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. (The corresponding dimensions are listed in ). 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 txt.
See also
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 (Exposition of Cohen's work)