The derivative map for diffeomorphism of disks: an example

Crowley, Diarmuid¹ ; Schick, Thomas² ; Steimle, Wolfgang³

¹ School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia
² Mathematisches Institut, Universität Göttingen, Göttingen, Germany
³ Institut für Mathematik, Universität Augsburg, Augsburg, Germany

Geometry & topology, Tome 27 (2023) no. 9, pp. 3699-3713.

Résumé

We prove that the derivative map $d:{\mathrm{Diff}}_{\partial }(D^k)\to {\mathrm{\Omega }}^k{\mathrm{SO}}_k$, 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:

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 ${ℝ}^k$–bundle (the rank of $E$ is $k=11$ and the base sphere is $S^{17}$).

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.

Détail

DOI : 10.2140/gt.2023.27.3699

Keywords: diffeomorphisms of discs, derivative map, topologically trivial vector bundle

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