Given two homeomorphisms $f_1, f_2 : X \to X$ a pseudoisotopy between $f_1$ and $f_2$ is a homeomorphism $F : X \times I \to X \times I$ with $F_0 = f_0$ and $F_1 = f_1$.
I would like an example of two homeomorphisms $f_1, f_2$ that are pseudoisotopic but not isotopic. Ideally, I would like such an example with $X$ a low dimensional manifold.
In dimension 3 such examples (connected sums of certain spherical space forms) do exist, see Corollary on page 364 of
S.Kwasik, R.Schultz, Pseudo-isotopies of 3-manifolds, Topology, Volume 35 (1996) 363-376.
In dimensions 1 and 2 there are no such examples (since homotopy implies isotopy for closed surfaces and circles, see D.B.A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 1966).