Quantifying discontinuity

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.

💡 Research Summary

The paper introduces a novel quantitative framework for studying non‑embeddability of compact spaces into Euclidean spaces. Classical non‑embeddability results (e.g., the van Kampen–Flores theorem, the non‑planarity of K₅, the Haefliger–Weber obstruction) are qualitative: they assert that no continuous injective map exists. However, an arbitrary injective map f into a lower‑dimensional Euclidean space can be made arbitrarily “flat” by scaling, so the usual modulus of discontinuity δ(f) fails to capture any meaningful obstruction.

To overcome this, the authors define a scale‑invariant modulus of discontinuity α(f). For a function f from a compact space X to ℝᵈ, consider the configuration space of ordered pairs of distinct points, Conf₂(X) = {(x,y) | x≠y}, equipped with the natural ℤ/2‑action swapping the coordinates. The map
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