operators on Z/kmanifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S1-actions on closed spin manifolds to the case of Z/k-manifolds. Résumé français On établit un théorème d’indice S-équivariant pour les opérateurs de Dirac sur des Z/k variétés. On donne une application de ce résultat, qui généralise le théorème d’Atiyah-Hirzebruch sur les actions de S aux Z/k variétés. Titre français Actions du cercle et Z/k variétés. §1. S-actions and the vanishing theorem Let X be a closed connected smooth spin manifold admitting a non-trivial circle action. A classical theorem of Atiyah and Hirzebruch [AH] states that Â(X) = 0, where Â(X) is the Hirzebruch Â-genus of X. In this Note we present an extension of the above result to the case of Z/k-manifolds, which were introduced by Sullivan in his studies of geometric topology. We recall the basic definition for completeness (cf. [F]). Definition 1.1 A compact connected Z/k-manifold is a compact manifold X with boundary ∂X, which admits a decomposition ∂X = ∪i=1(∂X)i into k disjoint manifolds and k diffeomorphism πi : (∂X)i → Y to a closed manifold Y . ∗Partially supported by the MOEC and the 973 project.