On automorphisms of high-dimensional solid tori

Bustamante, Mauricio; Randal-Williams, Oscar

doi:10.2140/gt.2024.28.1629

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On automorphisms of high-dimensional solid tori

Mauricio Bustamante and Oscar Randal-Williams

Geometry & Topology 28 (2024) 1629–1692

DOI: 10.2140/gt.2024.28.1629

Abstract

We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^{1} \times D^{2 n - 1}$ , for $2 n \geq 6$ , 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 a certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian $K$ –theory of the integral group ring of $π_{1} (S^{1})$ . We also show that these homotopy groups vanish rationally.

Keywords

diffeomorphisms, block diffeomorphisms, solid tori, Weiss fibre sequence, self-embeddings, surgery theory, infinite generation

Mathematical Subject Classification

Primary: 55R40, 57S05, 58D10

References

Publication

Received: 17 February 2021

Revised: 1 March 2022

Accepted: 18 October 2022

Published: 18 July 2024

Proposed: Steve Ferry

Seconded: Haynes R Miller, Nathalie Wahl

Authors

Mauricio Bustamante
Departamento de Matemáticas Universidad Católica de Chile Santiago Chile

Oscar Randal-Williams
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

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Volume 28, issue 4 (2024)