Re: Elementary embedding question

On 4 Oct 2006 06:01:20 -0400, lrudolph@xxxxxxxxx (Lee Rudolph) wrote:

pauldepstein@xxxxxxx writes:

Suppose a compact Hausdorff space has the property that every point has
a neighbourhood diffeomorphic to R ^ n. In other words, the space is a
smooth n-dimensional manifold.

You are making a category error (in both the Rylean and mathematical
senses of the phrase).

Paul: In case you didn't catch that, Lee's point is that
the condition you gave is _not_ the definition of "smooth
manifold". In fact "Suppose a compact Hausdorff space has
the property that every point has a neighbourhood
diffeomorphic to R ^ n" doesn't make sense, because there
is no notion of diffeomorphism for topological spaces.
("Category error": There _is_ of course a notion of
diffeomorphism, but in a different category than the
category that I might guess people call Top.)

Let me assume that you would be willing to
say "Suppose I have a smooth n-dimensional manifold M that is
compact and Hausdorff", and go on from there.

How do you prove that for sufficiently large N, the manifold can be
embedded in R ^ N.

I started to write out my understanding of the standard proof
and got bogged down in notation. It's written out nicely in
Munkres's textbook on differentiable manifolds.

I think that this is true for N = 2n but it's not
even obvious to me that any N works at all.

Once that you have some N, then the reduction to N = 2n+1 is
fairly simple (granting various machinery, e.g., Sard's Theorem):
given an n-manifold M in R^N, almost every orthogonal projection
from R^N to a (2n+1)-dimensional subspace V of R^N restricts to
an embedding of M into V by a general-position argument (the
failure of the projection to restrict to an injection is the
same as the existence of an affine (N-(2n+1))-subspace W
perpendicular to V that intersects M in 2 distinct points,
and the failure of the projection to be an immersion is
the same as the existence of such a W that is tangent to
M at some point, and "counting constants" shows that both
failures are infinitely unlikely to happen for the given
dimensions). You recall correctly that N = 2n is possible,
but for most M the reduction from 2n+1 to 2n cannot be
done locally; that is, the generic projection of the
n-manifold M into R^{2n} will be an immersion but not
an injection, and no small perturbation will change the
non-injectivity, so some global "surgery" (specifically,
the so-called "Whitney trick") will be required to change
the immersion to an embedding.

Lee Rudolph


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David C. Ullrich
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