The derivative map for diffeomorphism of disks: An example

Abstract

We prove that the derivative map $d:{\mathrm{Diff}}_{\partial }(D^k)\to {\Omega }^{k}S{O}_k$, 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 non-zero: $$d_{\ast }:{\pi }_5{\mathrm{Diff}}_{\partial }(D^{11})\to {\pi }_5{\Omega }^{11}S{O}_{11}\cong {\pi }_{16}S{O}_{11}$$ As a consequence we give a counter-example to a conjecture of Burghelea and Lashof and so give an example of a non-trivial vector bundle $E$ over a sphere which is trivial as a topological ${\mathbb{R}}^k$-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 those 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.

12 pages

References in corpus (1)

• On the high-dimensional geography problem

Cited by in corpus (2)

• Relative H-Principle and Contact Geometry
• Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$ (homological interpretation)