Editing 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>m>1</math>, any smooth <math>m</math>-dimensional [[manifold]] (required also to be [[Hausdorff space|Hausdorff]] and [[second-countable]]) has a one-to-one [[immersion (mathematics)|immersion]] in [[Euclidean space|Euclidean]] <math>2m</math>-space, and a (not necessarily one-to-one) immersion in <math>(2m-1)</math>-space. Similarly, every smooth <math>m</math>-dimensional manifold can be immersed in the <math>2m-1</math>-dimensional sphere (this removes the <math>m>1</math> constraint).

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

The conjecture that every ''n''-manifold immerses in <math>S^{2n-a(n)}</math> 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==
*[[Whitney embedding theorem]]

== 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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