Quantifying discontinuity

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.

17 pages, 3 figures. Revised version with updated author list; Sunhyuk Lim is no longer an author by mutual agreement