Aspherical space

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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. (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.)

Aspherical spaces are (directly from the definitions) Eilenberg-MacLane spaces.

1 Examples
2 Symplectically aspherical manifolds
3 See also
4 References

[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).

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.

Similarly, a product of any number of circles is aspherical.

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, and Γ be a lattice in G that acts freely on X. Then the locally symmetric space $\Gamma\backslash G/K$ 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.

[edit] 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

$\int_{S^2}f^*\omega=0$

for every continuous mapping

$f\colon S^2 \to M.$

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.^[1]

[edit] See also

Acyclic space

[edit] References

^ Robert E. Gompf, Symplectically aspherical manifolds with nontrivial π₂, Math. Res. Lett. 5 (1998), no. 5, 599–603. MR 1666848

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