H\"older-Topology of the Heisenberg group

TL;DR

This paper explores the H"older topology of the Heisenberg group, a sub-Riemannian manifold, discussing recent progress, related density questions for Sobolev maps, and ideas behind a Gromov conjecture involving linking numbers.

Contribution

It reviews recent advances in understanding the H"older topology of the Heisenberg group and discusses ideas related to Gromov's conjecture without providing definitive proof.

Findings

Progress on the analysis of H"older topology in the Heisenberg group

Discussion of density questions for Sobolev maps into the Heisenberg group

Insights into Gromov's conjecture based on linking number

Abstract

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.