Quantifying discontinuity

TL;DR

This paper introduces a measure of discontinuity for functions from compact spaces to Euclidean spaces, providing quantitative bounds on nonembeddability and extending topological Tverberg theorem results.

Contribution

It defines a scale-invariant modulus of discontinuity and derives lower bounds, offering a quantitative approach to classical nonembeddability and Tverberg theorems.

Findings

Established lower bounds for discontinuity in non-embeddable spaces

Provided quantified nonembeddability results of Haefliger--Weber type

Extended bounds to simplicial complexes related to topological Tverberg theorem

Abstract

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ 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 $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.