A path-connected topological space is aspherical if for every . For a connected CW complex, this is equivalent to its universal cover being a contractible space. Indeed, a covering map preserves all higher homotopy groups, while the universal cover is simply connected; the Whitehead theorem then makes its map to a point a homotopy equivalence.
The circle is aspherical because its universal cover is the real line. The torus is aspherical because its universal cover is . Their only nonzero positive-degree homotopy groups are their fundamental groups, and . In contrast, is not aspherical since .
For an aspherical CW complex with fundamental group , maps from a connected based CW complex are classified byExtending a map over successive higher cells encounters precisely the vanishing higher homotopy groups of , giving both existence and uniqueness up to based homotopy. For free homotopy classes, the homomorphisms are instead identified up to conjugation in .[ref]
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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.