Diffeomorphism groups of solid tori and the rational pseudoisotopy stable range

We compute the rational homotopy groups of the classifying space ${\mathrm{B}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(S^1\times D^{d-1})$ of the topological group of diffeomorphisms of $S^1\times D^{d-1}$ fixing the boundary for $d\ge 6$, in a range of degrees up until around $d$. This extends results of Budney-Gabai, Bustamante-Randal-Williams, and Watanabe. As consequences of this computation, we determine the rational pseudoisotopy stable range for compact spin manifolds with fundamental group $\mathbf{Z}$ of dimension $d\ge 6$ to be $[0,d-5]$, and compute in this range the rational homotopy groups of ${\mathrm{B}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(S^1\times N)$ for compact simply-connected spin $(d-1)$-manifolds $N$. Finally, by combining our results with work of Krannich-Randal-Williams and Kupers-Randal-Williams on ${\mathrm{B}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^d)$, we compute the rational homotopy groups of the space ${\mathrm{E}\mathrm{m}\mathrm{b}}_{\mathrm{\partial }}(D^{d-2},D^d)$ of long knots in codimension 2 for $d\ge 6$, again in the same range.