Isotopy Versions
A relatively ‘easy’ result is to prove that any two embeddings of a 1-manifold into are isotopic. This is proved using general position, which also allows to show that any two embeddings of an -manifold into are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
Wu proved that for, any two embeddings of an -manifold into are isotopic. This result is an isotopy version of the strong Whitney embedding theorem.
As an isotopy version of his embedding result, Haefliger proved that if is a compact -dimensional -connected manifold, then any two embeddings of into are isotopic provided . The dimension restriction is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in (and, more generally, -spheres in ). See further generalizations.
Read more about this topic: Whitney Embedding Theorem
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