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The derivative map for diffeomorphism of disks: an example

Diarmuid Crowley, Thomas Schick and Wolfgang Steimle

Geometry & Topology 27 (2023) 3699–3713
Abstract

We prove that the derivative map d: Diff (Dk) Ωk SO k, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for k = 11 we prove that the following homomorphism is nonzero:

d: π5 Diff (D11) π 5Ω11 SO 11 π16 SO 11.

As a consequence we give a counterexample to a conjecture of Burghelea and Lashof by giving an example of a nontrivial vector bundle E over a sphere which is trivial as a topological k–bundle (the rank of E is k = 11 and the base sphere is S17).

The proof relies on a recent result of Burklund and Senger which determines the homotopy 17–spheres bounding 8–connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres.

Keywords
diffeomorphisms of discs, derivative map, topologically trivial vector bundle
Mathematical Subject Classification
Primary: 57R50, 57S05
Secondary: 57R60
References
Publication
Received: 6 May 2021
Revised: 12 January 2022
Accepted: 26 March 2022
Published: 5 December 2023
Proposed: Stefan Schwede
Seconded: Ulrike Tillmann, Nathalie Wahl
Authors
Diarmuid Crowley
School of Mathematics and Statistics
University of Melbourne
Parkville, Victoria
Australia
https://www.dcrowley.net
Thomas Schick
Mathematisches Institut
Universität Göttingen
Göttingen
Germany
https://www.uni-math.gwdg.de/schick
Wolfgang Steimle
Institut für Mathematik
Universität Augsburg
Augsburg
Germany
https://www.uni-augsburg.de/de/fakultaet/mntf/math/prof/diff/team/wolfgang-steimle

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