# Aspherical space

In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups ${\displaystyle \pi _{n}(X)}$ equal to 0 when ${\displaystyle n>1}$.

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 ${\displaystyle p\colon E\to B}$ is any covering map, then E is aspherical if and only if B is aspherical.)

Each aspherical space X is, by definition, an Eilenberg–MacLane space of type ${\displaystyle K(G,1)}$, where ${\displaystyle G=\pi _{1}(X)}$ is the fundamental group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with the discrete topology).

## Symplectically aspherical manifolds

In the context of 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

${\displaystyle \int _{S^{2}}f^{*}\omega =\langle c_{1}(TM),f_{*}[S^{2}]\rangle =0}$

for every continuous mapping

${\displaystyle f\colon S^{2}\to M,}$

where ${\displaystyle c_{1}(TM)}$  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."