Symmetric products and finite subset spaces of absolute Lipschitz retracts
Abstract
We prove that the Qth 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.
Related
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BibTeX
@misc{kovalev2026symmetric,
title = {{Symmetric products and finite subset spaces of absolute Lipschitz retracts}},
author = {Leonid V. Kovalev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10976},
eprint = {2610.10976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10976},
}
RIS
TY - UNPB
TI - Symmetric products and finite subset spaces of absolute Lipschitz retracts
AU - Leonid V. Kovalev
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.10976
UR - https://arxiv.org/abs/2610.10976
AB - We prove that the Qth 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.
ER -