2025/05/27 by Hannah Alpert, Arka Banerjee, Alpert, Hannah +3
Computer Science · Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.DG #math.MG
paper · pdf · doi:10.48550/arxiv.2505.21126
openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We investigate the following question: Do there exist Riemannian polyhedra X such that the 1-Uryson width of their universal covers UW1(\widetildeX) is bounded but UW1(X) is arbitrarily large? We rule out two specific cases: when π1(X) is virtually cyclic and when X is a Riemannian surface. More specifically, we show that if X is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover \widetildeX. Precisely: UW1(X) ≤ 6 ⋅ UW1(\widetildeX). We show that if X is a Riemannian surface with boundary then UW1(X) ≤ UW1(\widetildeX). Furthermore, we show that if there exist spaces X for which UW1(\widetildeX) is bounded while UW1(X) is arbitrarily large, then such examples must already appear in low dimensions. In particular, such X can be found among Riemannian 2-complexes.