Whitney immersion theorem explained

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 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 ^{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 .

See also

• Whitney embedding theorem

References

• 10.2307/1971304. Ralph L.. Cohen. Ralph Louis Cohen. The immersion conjecture for differentiable manifolds. Annals of Mathematics. 1985. 237–328. 1971304. 122. 2. 0808220.
• Massey . William S. . William S. Massey. On the Stiefel-Whitney classes of a manifold . . 82 . 1 . 1960 . 10.2307/2372878 . 92–102 . 0111053. 2372878.

External links

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