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Hopf–Whitney Theorem
In mathematics, especially algebraic topology and homotopy theory, the Hopf–Whitney theorem is a result relating the homotopy classes between a CW complex and a multiply connected space with singular cohomology classes of the former with coefficients in the first nontrivial homotopy group of the latter. It can for example be used to calculate Cohomotopy set, cohomotopy as Sphere, spheres are multiply connected. Statement For a n-dimensional CW complex X and a n-1-connected space Y, the well-defined map: : [X,Y]\rightarrow H^n(X,\pi_n(Y)), [f]\mapsto f^*\iota with a certain cohomology class \iota\in H^n(Y,\pi_n(Y)) is an isomorphism. The Hurewicz theorem claims that the well-defined map \pi_n(Y)\rightarrow H_n(Y,\mathbb),[f]\mapsto f_*[S^n] with a fundamental class [S^n]\in H_n(S^n,\mathbb)\cong\mathbb is an isomorphism and that H_(Y,\mathbb)\cong 1, which implies \operatorname_\mathbb^1(H_(Y,\mathbb),\pi_n(Y))\cong 1 for the Ext functor. The Universal coefficient theorem th ... [...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] |
