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Bundle Gerbes
In mathematics, a bundle gerbe is a geometry, geometrical model of certain 1-gerbes with connection (mathematics), connection, or equivalently of a 2-class in Deligne cohomology. Topology U(1)-principal bundles over a space M (see circle bundle) are geometrical realizations of 1-classes in Deligne cohomology which consist of 1-form connection (mathematics), connections and 2-form curvatures. The topology of a U(1) bundle is classified by its Chern class, which is an element of H^2(M, \mathbb), the second integral cohomology of M. Gerbes, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of H^3(M, \mathbb), the third integral cohomology of ''M''. As a cohomology class in Deligne cohomology Recall for a smooth manifold M the p-th Deligne cohomology groups are defined by the Hyperhomology, hypercohomology of the complex \mathbb(q)_D^\infty = \underline(q) \to \mathcal_^0 \xrightarrow \mathcal_^1 \xrightarrow \cdots \xrightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
