On automorphisms of high-dimensional solid tori

TL;DR

This paper investigates the complex structure of the homotopy groups of diffeomorphism groups of high-dimensional solid tori, revealing their infinite generation in certain degrees and their rational vanishing, through advanced algebraic topology techniques.

Contribution

It provides new insights into the homotopy groups of diffeomorphism groups of high-dimensional solid tori, connecting them to Hermitian K-theory and analyzing their rational properties.

Findings

Homotopy groups are infinitely generated in specific degrees.

Homotopy groups vanish rationally.

Analysis uses the homotopy fibre of a linearisation map and Hermitian K-theory.

Abstract

We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 \times D^{2n-1}$, for $2n \geq 6$, in a range of degrees up to $n-2$. Our analysis relies on understanding the homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian $K$-theory of the integral group ring of $\pi_1(S^1)$. We also show that these homotopy groups vanish rationally.