Laurent GUILLAUME

Abstract: This paper is concerned with pseudodifferential calculus on mani\-folds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This groupoid has a natural geometric meaning as a holonomy groupoid of singular foliation, it is a singular leaf space in the sense of Androulidakis and Skandalis. We then show that the associated compactly supported pseudo-differential calculus coincides with Mazzeo and Melrose's phi-calculus and we introduce an extended algebra of smoothing operators that is shown to be stable under holomorphic functional calculus. This result allows the interpretation of phi-calculus as the pseudo-differential calculus associated with the holonomy groupoid of the singular foliation defined by the manifold with fibred corners. It is a key step to set the index theory of those singular manifolds in the noncommutative geometry framework.

Abstract: Tools from noncommutative geometry are used to study the index theory of pseudo-stratified manifolds. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with foliated boundary, then to every manifold with fibred corners a longitudinally smooth groupoid. We then show in the fibred case that the associated compactly supported pseudodifferential calculus coincides with Melrose’s phi-calculus and we introduce an extended algebra of smoothing operators that is shown to be stable under holomorphic functional calculus. Some elements of relative cyclic cohomology arising in higher index problems are defined over this extended algebra. Finally we show that the groupoid we built has a natural geometric meaning as a holonomy groupoid of singular foliation, it is an explicit example of a singular leaf space in the sense of Androulidakis and Skandalis. This result allows the conceptual interpretation of phi-calculus as the pseudodifferential calculus associated with the holonomy groupoid of the singular foliation defined by the manifold with fibred boundary.