Sphere theorem (3-manifolds)

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.

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 Jean-Loïc Batude (1971).

Another more general version (also called the projective plane theorem, and due to David B.

A. Epstein) is: Let

-invariant subgroup of

is a general position map such that

is any neighborhood of the singular set

satisfying quoted in (Hempel 1976, p. 54).