Quantum extensions of ordinary maps

Abstract

We define a loop to be quantum nullhomotopic if and only if it admits a nonempty quantum set of extensions to the unit disk. We show that the canonical loop in the unit circle is not quantum nullhomotopic, but that every loop in the real projective plane is quantum nullhomotopic. Furthermore, we apply Kuiper's theorem to show that the canonical loop admits a continuous family of extensions to the unit disk that is indexed by an infinite quantum space. We obtain these results using a purely topological condition that we show to be equivalent to the existence of a quantum family of extensions of a given map.