From: Heinz Spindler Subject: Q:homeomorphisms from S^3 to S^3 Date: 7 Dec 1999 10:00:07 -0600 Newsgroups: sci.math.research Keywords: topology of space of isomorphisms (various categories) of S^3 Hi everybody, Who can help me with the following problem. Let Aut_TOP(S^3) be the set of orientation preserving homeomorphisms from S^3 to S^3, equipped with the compact-open topology (also called C^0 topology). Is this space path-connected? Or what else is \pi_0(Aut_TOP(S^3)) ? The same question could be posed for Aut_DIFF(S^3), the set of orientation preserving diffeomorphisms, equipped with the C^\infty topology. What about the higher homotopy groups of these spaces? Thanks, heinz spindler ============================================================================== From: greg@math.ucdavis.edu (Greg Kuperberg) Subject: Re: Q:homeomorphisms from S^3 to S^3 Date: 7 Dec 1999 15:30:01 -0600 Newsgroups: sci.math.research In article <384CE548.D2EFA9C7@Mathematik.uni-osnabrueck.de>, Heinz Spindler wrote: >Who can help me with the following problem. >Let Aut_TOP(S^3) be the set of orientation preserving homeomorphisms >from S^3 to S^3, equipped with the compact-open topology (also called >C^0 topology). Is this space path-connected? Or what else is >\pi_0(Aut_TOP(S^3))? The set of homeomorphisms with this topology is generally considered to be too coarse (too few open sets) to be interesting. There is a process called coning available in both the TOP and PL categories; it gives a messy homeomorphism a "face lift" to produce something simple. I think that it is relatively easy to show using coning that Aut_TOP(S^n) is homotopy equivalent to the Lie group O(n+1). >The same question could be posed for >Aut_DIFF(S^3), the set of orientation preserving diffeomorphisms, >equipped with the C^\infty topology. What about the higher homotopy >groups of these spaces? This is much more interesting since in high dimensions Milnor found discrepancies between Aut_DIFF(S^n) and O(n+1). For n=3, Allen Hatcher proved that they are the same. I've heard that his paper is long and difficult and it might not be adequately verified, but at the same time no one has raised any concrete doubts. For n=2 they are the same by a result of Smale. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math Archive Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print * ============================================================================== From: greg@math.ucdavis.edu (Greg Kuperberg) Subject: Re: Q:homeomorphisms from S^3 to S^3. Answers and further questions Date: 9 Dec 1999 10:52:02 -0800 Newsgroups: sci.math.research In article <384F31C1.84F7517@uos.de>, Marc Nardmann wrote: >Greg Kuperberg wrote: >> There is a process called coning available in both the TOP and PL >> categories; it >> gives a messy homeomorphism a "face lift" to produce something simple. >> I think that it is relatively easy to show using coning that Aut_TOP(S^n) >> is homotopy equivalent to the Lie group O(n+1). >(In the introduction to [1], Cerf calls this the Feldbau conjecture and says >that there would be no reason to believe it's true for all n. Indeed:) This >conjecture implies that the RHS of Cerf's equation vanishes for all i and n. >But, as was noticed by Milnor in [6], the LHS does NOT vanish for all i and n >(take i=0 and n=6, for example). So the conjecture can't be true. The fallacy in my reasoning was also pointed out by e-mail from Danny Calegari. I was thinking of Top(B^n,S^(n-1)) (that is, homeomorphisms rel boundary). This is contractible by coning. If you are studying Diff_+(S^n) with the C^infty topology (or PL_+(S^n) with an analogous Lipschitz topology), then it is elementary to reduce the question to Diff(B^n,S^(n-1)); you just blow up a point in S^n to make it an S^(n-1) . But then C^infty topology prevents coning. So I guess there is no easy way out in any of the listed cases; if the topology on your favorite group of homeomorphisms is fine enough to allow blowing up a point, it ipso facto disallows coning. I can think of one facile case: Any two PL homeomorphisms of the sphere are connected by a level-preserving pseudoisotopy by coning. But whereas such a pseudoisotopy corresponds to a C^0 path in the Top case and a C^infty path in the Diff case, I do not know of a corresponding topology in the PL case. