Aspherical space
In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups πn(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.)
Each aspherical space X is, by definition, an Eilenberg-MacLane space K(G,1), where G = π1(X) is the fundamental group of X.
Also directly from the definition, an aspherical space is the classifying space of its fundamental group (considered to be a topological group when endowed with the discrete topology).
In the context of symplectic manifolds, the meaning of "aspherical" is a little bit different. Specifically, we say that a symplectic manifold (M,ω) is symplectically aspherical if and only if
for every continuous mapping
where denotes the first Chern class of an almost complex structure which is compatible with ω.
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