We prove that the derivative map $d \colon \mathrm{ Diff}_\partial(D^k) \to \Omega^kSO_k$, 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: $$ d_* \colon \pi_5\mathrm{ Diff}_\partial(D^{ 11}) \to \pi_{ 5}\Omega^{ 11}SO_{ 11} \cong \pi_{ 16}SO_{ 11} $$ 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.
Categories: math.GT math.AT