Foliations II

This is the second volume of the two-volume series with the title Foliations. It has three independent parts, describing three special topics in the theory of foliations: Analysis on foliated spaces, Characteristic classes of foliations and Foliated 3-manifolds. Each part contains a description of a topic in foliation theory and its relation to another field of contemporary mathematics. In the first part, the C*- algebras of foliated spaces are studied and some of the classical notions from Riemannian geometry (heat flow and Brownian motion) are generalized to foliated spaces. Necessary analytic background can be found in three appendices. The second part is devoted to characteristic classes and foliations. Here the reader can find constructions of exotic classes based on the Chern-Weil theorem, vanishing theorem for Godbillon-Vey classes and a discussion on obstructions to existence of a foliation transverse to the fibres of circle bundles over surfaces. In the third part, compact 3-manifolds foliated by surfaces are studied. Special methods of 3-manifolds topology yield existence theorems and further results unique for dimension three. There is an appendix with a proof and further discussion of Palmeiras theorem, which says that the only simply connected n-manifold foliated by leaves diffeomorphic to Rn-1 is Rn. The book contains a lot of interesting results and can be recommended to anybody interested in the topic.

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