Global Analysis on Foliated Spaces

Foliated spaces look locally like products but their global structure is generally not a product and tangential differential operators are correspondingly more complex. In the 1980s Alain Connes founded what is now known as noncommutative geometry and topology. One of the first results was his generalization of the Atiyah-Singer index theorem to compute the analytic index associated with a tangential (pseudo) - differential operator and an invariant transverse measure on a foliated manifold in terms of topological data on the manifold and the operator. This second edition presents a complete proof of this beautiful result generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis geometry and topology. The present edition has improved exposition an updated bibliography an index and additional material covering developments and applications since the first edition came out including the confirmation of the Gap Labeling Conjecture of Jean Bellissard.