(Non)-exoticness of a diffeomorphism of a sphere

Question

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 $S^1$.

Suppose now that $\phi:S^n\to S^n$ is a diffeomorphism such that:

• $\phi$ fixes $S^{n-2}$ pointwise: $\phi\big|_{S^{n-2}}= id\big|_{S^{n-2}}$.
• $\phi$ sends the hypersurfaces $S_t$ to themselves (it is not the identity though).

Question 1: is it true that $\phi$ is homotopic to an isometry of $S^n$ in $Diff(S^n)$?

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" $S^{n-k}=S^n\cap \{x_{i_1}=\ldots x_{i_k}=0\}$.

Question 2: is it true that $\phi$ is homotopic to an isometry of $S^n$ in $Diff(S^n)$?

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:

Question 3: is it true that $\pi_i(Diff(T^k))=0$ for every torus $T^k$ and every $i>0$?

Thanks in advance!

Answer 1

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.

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.

http://arxiv.org/abs/1204.6474

edit: For some basic results on the homotopy-type of $Diff((S^1)^n)$ see:

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