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.
The proof of this version of the theorem can be based on transversality methods, see.
Another more general version 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 of.
quoted in.