Torsion-freeness of the first Lipschitz homotopy group of H^1

The paper studies the universal Lipschitz path space for purely 2-unrectifiable spaces, with a key application to the Heisenberg group H^1. The authors develop lifting and universal properties that parallel aspects of classical covering space theory in a metric context.

The first Lipschitz homotopy group #1{1}(H^1,0) is known to be uncountably generated, but finer structural properties are not fully understood. Hajłasz posed the question of whether this group is torsion-free, and the authors indicate that techniques from this work will be used to address the question affirmatively in a sequel.