Bundle gerbes. (English) Zbl 0867.55019

Murray, M. K.

J. Lond. Math. Soc., II. Ser. 54, No. 2, 403-416 (1996).

Summary: Just as \(\mathbb{C}^\times\) principal bundles provide a geometric realisation of two-dimensional integral cohomology, gerbes or sheaves of groupoids, provide a geometric realisation of three-dimensional integral cohomology through their Dixmier-Douady class. An alternative related geometric realisation of three-dimensional cohomology is considered. It is called a bundle gerbe. Every bundle gerbe gives rise to a gerbe and most of the well-known examples of gerbes are bundle gerbes. The properties of bundle gerbes are discussed, in particular bundle gerbe connections and curvature and their associated Dixmier-Douady class.

MSC:

55R65

Generalizations of fiber spaces and bundles in algebraic topology

Keywords:

gerbes; sheaves of groupoids; integral cohomology