Aspherical space
= Aspherical space
{wiki}
= Aspherical
{synonym}
A path-connected <topological space> $X$ is <aspherical> if $\pi_n(X)=0$ for every $n\geq2$. For a connected <CW complex>, this is equivalent to its <universal cover> being a <contractible space>. Indeed, a <covering map> preserves all higher <homotopy groups>, while the <universal cover> is <simply connected>; the <Whitehead theorem> then makes its map to a <point> a <homotopy equivalence>.
The <circle> is aspherical because its <universal cover> is the <real line>. The <torus> is aspherical because its <universal cover> is $\mathbb R^2$. Their only nonzero positive-degree <homotopy groups> are their <fundamental groups>, $\mathbb Z$ and $\mathbb Z^2$. In contrast, $S^2$ is not aspherical since $\pi_2(S^2)\cong\mathbb Z$.
For an aspherical <CW complex> $Y$ with <fundamental group> $G$, maps from a connected based <CW complex> $X$ are classified by
$$
[X,Y]_*\cong\operatorname{Hom}(\pi_1X,G).
$$
Extending a map over successive higher <cells> encounters precisely the vanishing higher <homotopy groups> of $Y$, giving both existence and uniqueness up to based <homotopy>. For free <homotopy classes>, the <homomorphisms> are instead identified up to conjugation in $G$.<Algebraic Topology by Allen Hatcher>[§1.B, Proposition1B.9 and its proof, printed pages90–91]{ref}