Diffeomorphism groups of solid tori and the rational pseudoisotopy stable range

Abstract

We compute the rational homotopy groups of the classifying space $\mathrm{BDiff}_{\partial}(S^1 \times D^{d-1})$ of the topological group of diffeomorphisms of $S^1 \times D^{d-1}$ fixing the boundary for $d \geq 6$, 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 $d\geq 6$ to be $[0,d-5]$, and compute in this range the rational homotopy groups of $\mathrm{BDiff}_{\partial}(S^1 \times N)$ 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 $\mathrm{BDiff}_{\partial}(D^d)$, we compute the rational homotopy groups of the space $\mathrm{Emb}_{\partial}(D^{d-2}, D^d)$ of long knots in codimension 2 for $d \geq 6$, again in the same range.