(Non)-exoticness of a diffeomorphism of a sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:49:05Z http://mathoverflow.net/feeds/question/111000 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111000/non-exoticness-of-a-diffeomorphism-of-a-sphere (Non)-exoticness of a diffeomorphism of a sphere Marco Radeschi 2012-10-29T14:11:07Z 2012-10-29T22:57:07Z <p>Suppose you have a standard sphere $S^n$ and a "standard" $S^{n-2}\subset S^n$. I am really thinking about $S^{n}\subset \mathbb{R}^{n+1}$ the usual sphere, and $S^{n-2}=S^n\cap \{x_0=x_1=0\}$. Let $S^1$ be the circle "orthogonal" to $S^{n-2}$, i.e. $S^1=S^n\cap span\{x_0, x_1\}$. Then $S^n$ gets decomposed by the hypersurfaces $S_t:=S^{n-2}(\cos t)\times S^1(\sin t)$, i.e. the distance tubes around $S^{n-2}$ and $S^1$.</p> <p>Suppose now that $\phi:S^n\to S^n$ is a diffeomorphism such that:</p> <ul> <li>$\phi$ fixes $S^{n-2}$ pointwise: <code>$\phi\big|_{S^{n-2}}= id\big|_{S^{n-2}}$</code>.</li> <li>$\phi$ sends the hypersurfaces $S_t$ to themselves (it is not the identity though).</li> </ul> <p><strong>Question 1:</strong> is it true that $\phi$ is homotopic to an isometry of $S^n$ in $Diff(S^n)$?</p> <p>Here is a (probably) much stronger assumption on $\phi$: fix a basis $x_0,\ldots x_n$ of $\mathbb{R}^{n+1}$, and suppose that $\phi:S^n\to S^n$ preserves any subsphere "main subsphere" $S^{n-k}=S^n\cap \{x_{i_1}=\ldots x_{i_k}=0\}$.</p> <p><strong>Question 2:</strong> is it true that $\phi$ is homotopic to an isometry of $S^n$ in $Diff(S^n)$?</p> <p>Regarding this second question, my approach was to start deforming $\phi$ to be an isometry on the smallest "main subspheres", and hopefully going up in dimension, but this requires me to know that $\pi_i(Diff(T^k))=0$, $i>0$, where $T^k$ is a $k$-dimensional torus. So here is a third, kind of related, question:</p> <p><strong>Question 3:</strong> is it true that $\pi_i(Diff(T^k))=0$ for every torus $T^k$ and every $i>0$?</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/111000/non-exoticness-of-a-diffeomorphism-of-a-sphere/111023#111023 Answer by Ryan Budney for (Non)-exoticness of a diffeomorphism of a sphere Ryan Budney 2012-10-29T19:25:29Z 2012-10-29T22:57:07Z <p>What definition of torus are you using Marco? The usual definition is $T^k = (S^1)^k$ but from the the way you've structured things above it looks like you're using the convention $T^k = S^1 \times S^{k-1}$. Either way, the homotopy-groups of these diffeomorphism groups are generally not trivial as they generally contain plenty of torsion. $Diff(S^1 \times S^{k-1})$ contains $SO_2 \times SO_k \times \Omega SO_k$ for example. </p> <p>Question 1 is a standard pseudo-isotopy type question. Have you looked up the literature on pseudo-isotopy diffeomorphisms of $S^1 \times S^{n-2}$ ? For example, there's a recent arXiv paper of Crowley and Schick which states that there's elements of $Diff(S^n)$ with large Gromoll Degree, meaning they can be put into positions like in your questions 1 and 2, yet they're non-trivial diffeomorphisms of the sphere. </p> <p><a href="http://arxiv.org/abs/1204.6474" rel="nofollow">http://arxiv.org/abs/1204.6474</a></p> <p>edit: For some basic results on the homotopy-type of $Diff((S^1)^n)$ see:</p> <p>Hatcher, A. E. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 3--21, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. (Reviewer: Gerald A. Anderson) 57R52</p>