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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] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
