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