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.)
- Using the second of above definitions we easily see that all orientable compact surfaces of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover).
- It follows that all non-orientable surfaces, except the real projective plane, are aspherical as well, as they can be covered by an orientable surface genus 1 or higher.
- Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H3, hence aspherical.
- Let X = G/K be a Riemannian symmetric space of negative type, and Γ be a lattice[disambiguation needed] in G that acts freely on X. Then the locally symmetric space is aspherical.
- The Bruhat-Tits building of a simple algebraic group over a field with a discrete valuation is aspherical.
- Metric spaces with nonpositive curvature in the sense of Aleksandrov (locally CAT(0) spaces) are aspherical. In the case of Riemannian manifolds, this follows from the Cartan–Hadamard theorem, which has been generalized to geodesic metric spaces by Gromov and Ballmann. This class of aspherical spaces subsumes all the previously given examples.
- Any nilmanifold is aspherical.
Symplectically 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
By Stokes' theorem, we see that symplectic manifolds which are aspherical are also symplectically aspherical manifolds. However, there do exist symplectically aspherical manifolds which are not aspherical spaces.
Some references 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."
- Robert E. Gompf, Symplectically aspherical manifolds with nontrivial π2, Math. Res. Lett. 5 (1998), no. 5, 599–603. MR 1666848
- Jarek Kedra, Yuli Rudyak, and Aleksey Tralle, Symplectically aspherical manifolds, J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21. MR 2402905
- Bridson, Martin R.; Haefliger, André, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN 3-540-64324-9 MR 1744486
- Aspherical manifolds on the Manifold Atlas.