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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 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 {{Harv|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 {{harvs|first=Ralph|last=Cohen|authorlink=Ralph Louis Cohen|year=1985|txt}}.

See also

References

  • {{cite journal

|doi=10.2307/1971304

|first=Ralph L.

|last=Cohen

|author-link=Ralph Louis Cohen

|title=The immersion conjecture for differentiable manifolds

|journal=Annals of Mathematics

|year=1985

|pages=237–328

|jstor=1971304

|volume=122

|issue=2

|mr=0808220

}}

  • {{cite journal | last=Massey | first=William S. | author-link=William S. Massey| title=On the Stiefel-Whitney classes of a manifold | journal=American Journal of Mathematics | volume=82 | issue=1 | year=1960 | doi=10.2307/2372878 | pages=92–102 | mr=0111053|jstor=2372878}}

External links

  • {{cite thesis|url=http://maths.swan.ac.uk/staff/jhg/papers/thesis-final.pdf |title=Stiefel-Whitney Characteristic Classes and the Immersion Conjecture|first=Jeffrey|last=Giansiracusa|year= 2003}} (Exposition of Cohen's work)

Category:Theorems in differential topology

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