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. ============================================================================== 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. ============================================================================== 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.