In differential topology, the **Whitney immersion theorem** (named after Hassler Whitney) states that for , any smooth -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean -space, and a (not necessarily one-to-one) immersion in -space. Similarly, 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, dimension counting): two *m*-dimensional manifolds in intersect generically in a 0-dimensional space.

## Further results

William S. Massey (Massey 1960) 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**. This conjecture was eventually solved in the affirmative by Ralph Cohen (Cohen 1985).

## See also

## References

- Cohen, Ralph L. (1985). "The immersion conjecture for differentiable manifolds".
*Annals of Mathematics*.**122**(2): 237–328. doi:10.2307/1971304. JSTOR 1971304. MR 0808220. - Massey, William S. (1960). "On the Stiefel-Whitney classes of a manifold".
*American Journal of Mathematics*.**82**(1): 92–102. doi:10.2307/2372878. JSTOR 2372878. MR 0111053.

## External links

- Giansiracusa, Jeffrey (2003).
*Stiefel-Whitney Characteristic Classes and the Immersion Conjecture*(PDF) (Thesis). (Exposition of Cohen's work)