H\"older-Topology of the Heisenberg group

The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions. We report on some recent progress in the Analysis of the H\"older topology of the Heisenberg group, some related and some unrelated to density questions for Sobolev maps into the Heisenberg group. In particular we describe the main ideas behind a result by Haj\l{}asz, Mirra, and the author regarding Gromov's conjecture, which is based on the linking number. We do not prove or disprove the Gromov Conjecture.