# Whitney immersion theorem

In differential topology, the Whitney immersion theorem states that for ${\displaystyle m>1}$, any smooth ${\displaystyle m}$-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean ${\displaystyle 2m}$-space, and a (not necessarily one-to-one) immersion in ${\displaystyle (2m-1)}$-space. Similarly, every smooth ${\displaystyle m}$-dimensional manifold can be immersed in the ${\displaystyle 2m-1}$-dimensional sphere (this removes the ${\displaystyle m>1}$ constraint).

The weak version, for ${\displaystyle 2m+1}$, is due to transversality (general position, dimension counting): two m-dimensional manifolds in ${\displaystyle \mathbf {R} ^{2m}}$ intersect generically in a 0-dimensional space.

## Further results

Massey went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in ${\displaystyle S^{2n-a(n)}}$ where ${\displaystyle a(n)}$ is the number of 1's that appear in the binary expansion of ${\displaystyle n}$. 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 ${\displaystyle S^{2n-1-a(n)}}$. The conjecture that every n-manifold immerses in ${\displaystyle S^{2n-a(n)}}$ became known as the Immersion Conjecture which was eventually solved in the affirmative by Ralph Cohen (Cohen 1985).