In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups
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.
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).
In the context of symplectic manifolds, the meaning of "aspherical" is a little bit different.
denotes the first Chern class of an almost complex structure which is compatible with ω.
[1] Some references[2] drop the requirement on c1 in their definition of "symplectically aspherical."
However, it is more common for symplectic manifolds satisfying only this weaker condition to be called "weakly exact."