Whitney immersion theorem

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 $$\mathbf{R}^{2m}$$ 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 $$S^{2n-1-a(n)}$$.

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.