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 (Massey 1960) 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 Ralph Cohen (Cohen 1985).

References

Cohen, Ralph L. (1985). "The immersion conjecture for differentiable manifolds". Annals of Mathematics. 122 (2): 237–328. doi:10.2307/1971304. JSTOR 1971304. MR 0808220.
Massey, William S. (1960). "On the Stiefel-Whitney classes of a manifold". American Journal of Mathematics. 82 (1): 92–102. doi:10.2307/2372878. JSTOR 2372878. MR 0111053.

External links

Giansiracusa, Jeffrey (2003). Stiefel-Whitney Characteristic Classes and the Immersion Conjecture (PDF) (Thesis). (Exposition of Cohen's work)

Whitney immersion theorem

Further results

See also

References

External links