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 formula_1 be an orientable 3-manifold such that formula_2 is not the trivial group. Then there exists a non-zero element of formula_2 having a representative that is an embedding formula_4. 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 formula_1 be any 3-manifold and formula_6 a formula_7-invariant subgroup of formula_2. If formula_9 is a general position map such that formula_10 and formula_11 is any neighborhood of the singular set formula_12, then there is a map formula_13 satisfying quoted in .