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 π2(M) is not the trivial group. Then there exists a non-zero element of π2(M) having a representative that is an embedding 
.
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 π1(M)-invariant subgroup of π2(M). If 
is a general position map such that ![[f]\notin N](../../../math/4/6/4/464d68d51a4ca9db0a18d04549f99ed0.png)
and U is any neighborhood of the singular set Σ(f), then there is a map 
satisfying
![[g]\notin N](../../../math/4/d/7/4d762ecd1d92c44c441ab01a84f98a6b.png)
,
,
is a covering map, and- g(S2) 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.
- Epstein, D. B. A. (1961). "Projective planes in 3-manifolds". Proceedings of the London Mathematical Society, III Ser. 11: 469–484.
- Hempel, J. (1978). 3-manifolds. Princeton University Press.
- C. Papakyriakopoulos (1957). "On Dehn's lemma and asphericity of knots". Annals of Mathematics 66: 1–26.
- Whitehead, J. H. C. (1958). "On 2-spheres in 3-manifolds". Bulletin of the American Mathematical Society 64: 161–166.