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math Archive Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print * ============================================================================== From: baez@math.ucr.edu (john baez) Subject: Re: Q: homeomorphisms from S^3 to S^3 Date: 8 Dec 1999 00:30:03 -0600 Newsgroups: sci.math.research In article <384CE548.D2EFA9C7@Mathematik.uni-osnabrueck.de>, Heinz Spindler wrote: >Who can help me with the following problem. >Let Aut_TOP(S^3) be the set of orientation preserving homeomorphisms >from S^3 to S^3, equipped with the compact-open topology (also called >C^0 topology). Is this space path-connected? Or what else is >\pi_0(Aut_TOP(S^3)) ? The same question could be posed for >Aut_DIFF(S^3), the set of orientation preserving diffeomorphisms, >equipped with the C^\infty topology. What about the higher homotopy >groups of these spaces? I'm pretty sure the space of orientation preserving diffeomorphisms of S^3, with its C^infinity topology, is homotopy equivalent to SO(3). This would imply all its homotopy groups equal those of SO(3). If want to be sure I'm not misremembering, you might try: Cerf J, Sur les Diffeomorphisms de la Sphere de Dimension Trois $(\Gamma_4 = 0)$, Springer Lecture Notes in Mathematics #53, Springer-Verlag, Berlin, 1968. ============================================================================== From: baez@math.ucr.edu (john baez) Subject: Re: Q: homeomorphisms from S^3 to S^3 Date: 8 Dec 1999 17:00:02 -0600 Newsgroups: sci.math.research In article <199912080520.VAA26561@charity.ucr.edu>, john baez wrote: >I'm pretty sure the space of orientation preserving diffeomorphisms of >S^3, with its C^infinity topology, is homotopy equivalent to SO(3). >This would imply all its homotopy groups equal those of SO(3). If >want to be sure I'm not misremembering, you might try: > >Cerf J, Sur les Diffeomorphisms de la Sphere de Dimension Trois >$(\Gamma_4 = 0)$, Springer Lecture Notes in Mathematics #53, >Springer-Verlag, Berlin, 1968. Oops - the fact is apparently true, but this is the wrong reference! This paper just shows that the space of orientation preserving diffeomorphisms of S^3, with its C^infinity topology, is connected. The stronger result is due to Hatcher, as Greg Kuperberg pointed out. Thanks go to Ryan Budney for catching this error. ============================================================================== From: Linus Kramer Subject: Re: Q:homeomorphisms from S^3 to S^3 Date: 8 Dec 1999 09:00:04 -0600 Newsgroups: sci.math.research Heinz Spindler wrote: > > Hi everybody, > > Who can help me with the following problem. > Let Aut_TOP(S^3) be the set of orientation preserving homeomorphisms > from S^3 to S^3, equipped with the compact-open topology (also called > C^0 topology). Is this space path-connected? Or what else is > \pi_0(Aut_TOP(S^3)) ? The same question could be posed for > Aut_DIFF(S^3), the set of orientation preserving diffeomorphisms, > equipped with the C^\infty topology. What about the higher homotopy > groups of these spaces? > > Thanks, > heinz spindler Let Diff(S^3) denote the diffeomorphism group of S^3 (with the C^\infty topolgy), Top(S^3) its homeomorphism group (c-o-topology) and O(4) the rotation group in R^4. There are natural maps O(4) --> Diff(S^3) --> Top(S^3). Hatcher proved [Ann. Math. 117] that both maps are homotopy equivalences. This settled the so-called Smale conjecture, and it's a very difficult theorem! Also, the result is not true any more for spheres of higher dimensions. Linus Kramer -- Linus Kramer Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg Germany E-mail: kramer@mathematik.uni-wuerzburg.de http://www.mathematik.uni-wuerzburg.de/~kramer