Aspherical space
From Wikipedia, the free encyclopedia
In topology, an 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: 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.
Aspherical spaces are (directly from the definitions) Eilenberg-MacLane spaces.
[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, 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
- 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".