Whitney immersion theorem
In differential topology, the Whitney immersion theorem 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).
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 (Cohen 1985).
- Cohen, Ralph L. (1985), "The Immersion Conjecture for Differentiable Manifolds", The Annals of Mathematics, Annals of Mathematics, 122 (2): 237–328, doi:10.2307/1971304, JSTOR 1971304
- Stiefel-Whitney Characteristic Classes and the Immersion Conjecture, by Jeffrey Giansiracusa, 2003
- Exposition of Cohen's work
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