Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of 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 be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding . This statement may be strengthened to show that the embedding is piecewise linear .

The proof of this version of the theorem can be based on transversality methods, see .

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

Let be any 3-manifold and a -invariant subgroup of . If is a general position map such that and is any neighborhood of the singular set , then there is a map satisfying

1. ,
2. ,
3. is a covering map, and
4. is a 2-sided submanifold (2-sphere or projective plane) of .

quoted in .

References