Geometry of Characteristic Classes (Translations of Mathematical Monographs) by S. Morita (PDF)

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Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional. The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichmüller theory. In this book Morita presents an introduction to the modern theories of characteristic classes.

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⭐The theory of characteristic classes goes back to the 1930’s, and is a subject that is one of the most poorly motivated from a didactic point of view. The literature on the subject is vast, but there are few works that explain the theory in a way that is transparent and helpful to those who need to apply it, such as physicists in the areas of high energy physics and quantum field theory. Individuals who really desire an in-depth real understanding of characteristic classes, rather than a purely formal one, will find the road difficult. There does not seem to be anywhere in the literature a treatment of characteristic classes that gives genuine insight on why these objects work as well as they do for the jobs they are supposed to do. This book is no exception to this, but it does serve to introduce the reader to the theory of characteristic classes as it has been developed in the last few decades. The first chapter is an introduction to de Rham homotopy theory, and its purpose is to outline the approach of using differential forms rather than cohomology groups to study the homotopy type of a smooth manifold. After a brief overview of Postnikov decompositions, the author discusses the rational homotopy type of a topological space. This is a strategy that is based on ignoring torsion and viewing the homotopy group as being over the rational numbers. This makes computation of the homotopy type much easier. The rational spaces have fundamental groups that are Lie groups over the rationals and each higher homotopy group is a vector space over the rationals. This theory is basically due to D. Sullivan and his main results are proved in this chapter. The interesting part of this discussion is the construction of de Rham complexes over simplicial complexes instead of smooth manifolds. The author shows in detail how the rational cohomology can be computed for these types of objects. He then discusses the cohomology of groups in terms of Eilenberg-Maclane spaces, and defines what is called the Malcev completion of a finitely generated group. This is basically a rational nilpotent completion of the group, and is used in the chapter to give the connection between fundamental groups and differential forms. In chapter 2, the author discusses characteristic classes over flat bundles. The author shows very effectively how the fundamental group of the base space of a flat bundle can introduce twist in the flat bundle. The Chern-Weil theory, which allows one to study the global structure of principal bundles with a given Lie group as a structure group, is reviewed in the first part of the chapter. The author also gives a very interesting characterization of flatness in terms of holonomy homomorphisms. Then after a brief overview of the cohomology of Lie algebras, he defines characteristic classes of flat product bundles. Since a given principal bundle may not be trivial, defining characteristic classes for a general flat principal bundle takes more work, which the author does nicely in what follows. The Chern-Simons theory, so important in physical applications, is then discussed as a specialization of the Chern-Weil theory. The author then returns to the consideration of characteristic classes of flat bundles via Gelfand-Fuks cohomology. This is the cohomology of the subcomplex of the continuous cochains of the smooth vector fields of a smooth manifold, and the author shows how it acts as a characteristic class for flat product bundles. The author moves on to the characteristic classes of foliations in chapter 3. This is a topic very important to those who study dynamical systems, and the author gives a pretty good overview of how characteristic classes can measure the global properties of foliations. He reviews briefly what foliations are, and then defines the Godbillon-Vey characteristic class for a foliation of a smooth manifold. The author gives an example of foliation for the Lie group PSL(2, R), and shows the existence of a non-trivial Godbillon-Vey class for a collection of torsion-free discrete subgroups of this group. The Bott vanishing theorem is proved, and then the author shows how to relate the Gelfand-Fuks cohomology to the characteristic classes of foliations via the Weil algebra. Then the author gives a fascinating discussion on how to find “discontinuous invariants” induced by a real cohomology class. The discussion is very interesting given the disparity between real and integer cohomology classes. He then returns to the consideration of the characteristic classes of flat bundles where now the Lie group is replaced by a given closed smooth manifold. In the last chapter of the book, the author discusses the characteristic classes of surface bundles, where the genus of the surface is greater than or equal to 2. String theorists will appreciate the discussion, as it goes into the mapping class group of surfaces, the Teichmuller modular group, and how they act on the homology group of surfaces. The author shows explicitly how to construct surface bundles with non-trivial characteristic classes, and gives a brief outline of the proof that these characteristic classes are algebraically independent. The author ends the book with a discussion of future research problems for the interested reader, along with a short list of references.

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