Lifts of continuous and Hölder alpha curves in the configuration space MN/SN$M^N/S_N$

Abstract In this paper, we study the quotient space of equivalence classes of ‐tuples in a metric space , equipped with the metric induced by the minimal total pairing distance. Given a continuous path , we prove that there exist continuous functions such that for every , the multiset represents . That is, lifts to a continuous path in . Furthermore, we prove that if is Hölder continuous of order , the lift is Hölder continuous of the same order. The proof proceeds by induction on , with the fundamental step of showing that local lifts exist around all configurations, including those with point collisions.