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| In [[differential topology]], the '''Whitney immersion theorem''' 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).
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| 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. | |
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| ==Further Results==
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| Massey went on to prove that every n-dimensional manifold is 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''' which was eventually solved in the affirmative by Ralph Cohen {{Harv|Cohen|1985}}.
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| ==See also==
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| *[[Whitney embedding theorem]]
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| == References ==
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| * {{citation
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| |doi=10.2307/1971304
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| |first=Ralph L.
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| |last=Cohen
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| |title=The Immersion Conjecture for Differentiable Manifolds
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| |journal=The Annals of Mathematics
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| |year=1985
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| |pages=237–328
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| |jstor=1971304
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| |volume=122
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| |issue=2
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| |publisher=Annals of Mathematics
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| }}
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| == External links ==
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| * [http://maths.swan.ac.uk/staff/jhg/papers/thesis-final.pdf Stiefel-Whitney Characteristic Classes and the Immersion Conjecture], by Jeffrey Giansiracusa, 2003
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| *:Exposition of Cohen's work
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| [[Category:Theorems in differential topology]]
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| {{topology-stub}}
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