In topology, a branch of mathematics, an aspherical space is a topological space with all higher homotopy groups equal to–{0}. 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.

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