Whitney immersion theorem

In differential topology, the Whitney immersion theorem (named after Hassler Whitney) 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 $\mathbf^$ intersect generically in a 0-dimensional space.

Further results

William S. Massey went on to prove that every ''n''-dimensional manifold is cobordant to a manifold that immerses in $S^$ 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 $S^$.

The conjecture that every ''n''-manifold immerses in $S^$ became known as the immersion conjecture. This conjecture was eventually solved in the affirmative by .

See also

*Whitney embedding theorem

References

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External links

* (Exposition of Cohen's work)

Theorems in differential topology

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