In differential topology, the Whitney immersion theorem 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 2 m -space, and a (not necessarily one-to-one) immersion in ( 2 m − 1 ) -space. Similarly, every smooth m -dimensional manifold can be immersed in the 2 m − 1 -dimensional sphere (this removes the m > 1 constraint).
The weak version, for 2 m + 1 , is due to transversality (general position, dimension counting): two m-dimensional manifolds in R 2 m intersect generically in a 0-dimensional space.
Massey went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in S 2 n − 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 S 2 n − 1 − a ( n ) . The conjecture that every n-manifold immerses in S 2 n − a ( n ) became known as the Immersion Conjecture which was eventually solved in the affirmative by Ralph Cohen (Cohen 1985).
