Whitney embedding theorem Totally Explained

Everything about Whitney Embedding Theorem totally explained

In mathematics, particularly in differential topology,there are two Whitney embedding theorems:

The strong Whitney embedding theorem states that any connected smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean $2m$ -space. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of even dimension $m$ can't be embedded into Euclidean ( $2m-1$ )-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).

The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m>2n. Whitney similarly proved that such a map could be approximated by an immersion provided m>2n-1. This last result is sometimes called the weak Whitney immersion theorem.

A little about the proof

The general outline of the proof is to start with an immersion

f:M	omathbb R^

are isotopic provided

2k+2 leq n

. Haefliger went on to give examples of non-trivially embedded 3-spheres in

R^6

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