Sphere theorem (3-manifolds)

Short description: On when elements of the 2nd homotopy group of a 3-manifold can be embedded spheres

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 [math]\displaystyle{ M }[/math] be an orientable 3-manifold such that [math]\displaystyle{ \pi_2(M) }[/math] is not the trivial group. Then there exists a non-zero element of [math]\displaystyle{ \pi_2(M) }[/math] having a representative that is an embedding [math]\displaystyle{ S^2\to M }[/math].

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 [math]\displaystyle{ M }[/math] be any 3-manifold and [math]\displaystyle{ N }[/math] a [math]\displaystyle{ \pi_1(M) }[/math]-invariant subgroup of [math]\displaystyle{ \pi_2(M) }[/math]. If [math]\displaystyle{ f\colon S^2\to M }[/math] is a general position map such that [math]\displaystyle{ [f]\notin N }[/math] and [math]\displaystyle{ U }[/math] is any neighborhood of the singular set [math]\displaystyle{ \Sigma(f) }[/math], then there is a map [math]\displaystyle{ g\colon S^2\to M }[/math] satisfying

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

quoted in (Hempel 1976).

References

• Batude, Jean-Loïc (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable". Annales de l'Institut Fourier 21 (3): 151–172. doi:10.5802/aif.383. http://www.numdam.org/article/AIF_1971__21_3_155_0.pdf. 

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

• Hempel, John (1976). 3-manifolds. Annals of Mathematics Studies. 86. Princeton, NJ: Princeton University Press. 

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

• 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.