Symmetric products and finite subset spaces of absolute Lipschitz retracts

Abstract

We prove that the $Q$th symmetric product of an absolute $\lambda$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)\lambda$-Lipschitz retract. This answers a question of De Pauw about the target spaces of Almgren's $Q$-valued maps. We also prove that the space of nonempty subsets with at most $n$ points, equipped with the Hausdorff metric, is an absolute $(6n-5)\lambda$-Lipschitz retract. The main tool is a reconstruction theorem for bounded products of metric stars with finitely many rays, modulo a finite permutation group: these quotients are absolute $3$-Lipschitz retracts, independently of the group and the number of coordinates. A variant of this theorem, in which unordered sets of sign vectors take the place of the group action, handles finite subset spaces.

arXiv commentV2 covers finite subset spaces as well