A path-connected space is aspherical when its higher homotopy groups vanish, equivalently when its universal cover is contractible. An aspherical space with fundamental group is a , so based homotopy classes of maps into it are controlled by homomorphisms of fundamental groups.
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Aspherical space is a term used in topology, a branch of mathematics that studies the properties of space that are preserved under continuous transformations. Specifically, an aspherical space is a manifold (or more generally, a topological space) whose universal covering space is contractible. This means that the universal cover does not have any "holes"; it can be continuously shrunk to a point without leaving the space.