Whitney immersion theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In differential topology, the Whitney immersion theorem states that for , any smooth -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean -space, and a (not necessarily one-to-one) immersion in -space. Similarly, every smooth -dimensional manifold can be immersed in the -dimensional sphere (this removes the constraint).

The weak version, for , is due to transversality (general position, dimension counting): two m-dimensional manifolds in 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 where is the number of 1's that appear in the binary expansion of . 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 . The conjecture that every n-manifold immerses in became known as the Immersion Conjecture which was eventually solved in the affirmative by Ralph Cohen (Cohen 1985).

See also[edit]

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 

External links[edit]