Lifts of continuous and Hölder alpha curves in the configuration space MN/S_N

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