# Easiest example where pseudo-isotopy fails to be the same as isotopy?

This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is isotopic to the identity if there is a continuous one-parameter family of diffeomorphisms $$f_t: M \to M$$ paramatrized by $t \in [0,1]$ such that $f_0 = id$ and $f_1 = f$.

We will say that $f$ is pseudoisotopic to the identity if there is a diffeomorphism $$F: M \times I \to M \times I$$ which restricts to the identity map on the top face $M \times \{ 0\}$ and to $f$ on the bottom face $M \times \{1\}$.

isotopy implies psuedoisotopy, but not necessarily the other way around. Cerf famously proved that in high dimensions if M is also simply connected then psuedoisotopy is the same as isotopy.

I would like to better understand how this can fail. What is the easiest example where psuedoisotopy differs from isotopy?

In high dimensions ($\geq 5$) the most basic examples arise on manifolds with nonempty boundary, where one requires that diffeomorphisms restrict to the identity on the boundary. The simplest case is a diffeomorphism of $S^1\times D^{n-1}$ that is pseudoisotopic to the identity but not isotopic to the identity (always fixing the boundary). I described this construction in a paper called "Concordance spaces, higher simple-homotopy theory, and applications" that appeared in AMS Proc. Symp. Pure Math. 32 (1978). The paragraphs describing the construction are copied almost verbatim below, and a scanned version of the paper is available on my webpage. The construction is due originally to Tom Farrell (unpublished) but the fact that it has the properties claimed is an application of an old theorem of mine that had a gap in its proof that was later filled in by Kiyoshi Igusa.
To get examples in closed manifolds $M$ one can embed $S^1\times D^{n-1}$ in $M$ so as to represent a nontrivial element of $\pi_1(M)$, then extend the diffeomorphism via the identity map on the complement. The resulting diffeomorphism of $M$ may or may not be isotopic to the identity, but there are some cases when it is definitely not isotopic to the identity, e.g., $M=S^1 \times S^{n-1}$ or $M=T^n$, the $n$-torus. The latter case is described in the paper, while the former case is more elementary using the fact that the space of embeddings $S^1\to S^1\times S^{n-1}$ homotopic to a given embbedding is simply-connected if $n$ is not too small.
Now for the construction of the diffeomorphism $f\colon S^1\times D^{n-1} \to S^1\times D^{n-1}$, as described in the paper:
To construct $f$ we will perform two embedded surgeries on the interior of the codimension one slice $D^{n-1}_0 = * \times D^{n-1}$, producing a new disk $D^{n-1}_1\subset S^1\times D^{n-1}$ with $\partial D^{n-1}_0=\partial D^{n-1}_1$ and with the complement of $D^{n-1}_1$ still an $n$-ball, so that $D^{n-1}_1 = f(D^{n-1}_0)$ for some homeomorphism (in fact diffeomorphism) $f$ of $S^1\times D^{n-1}$ fixing the boundary.
In a neighborhood of $D^{n-1}_0$ label the two sides of $D^{n-1}_0$ as $+$ and $-$. On the $+$ side, attach an embedded $i$-handle $D^i\times D^{n-i}$ to $D^{n-1}_0$ in the trivial way. This effects a surgery on $D^{n-1}_0$ to $\chi(D^{n-1}_0)$, say. We could undo the effect of this surgery by now attaching an embedded $(i+1)$-handle $D^{i+1}\times D^{n-i-1}$ on the $+$ side of $\chi(D^{n-1}_0)$, in the trivial way so that the surgered $\chi(D^{n-1}_0)$ would be an $(n-1)$-disk isotopic to $D^{n-1}_0$. (All of this would occur near $D^{n-1}_0$.) The Farrell construction is to take instead a new embedding of the $(i+1)$-handle, but attached to $\chi(D^{n-1}_0)$ in the same way so that $\chi(D^{n-1}_0)$ is again surgered to a disk, this time the desired $D^{n-1}_1$. The new $(i+1)$-handle is obtained from the old one by replacing the old core $D^{i+1}$ by $D^{i+1}\# S^{i+1}$, the (interior) connected sum with a certain $S^{i+1}\subset (S^1\times D^{n-1})-\chi(D^{n-1}_0)$. This $S^{i+1}$ is constructed as follows. The core $D^i$ of the $i$-handle can be completed to a sphere $S^i$ by adding another $i$-disk on the $-$ side of $D^{n-1}_0$. In a neighborhood of this $S^i$, embed $S^{i+1}$ so as to represent the Hopf map $S^{i+1}\to S^i$. For this we must assume $i\geq 2$ and $n$ large enough to get $S^{i+1}$ actually embedded. Finally, to form the connected sum of $D^{i+1}$, which is on the $+$ side of $\chi(D^{n-1}_0)$, with $S^{i+1}$, which is on the $-$ side, we connect $D^{i+1}$ to $S^{i+1}$ by an arc which circles around the $S^1$ factor of $S^1\times D^{n-1}$.
The construction actually gives a concordance from $D^{n-1}_0$ to $D^{n-1}_1$ in $S^1\times D^{n-1} \times I$, namely the trace of the two surgeries. This then extends to a concordance (pseudoisotopy) between $f$ and the identity, using the h-cobordism theorem to guarantee that the complement of the codimension-one concordance is a product (or maybe this can be seen directly).