The derivative map for diffeomorphism of disks: An example

We prove that the derivative map $d \colon $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}$_\partial(D^k) \to \Omega^kSO_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: $$d_{\ast }\text{\hspace{0.17em}⁣}:{\pi }_5{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^{11})\to {\pi }_5{\mathrm{\Omega }}^{11}S{O}_{11}\cong {\pi }_{16}S{O}_{11}$$ 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.