In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, and a (not necessarily one-to-one) immersion in (2m-1)-space. Similarly, every smooth m-dimensional manifold can be immersed in the 2m-1-dimensional sphere (this removes the m>1 constraint). The weak version, for 2m+1, 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 went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in S^{2n-a(n)} where a(n) is the number of 1's that appear in the binary expansion of n. 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 S^{2n-a(n)} became known as the immersion conjecture. This conjecture was eventually solved in the affirmative by.