Newsgroups: sci.math.research
From: palais@binah.cc.brandeis.edu
Subject: Re: Diffeomorphism question
Date: Sun, 20 Sep 1992 13:58:37 GMT
John Baez asks:
>> Let's call the "standard" embedding of S^2 in S^3 that given
>> by w = 0,where S^3 is given by w^2 + x^2 + y^2 + z^2 = 1
>> in R^4. Question: given a smooth embedding of S^2 in S^3,
>> can we compose it with a diffeomorphism of S^3 to obtain the
>> "standard" embedding? I sure hope so.
It is quite easy to show that, for any compact, smooth
n-dimensional manifold, M, the group Diff(M) acts transitively on
the space E(D^k,M) of smooth embeddings of the k-disk in M.
(well, to be precise, if k=n and M is orientable, then one has to
assume that there exists an orientation reversing diffeomorphism
of M---if not then E(D^k,M) clearly has two orbits, depending on
which orientation is induced by the embedding). For a reference see
Extending diffeomorphisms, Proc. Amer. Math. Soc. 11(1960), 274-277,
or On the local triviality of the restriction map for embeddings,
Comm. Math. Helv. 34 (1960), 306-312. It follows that your question
is equivalent to asking whether any smooth embedding of the 2-sphere
in the 3-sphere is the restriction of an embedding of a 2-disk---
i.e., the Schoenflies Problem---which is well known has a positive
answer.
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Newsgroups: sci.math.research
From: "John C. Baez"
Subject: Re: Diffeomorphism question
Date: Mon, 21 Sep 1992 03:58:42 GMT
Just to let everyone know, my question about two-spheres in
three-spheres was answered by Richard Palais and Dan Asimov. More
generally, the theorem is that if one has any S^n smoothly embedded in
S^{n+1}, one can arrange by a diffeomorphism of S^{n+1} to turn this
into your favorite "standard" embedding. I.e. all embeddings "look
alike" up to diffeomorphisms of S^{n+1}. The proof proceeds in two
steps. First note that any two embedded D^{n+1}'s "look alike" in this
sense as long as their orientations match: Extending diffeomorphisms,
Proc. Amer. Math. Soc. 11(1960), 274-277, and On the local triviality of
the restriction map for embeddings, Comm. Math. Helv. 34 (1960),
306-312. Then apply the Schoenflies theorem that says that any smoothly
embedded S^n in S^{n+1} bounds two embedded D^{n+1}'s. (This is like
the Jordan curve theorem for merely continuous embeddings of S^n, which
however fails for n>1 due to the Alexander horned sphere and other
monsters.)
This is important in quantum gravity since, as we all know, space is
S^3. (Well, at least it's convenient.) This means that any two ways
of chopping the world in half with an S^2 are the same up to
diffeomorphism.
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Newsgroups: sci.math.research
From: ruberman@binah.cc.brandeis.edu
Subject: Re: Diffeomorphism question
Date: Wed, 23 Sep 1992 13:49:44 GMT
In article <5749@osc.COM>, osc!jgk@amd.com (Joe Keane) writes:
>
>In article <1992Sep18.022831.8432@galois.mit.edu> jbaez@BOURBAKI.MIT.EDU (John
>C. Baez) writes:
>>Let's call the "standard" embedding of S^2 in S^3 that given by w = 0,
>>where S^3 is given by w^2 + x^2 + y^2 + z^2 = 1 in R^4. Question:
>>given a smooth embedding of S^2 in S^3, can we compose it with a
>>diffeomorphism of S^3 to obtain the "standard" embedding? I sure hope
>>so.
>
>In fact it should be true even if you use S^4 instead of S^3. Basically,
>there aren't enough extra dimensions to do anything tricky. I'll leave a
>proof in terms of diffeomorphisms to the topologists though. On the other
>hand, you can embed S^2 in S^5 in arbitrarily complicated ways.
>--
>Joe Keane, amateur mathematician
>jgk@osc.com (uunet!amdcad!osc!jgk)
>
Since this point seems to have come up a few times--the problem of whether
every (smooth) S^3 in S^4 is standard (via a diffeomorphism, or perhaps an
isotopy) is still unsolved, and is known as the Schoenfliess problem. The
corresponding problem in ambient dimension >= 5 is solved. In dimension 5
one uses the classification of simply connected d5-manifolds to conclude that
both complementary pieces are 5-balls. In higher dimensions the analogous
fact follows directly from the h-cobordism theorem. In both cases one then
applies the (relatively easy) fact that any two discs in a smooth manifold
are isotopic.
D. Ruberman
PS The last line in the quoted post is not right either; any S^2 in S^5 is
unknotted. ON the other hand, knots of S^2 in S^4 can be interesting.