Lipschitz homotopy groups of Heisenberg groups

Presenting a joint work with Hajlasz and Tyson, I will introduce some basics of the Heisenberg group. Since the Heisenberg-group is homeomorphic to the euclidean space, classical Homotopy groups are all zero, so it seems to be more reasonable to consider Lipschitz homotopy groups.

Then I will present an argument about non-triviality of certain Homotopy groups, by relating Homotopy groups of spheres and the Heisenberg group, and extending the Hopf invariant. As a consequence, one can also prove that Lipschitz mappings are not dense in certain the Sobolev spaces.