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arXiv · 2610.10976

Symmetric products and finite subset spaces of absolute Lipschitz retracts

Leonid V. Kovalev
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Abstract

We prove that the $Q$th symmetric product of an absolute $λ$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)λ$-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 $(6n-5)λ$-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.

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Leonid V. Kovalev. 2026-10-11. Symmetric products and finite subset spaces of absolute Lipschitz retracts. https://arxiv.org/abs/2610.10976

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