Diffeomorphism groups of solid tori and the rational pseudoisotopy stable range

TL;DR

This paper computes the rational homotopy groups of classifying spaces of diffeomorphisms of certain manifolds, extending previous results and determining the stable range for rational pseudoisotopy for specific manifolds, with applications to knot spaces.

Contribution

It extends the computation of rational homotopy groups of diffeomorphism classifying spaces to higher dimensions and specific manifolds, establishing the rational pseudoisotopy stable range.

Findings

Rational homotopy groups of $ ext{BDiff}_{oundary}(S^1 imes D^{d-1})$ computed for $d geq 6$

Rational pseudoisotopy stable range determined as $[0, d-5]$ for certain manifolds

Rational homotopy groups of embedding spaces of long knots in codimension 2 computed for $d geq 6$

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.