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 $M$ 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 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 $\pi_1(M)$-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

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

quoted in Hempel (p. 54)

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.

• Epstein, D. B. A. (1961). "Projective planes in 3-manifolds". Proceedings of the London Mathematical Society, III Ser. 11 (1): 469–484. doi:10.1112/plms/s3-11.1.469.

• Hempel, J. (1978). 3-manifolds. Princeton University Press.

• C. Papakyriakopoulos (1957). "On Dehn's lemma and asphericity of knots". Annals of Mathematics (Annals of Mathematics) 66 (1): 1–26. doi:10.2307/1970113. JSTOR 1970113.

• Whitehead, J. H. C. (1958). "On 2-spheres in 3-manifolds". Bulletin of the American Mathematical Society 64 (4): 161–166. doi:10.1090/S0002-9904-1958-10193-7.