Whitney immersion theorem

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In differential topology, the Whitney immersion theorem 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[edit]

Massey 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 which was eventually solved in the affirmative by Ralph Cohen (Cohen 1985).

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References[edit]

  • Cohen, Ralph L. (1985), "The Immersion Conjecture for Differentiable Manifolds", The Annals of Mathematics (Annals of Mathematics) 122 (2): 237–328, doi:10.2307/1971304, JSTOR 1971304 

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