Quantifying discontinuity

2025/11/10 by Henry Adams, Adams, Henry, Florian Frick +8 · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2511.07636

Abstract

Given a compact space X that does not admit an embedding (an injective continuous function) into ℝ^d, we study the ''degree'' of discontinuity that any injective function X → ℝ^d must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost r-embedding in ℝ^d, thus obtaining a quantified version of the topological Tverberg theorem.

Citations

• The G-Gromov-Hausdorff Distance and Equivariant Topology
• Vietoris thickenings and complexes are weakly homotopy equivalent
• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
• Vietoris-Rips Metric Thickenings of the Circle
• The topology of projective codes and the distribution of zeros of odd maps
• Best constants for Lipschitz embeddings of metric spaces into c₀

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• An optimal Brouwer's fixed point theorem for discontinuous functions

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