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 ${\displaystyle n\not =1}$.

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).