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The
derivative map for diffeomorphism of disks: an example
Diarmuid Crowley, Thomas Schick and Wolfgang Steimle |
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Geometry & Topology 27 (2023) 3699–3713
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Abstract |
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We prove that the derivative map , defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for we prove that the following homomorphism is nonzero: As a consequence we give a counterexample to a conjecture of Burghelea and Lashof by giving an example of a nontrivial vector bundle over a sphere which is trivial as a topological –bundle (the rank of is and the base sphere is ). The proof relies on a recent result of Burklund and Senger which determines the homotopy –spheres bounding –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
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Mathematical Subject Classification
Primary: 57R50, 57S05
Secondary: 57R60
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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
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Authors |
| Diarmuid Crowley | |
| School of Mathematics and
Statistics University of Melbourne Parkville, Victoria Australia |
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| https://www.dcrowley.net | |
| Thomas Schick | |
| Mathematisches Institut Universität Göttingen Göttingen Germany |
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| https://www.uni-math.gwdg.de/schick | |
| Wolfgang Steimle | |
| Institut für Mathematik Universität Augsburg Augsburg Germany |
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| https://www.uni-augsburg.de/de/fakultaet/mntf/math/prof/diff/team/wolfgang-steimle | |
| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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