Template:Short description 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 and second-countable) has a one-to-one immersion in 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 (general position, dimension counting): two m-dimensional manifolds in <math>\mathbf{R}^{2m}</math> intersect generically in a 0-dimensional space.
Further resultsEdit
William S. Massey Template:Harv 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. This conjecture was eventually solved in the affirmative by Template:Harvs.
See alsoEdit
ReferencesEdit
External linksEdit
- Template:Cite thesis (Exposition of Cohen's work)