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Quantifying discontinuity

Henry Adams, Florian Frick, Michael Harrison, Sunhyuk Lim, Nikola Sadovek, Matt Superdock

TL;DR

The paper develops a scale-invariant modulus of discontinuity $\alpha$ for injective maps $f:X\to\mathbb{R}^d$ to quantify nonembeddability, connecting this obstruction to Haefliger–Weber-type results and Vietoris–Rips topology. It provides general lower bounds $\alpha(f)\ge r_{d-1}=\arccos(-1/d)$ when no $\mathbb{Z}/2$-equivariant obstruction exists, and refined bounds $\alpha(f)\ge c_{d-1,k}$ via the coindex of the configuration space; it also clarifies when $\alpha$ detects discontinuity. Extending to almost $r$-embeddings, the paper defines $\alpha^{(r)}$ and proves quantified topological Tverberg theorems: for prime-power $r$, almost $r$-injective maps $\Delta_{(r-1)(d+1)}\to\mathbb{R}^d$ obey $\alpha^{(r)}(f)\ge \arccos(-1/(d(r-1)))$, revealing a robust, geometric obstruction framework. By relating $\alpha^{(r)}$ to $\delta(f)$ and $\kappa^{(r)}(f)$, the authors derive explicit tradeoffs between discontinuity and near-violation of almost $r$-injectivity, yielding concrete quantitative bounds in the Tverberg context. Overall, the work furnishes scale-free obstructions to embeddings and almost-embeddings with potential impact on combinatorial topology and metric geometry.

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.

Quantifying discontinuity

TL;DR

The paper develops a scale-invariant modulus of discontinuity for injective maps to quantify nonembeddability, connecting this obstruction to Haefliger–Weber-type results and Vietoris–Rips topology. It provides general lower bounds when no -equivariant obstruction exists, and refined bounds via the coindex of the configuration space; it also clarifies when detects discontinuity. Extending to almost -embeddings, the paper defines and proves quantified topological Tverberg theorems: for prime-power , almost -injective maps obey , revealing a robust, geometric obstruction framework. By relating to and , the authors derive explicit tradeoffs between discontinuity and near-violation of almost -injectivity, yielding concrete quantitative bounds in the Tverberg context. Overall, the work furnishes scale-free obstructions to embeddings and almost-embeddings with potential impact on combinatorial topology and metric geometry.

Abstract

Given a compact space that does not admit an embedding (an injective continuous function) into , we study the ''degree'' of discontinuity that any injective function 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 -embedding in , thus obtaining a quantified version of the topological Tverberg theorem.

Paper Structure

This paper contains 8 sections, 25 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Injective functions
  4. Scale-invariant modulus of discontinuity $\alpha$
  5. Suitability of $\alpha$ for detecting discontinuity
  6. Van Kampen--Flores theorem
  7. Almost r-embeddings
  8. Relating $\alpha^{(r)}$ to the modulus of discontinuity

Key Result

Theorem 1.1

Any injective function $f \colon \mathbb{R}\mathrm{P}^{2^k} \to \mathbb{R}^{2^{k+1}-1}$ satisfies $\alpha(f) \ge r_{2^{k+1}-2} = \arccos{\left(\frac{-1}{2^{k+1}-1}\right)}$.

Figures (3)

  • Figure 1: Two close pairs $(x,y), (x',y') \in \mathrm{Conf}_2(X)$ which induce a large angle between $f(x)-f(y)$ and $f(x')-f(y')$.
  • Figure 2: Discontinuity of $f$ at $x$.
  • Figure 3: The three vectors in the triple $\mathrm{Conf}_3^{\Delta}(f)(x_1,x_2,x_3)$ for $r=3$ and $d=2$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Quantified topological Tverberg
  • Theorem 2.1: Theorems 1.3 and 7.6 of GH-BU-VR
  • Theorem 3.1: Haefliger Haefliger, Weber Weber
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 35 more