Subject: Response to Rudyak's question Re: question about diffeomorphisms
From: Ryan BUDNEY
Date: Thu, 29 Mar 2007 15:05:50 +0200 (CEST)
Dear Yuli,
Cerf's pseudoisotopy theorem covers your question in high dimensions:
J. CERF, La stratification naturelle des espaces des fonctions
différentiables réelles et le théorème de pseudoïsotopie.
Publications Math. I.H.E.S., Vol. 39.
It states that \pi_0 Diff^+(D^n) is trivial for n > 5.
In low-dimensions, \pi_0 Diff^+(S^{n-1}) is known to be trivial for n=3 by
Smale, and n=4 by Cerf's earlier work.
J. CERF, Sur les difféomorphismes de la sphère de dimension trois,
Lecture Notes in Math., 53, Springer, 1968.
Smale, Stephen Diffeomorphisms of the $2$-sphere. Proc. Amer. Math. Soc.
10 1959 621--626.
I hope that helps.
-Ryan Budney
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Subject: Re: question about diffeomorphisms
From: Tom Goodwillie
Date: Thu, 29 Mar 2007 08:21:01 -0500
For a diffeomorphism of the sphere, extending to the disk is the same as
being pseudoisotopic to the identity. So another way to ask the question
is, does pseudoisotopy (not) always imply isotopy for the (n-1)-sphere?
If n is not too small (I forget the exact range) then it is a theorem of
Cerf that pseudoisotopy implies isotopy for any 1-connected smooth
n-manifold. So my answer to Rudi's question is that n would have to be
quite small for a counterexample.
By the way, Cerf's theorem can be thought of an h-cobordism theorem for
1-parameter families of 1-connected h-cobordisms, and the idea of Cerf's
proof was 1-parameter Morse theory. Hatcher and Wagoner later pushed the
same idea further and dealt with the non-simply-connected case, finding an
obstruction related to $K_2(\Bbb Z\pi_1(M))$ but also involving
$\pi_2(M)$. From a more modern perspective, the obstruction lives in
$\pi_2A(M)$,where $A$ is Waldhausen's $K$-theory functor; or rather it
lives in the quotient of that group by a summand isomorphic to the second
unreduced stable homotopy group of $M$.
TG
> From: Yuli Rudyak
> Date: Wed, 28 Mar 2007 22:49:15 -0400 (EDT)
>
> I have a question for the list
>
> There is a well-known exact sequence
> $\pi_0(Diff^+D^n) \to \pi_0(Diff^+S^{n-1}) \to \Gamma_n \to 0$
> where $\Gamma_n$ is the group of twisted $n$-spheres and Diff^+ denotes
the
> group of orientation-preserving diffeomorphisms.
>
> Question: Does somebody know the values (or a value) of $n$ such that
the first
> map is non-zero. In other words, are there self-diffeomorphisms of a
sphere that
> extends to the disk but are not isotopic to the identity?
>
> Yuli
>
> Dr. Yuli B. Rudyak
> Department of Mathematics
> University of Florida
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Subject: Re: question about diffeomorphisms
From: Johannes Ebert
Date: Thu, 29 Mar 2007 18:56:21 +0200 (CEST)
I think, for n > 5, the first map is always the zero map. Proof: Let f be
a self-diffeomorphism of D^n and let g be the restriction of f to
S^{n-1}. We want to show that g is isotopic to the identity.
Isotopy
classes of embeddings of a disc in euclidean space are easy to understand:
there are precisely two of them. Now let E be a small disc in D^n centered
at 0. The map f restricts to an embedding of E into the interior of D^n.
This embedding is orientation-preserving, so it must be isotopic to the
identity. It is then automatically ambiently isotopic. Using this
construction, one can arrange that f fixes the 1/2-ball.
In other words, g is pseudoisotopic to the identity. By Cerf's
theorem, it
follows that g is isotopic to the identity.
Or is there something wrong with this argument??
For n < 5, the first map is also the zero map, because Diff^+
(S^{n-1}) is connected in these dimensions. This is quite obvious if n
=1,2. Smale proved that Diff^+(S2) \simeq SO(3) and Hatcher that Diff^+
(S3) \simeq SO(4).
Thus the only remaining case is n = 5.
Best,
Johannes Ebert