Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.

One example is the following:

Let ${\displaystyle M}$ be an orientable 3-manifold such that ${\displaystyle \pi _{2}(M)}$ is not the trivial group. Then there exists a non-zero element of ${\displaystyle \pi _{2}(M)}$ having a representative that is an embedding ${\displaystyle S^{2}\to M}$.

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