Degree 3 Cohomology: The Dixmier-Douady Theory

Abstract

Much of this book is devoted to a geometric description of the degree 3 cohomology H ³(M,ℝ) of a manifold M. Recall in the case of degree 2 cohomology, a theorem of Weil and Kostant (Corollary 2.1.4) which asserts that H ²(M,ℝ) is the group of isomorphism classes of line bundles over M. Moreover, given a line bundle L, the corresponding class c ₁(L) in H ² (M,ℝ) is represented by $\frac{1}{{2\pi \sqrt { - 1} }}$ · K, where K is the curvature of a connection on L. We wish to find a similar description for H ³(M,ℝ), which is a more difficult task. One theory, due to Dixmier and Douady [D-D], involves so-called continuous fields of elementary C*-algebras. We will describe this theory in the present chapter and develop, in particular, the notion of curvature, which will be a closed 3-form associated to such a field of C*-algebras.