Aspherical space

In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups π_n(X) equal to 0 when n>1.

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. Also directly from the definitions, aspherical spaces are classifying spaces of their fundamental groups.

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 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.

Similarly, a product of any number of circles is aspherical. As is any complete, Riemannian flat manifold.

Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H³, hence aspherical. As is any n-manifold whose universal covering space is hyperbolic n-space Hⁿ.

Let X = G/K be a Riemannian symmetric space of negative type, 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.

The complement of a knot in S³ is aspherical, by the sphere theorem

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

\int_{S^2}f^*\omega=\langle c_1(TM),f_*[S^2]\rangle=0

for every continuous mapping

f\colon S^2 \to M,

where $c_1(TM)$ denotes the first Chern class of an almost complex structure which is compatible with ω.

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]

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."

Notes

↑ 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

References

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

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