The derivative map for diffeomorphism of disks: an example

Abstract

We prove that the derivative map d W Diff @ .D k / ! k SO k , defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for k D 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 D 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. 57R50, 57S05; 57R60