Sphere theorem (3-manifolds)

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In the topology of 3-manifolds, the sphere theorem denotes a family of statements which show us how the image of a 2-sphere, under a continuous map into a 3-manifold, may behave. One example is the following:

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

The proof of this version can be based on transversality methods, see Batude below.

Another more general version (also called the projective plane theorem due to Epstein) is:

Let $M$ be any 3-manifold and $N$ a $π 1 (M)$ -invariant subgroup of $π 2 (M)$ . 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 $Σ(f)$ , then there is a map $g\colon S^2\to M$ satisfying

$[g]\notin N$ ,
$g(S^2)\subset f(S^2)\cup U$ ,
$g\colon S^2\to g(S^2)$ is a covering map, and
$g (S 2)$ is a 2-sided submanifold (2-sphere or projective plane) of $M$ .

quoted in Hempel (p. 54)

[edit] References

Batude, J. L. (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable". Annales de l'Institut Fourier 21 (3): 151–172.