Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classicallv in the Gauss-Bonnet theorem and its generalizations. which culminate in the Ativah-Singer Index Theorem [ASI] which places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topologie al properties of the manifold. The Ativah-Singer Index Theorem has been generalized in several directions. notably by Atiyah-Singer to an index theorem for families [AS4]. The typical setting here is given by a family of elliptic operators (Pb) on the total space of a fibre bundle P = F_M_B. where is defined the Hilbert space on Pb 2 L 1p -llbl.dvollFll. In this case there is an abstract index class indlPI E ROIBI. Once the problem is properly formulated it turns out that no further deep analvtic information is needed in order to identify the class. These theorems and their equivariant counterparts have been enormously useful in topology. geometry. physics. and in representation theory.

Klappentext

This book develops a variety of aspects of analysis and geometry on foliated spaces which should be useful in many situations. These strands are then brought together to provide a context and to expose Connes` index theorem for foliated spaces, a theorem which asserts the equality of the analytic and the topological index (two real numbers) which are associated to a tangentially elliptic operator. An additional purpose of this exposition is preparing the way towards the more general index theorem of Connes and Skandalis. This index theorem describes the abstract index class in KO (CR*(G(M))), the index group of the C*-algebra of the foliated space, and is necessarily substantially more abstract, while the tools used here are relatively elementary and straightforward, and are based on the heat equation method.

Inhalt
I. Locally Traceable Operators.- II. Foliated Spaces.- III. Tangential Cohomology.- IV. Transverse Measures.- V. Characteristic Classes.- VI. Operator Algebras.- VII. Pseudodifferential Operators.- VIII. The Index Theorem.- Appendices.- C: Positive Scalar Curvature Along the Leaves.- References.