Whitney immersion theorem

Short description: On immersions of smooth m-dimensional manifolds in 2m-space and (2m-1) space

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

The weak version, for [math]\displaystyle{ 2m+1 }[/math], is due to transversality (general position, dimension counting): two m-dimensional manifolds in [math]\displaystyle{ \mathbf{R}^{2m} }[/math] 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 [math]\displaystyle{ S^{2n-a(n)} }[/math] where [math]\displaystyle{ a(n) }[/math] is the number of 1's that appear in the binary expansion of [math]\displaystyle{ n }[/math]. 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 [math]\displaystyle{ S^{2n-1-a(n)} }[/math].

The conjecture that every n-manifold immerses in [math]\displaystyle{ S^{2n-a(n)} }[/math] became known as the immersion conjecture. This conjecture was eventually solved in the affirmative by Ralph Cohen (1985).

See also

• Whitney embedding theorem

References

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

External links

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