Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos 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 $M$ be an orientable 3-manifold such that $\pi _{2}(M)$ is not the trivial group. Then there exists a non-zero element of $\pi _{2}(M)$ having a representative that is an embedding $S^{2}\to M$ .

The proof of this version of the theorem can be based on transversality methods, see Jean-Loïc Batude (1971).

Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is:

Let $M$ be any 3-manifold and $N$ a $\pi _{1}(M)$ -invariant subgroup of $\pi _{2}(M)$ . If $f\colon S^{2}\to M$ is a general position map such that $[f]\notin N$ and $U$ is any neighborhood of the singular set $\Sigma (f)$ , then there is a map $g\colon S^{2}\to M$ satisfying