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 π 2 ( M ) is not the trivial group. Then there exists a non-zero element of π 2 ( M ) having a representative that is an embedding S 2 → 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 π 1 ( M ) -invariant subgroup of π 2 ( M ) . If f : S 2 → M is a general position map such that [ f ] ∉ N and U is any neighborhood of the singular set Σ ( f ) , then there is a map g : S 2 → M satisfying

1. [ g ] ∉ N ,
2. g ( S 2 ) ⊂ f ( S 2 ) ∪ U ,
3. g : S 2 → 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)