The Case of Manifolds

Abstract

In this chapter we review briefly some of the fundamental results of the classical theory of indices of vector fields and characteristic classes of smooth manifolds. These were first defined in terms of obstructions to the construction of vector fields and frames. In the case of a vector field the Poincarée – Hopf Theorem says that Euler.Poincarée characteristic is the obstruction to constructing a nonzero vector field tangent to a compact manifold. Extension of this result to frames yields to the definition of Chern classes from the viewpoint of obstruction theory. There is another important point of view for defining characteristic classes on the differential geometry side, this is the Chern – Weil theory. Sections 3 and 4 provide an introduction to that theory and the corresponding definition of Chern classes. Finally, Sect. 5 sets up one of the key features of this monograph: the interplay between localization via obstruction theory, which yields to the classical relative characteristic classes, and localization via Chern – Weil theory, which yields to the theory of residues. This is one way of thinking of the Poincarée – Hopf Theorem and its generalizations. Throughout the book, M will denote either a complex manifold of (complex) dimension m, or a C⋡ manifold of (real) dimension m’.