Immersion (mathematics) - Classification

Classification

Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n+1 every map f : Mm → Nn of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem.

Stephen Smale expressed the regular homotopy classes of immersions f : Mm → Rn as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence.

Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any m-dimensional manifold Mm in any n-dimensional manifold Nn.

The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.