Degree 3 Cohomology: The Dixmier-Douady Theory

  • Jean-Luc Brylinski
Part of the Progress in Mathematics book series (MBC)


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.


Exact Sequence Vector Bundle Line Bundle Open Covering Heisenberg Group 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Jean-Luc Brylinski
    • 1
  1. 1.Department of MathematicsThe Penn State UniversityUniversity ParkUSA

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