In mathematics, in the topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 of 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

s, 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

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 .

The proof of this version can be based on transversality

Transversality theorem

In differential topology, the transversality theorem, also known as the Thom Transversality Theorem, is a major result that describes the transversal intersection properties of a smooth family of smooth maps...

 methods, see Batude below.

Another more general version (also called the projective plane theorem due to Epstein) is:

Let be any 3-manifold and a -invariant

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 subgroup of . If is a general position

General position

In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...

 map such that and is any neighborhood of the singular set , then there is a map satisfying

1. ,
2. ,
3. is a covering map
   Covering map
   In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
   
   , and
4. is a 2-sided
   2-sided
   In topology, a compact codimension one submanifold F of a manifold M is said to be 2-sided in M when there is an embeddingwith h=x for each x\in F andIn other words, if its normal bundle is trivial....
   
    submanifold (2-sphere or projective plane
   Projective plane
   In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
   
   ) of .

quoted in Hempel (p. 54)