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Classifying Space For O
In mathematics, the classifying space for the orthogonal group O(''n'') may be constructed as the Grassmannian of ''n''-planes in an infinite-dimensional real space \mathbb^\infty. Cohomology ring The cohomology ring of \operatorname(n) with coefficients in the Field (mathematics), field \mathbb_2 of GF(2), two elements is generated by the Stiefel–Whitney class, Stiefel–Whitney classes:Hatcher 02, Theorem 4D.4. : H^*(\operatorname(n);\mathbb_2) =\mathbb_2[w_1,\ldots,w_n]. Infinite classifying space The canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) induce canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) on their respective classifying spaces. Their respective colimits are denoted as: : \operatorname :=\lim_\operatorname(n); : \operatorname :=\lim_\operatorname(n). \operatorname is indeed the classifying space of \operatorname. See also * classifying space for U(n), Classifying space for U(''n'') * Classifying space for SO( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
