Sphere theorem for three-manifolds
= Sphere theorem for three-manifolds
If an orientable <three-manifold> has nonzero $\pi_2$, it contains an embedded essential two-sphere. Consequently a closed orientable <irreducible three-manifold> with infinite fundamental group is <aspherical>: its simply connected noncompact universal cover has zero second homotopy and top ordinary homology, hence is acyclic and contractible by the <Hurewicz theorem> and <Whitehead theorem>.