Getting a self-homeomorphism of the cylinder from a self-homeomorphism of the circle

Question

Given an orientation-preserving homeomorphism of the circle $$f : S^1 \rightarrow S^1,$$ I want to define a homeomorphism of the cylinder $$F : S^1 \times \mathbb{I} \rightarrow S^1 \times \mathbb{I}$$ such that for all $x \in S^1$, we have: $F(x,0) = (f(x),0)$ and $F(x,1) = (x,1)$.

The same goes with 'homeomorphism' replaced by 'diffeomorphism.' It seems likely that this is possible, but I haven't been able to find an explicit definition. Ideas, anyone?

Answer 1

This is possible. Let $e:\mathbb{R}\to S^1$ be the covering map $e(s)=\exp(2\pi i s)$. We can then lift $f$ to a map $g:\mathbb{R}\to \mathbb{R}$ such that $eg=fe$. The assumption that $f$ is an orientation-preserving homeomorphism implies that $g$ is strictly increasing with $g(s+1)=g(s)+1$ for all $s$. Now we obtain $F$ by just interpolating linearly between $g$ and the identity. That is, we define $$F(e(s),t)=(e(ts+(1-t)g(s)),t).$$ It is easy to verify this is a homeomorphism; the key point is that for any $t$, $h_t(s)=ts+(1-t)g(s)$ is again a strictly increasing map with $h_t(s+1)=h_t(s)$.

If $f$ was not just a homeomorphism but a diffeomorphism, then $g$ will be a diffeomorphism (as will all the maps $h_t$), and it follows easily that $F$ will also be a diffeomorphism.

Answer 2

This is an addendum to Eric Wofsey's answer, regarding what happens in higher dimensions: Given an orientation-preserving homeomorphism (diffeomorphims) $f: S^n\to S^n$, is there a homeomorphism (diffeomorphism) $$F: S^n\times I\to S^n\times I$$ such that $F(x,0)=(f(x),0), F(x,1)=(x,1)$?

The answer in the setting of homeomorphisms is positive for all $n$: Every orientation-preserving homeomorphism $f: S^n\to S^n$ is homotopic to the identity and, hence (Alexander, et al, see this Mathoverflow discussion) isotopic to the identity. This isotopy $f_t$ yields a homeomorphism $F(x,t)=(f_t(x),t)$, $S^n\times I\to S^n\times I$.

In the setting of diffeomorphisms, the answer is much more interesting, it amounts to the question of concordance of (orientation-preserving) diffeomorphisms $S^n\to S^n$ to the identity map. The space of concordance classes forms an abelian group, called $\Gamma^n$. This group is trivial for all $n\le 5$ and, hence, a diffeomorphic extension $F: S^n\times I\to S^n\times I$ always exists in this range ($n=1$ is a very special case). However, for $n=6$ the group $\Gamma^6$ is nontrivial and has order 28. In particular, there exists a diffeomorphism $f: S^6\to S^6$ for which a diffeomorphism $F: S^6\times I\to S^6\times I$ as above does not exist.

For other values of $n\ge 7$, these groups are well-studied (Kervaire-Milnor et al), see for instance this wikipeda article.

There is a bijection (for $n\ne 3$) between $\Gamma^n$ and the group $\Theta_{n+1}$ of smooth structures on $S^{n+1}$ (under the connected sum). For instance, the 27 concordance classes on $S^6$ correspond to the 27 exotic seven-dimensional spheres.