In differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, the Whitney immersion theorem states that for , any smooth -dimensional manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 can be immersed in Euclidean

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 -space. Equivalently, every smooth -dimensional manifold can be immersed in the -dimensional sphere (this removes the constraint).

The weak version, for , is due to transversality (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...

, dimension counting): two m-dimensional manifolds in intersect generically in a 0-dimensional space.

Further Results

Massey went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in where is the number of 1's that appear in the binary expansion of . In the same paper, Massey proved that for every n there is manifold (which happens to be a product of real projective spaces) that does not immerse in . The conjecture that every n-manifold immerses in became known as the Immersion Conjecture which was eventually solved in the affirmative by Ralph Cohen .

External links

• Stiefel-Whitney Characteristic Classes and the Immersion Conjecture, by Jeffrey Giansiracusa, 2003

  Exposition of Cohen's work