sphere theorem

Theorem 1 (sphere theorem).

If $M$ is a differentiable orientable 3-manifold such that ${\pi }_{\mathrm{2}}\mathit{}\mathrm{(}M\mathrm{)}$ is not trivial, then there exists an embedding $S^{\mathrm{2}}\mathrm{\to }M$ such that its image homotopy class is not equal to zero.

This theorem was established and proved by C. Papakyriakopoulos in 1957.