Homotopy theory | Homology theory | Algebraic topology

Aspherical space

In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups equal to 0 when . 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 is any covering map, then E is aspherical if and only if B is aspherical.) Each aspherical space X is, by definition, an Eilenberg–MacLane space of type , where is the fundamental group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with the discrete topology). (Wikipedia).

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Related pages

Topological space | Nilmanifold | Topology | Almost complex manifold | Lattice (discrete subgroup) | Stokes' theorem | Topological group | CW complex | Knot (mathematics) | Acyclic space | Cartan–Hadamard theorem | Classifying space | Eilenberg–MacLane space | Hyperbolic 3-manifold | Discrete valuation | Connected space | Whitehead conjecture | Riemannian manifold | Fundamental group | Algebraic group | Covering space | Chern class | Symplectic manifold | Projective plane | Sphere theorem (3-manifolds) | Surface (topology) | Aleksandr Danilovich Aleksandrov | Circle | Product topology | Essential manifold