Whitney immersion theorem

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In differential topology, the Whitney immersion theorem states that for m > 1, any smooth m-dimensional manifold can be immersed in Euclidean 2m − 1-space. Equivalently, every smooth m-dimensional manifold can be immersed in the 2m − 1-dimensional sphere (this removes the m > 1 constraint).

[edit] Further Results

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 at the Immersion Conjecture which was eventually solved by Ralph Cohen.

[edit] See also

• Whitney embedding theorem