Aspherical space

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In topology, aspherical space is a topological space with all higher homotopy groups equal to {0}.

If one works with CW complexes, one can reformulate this condition: 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.

[edit] Examples

Using the second of above definitions we easily see that all orientable compact surfaces of genus greater than 0 are aspherical (as they have R² as a universal cover).

Similarly, product of any number of circles is aspherical.

[edit] Symplectic aspherical manifolds

If one deals with symplectic manifolds, "aspherical" has a little bit different meaning. Namely, we say that a symplectic manifold (M,ω) is symplectically aspherical if and only if

for every continuous mapping

f: S² → M.

By Stokes' theorem, we see that symplectic manifolds which are aspherical with respect to "aspherical space" definition are also symplectically aspherical manifolds. However, there do exist symplectically aspherical manifold which are not aspherical spaces ^{[citation needed]}.

In everyday life one often uses term "symplectic aspherical manifold" instead of "symplectically aspherical manifold".