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In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups π''n''(''X'') equal to 0 when ''n''>1. If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence of a fibration that higher homotopy groups of a space and its universal cover are same. (By the same argument, if ''E'' is a path-connected space and ''p'': ''E'' → ''B'' is any covering map, then ''E'' is aspherical if and only if ''B'' is aspherical.) Aspherical spaces are, directly from the definitions, Eilenberg-MacLane spaces. Also directly from the definitions, aspherical spaces are classifying spaces of their fundamental groups. ==Examples== * Using the second of above definitions we easily see that all orientable compact surfaces of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover). * It follows that all non-orientable surfaces, except the real projective plane, are aspherical as well, as they can be covered by an orientable surface genus 1 or higher. * Similarly, a product of any number of circles is aspherical. As is any complete, Riemannian flat manifold. * Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H3, hence aspherical. As is any n-manifold whose universal covering space is hyperbolic n-space Hn. * Let ''X'' = ''G''/''K'' be a Riemannian symmetric space of negative type, and Γ be a lattice in ''G'' that acts freely on ''X''. Then the locally symmetric space is aspherical. * The Bruhat-Tits building of a simple algebraic group over a field with a discrete valuation is aspherical. * The complement of a knot in S3 is aspherical, by the sphere theorem * Metric spaces with nonpositive curvature in the sense of Aleksandrov (locally CAT(0) spaces) are aspherical. In the case of Riemannian manifolds, this follows from the Cartan–Hadamard theorem, which has been generalized to geodesic metric spaces by Gromov and Ballmann. This class of aspherical spaces subsumes all the previously given examples. * Any nilmanifold is aspherical. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Aspherical space」の詳細全文を読む スポンサード リンク
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