# 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.

## Keywords

Exact Sequence Vector Bundle Line Bundle Open Covering Heisenberg Group## Preview

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