Whitney immersion theorem

In differential topology, the Whitney immersion theorem states that for, any smooth -dimensional manifold has a one-to-one immersion in Euclidean -space, and a immersion in -space. Similarly, every smooth -dimensional manifold can be immersed in the -dimensional sphere.
The weak version, for, is due to transversality : two m-dimensional manifolds in intersect generically in a 0-dimensional space.

Further results

William S. 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 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.