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Borsuk-Ulam type theorems for metric spaces

Arseniy Akopyan, Roman Karasev, Alexey Volovikov

TL;DR

This work extends classical Borsuk–Ulam and Hopf-type results to metric and open-manifold settings by employing Gromov’s contraction in the space of cycles and short path maps. The authors construct a cycle-valued map $f^c$ and leverage its nontrivial homology class to force coincidences $f(x)=f(y)$ with controlled distance, yielding a broad Borsuk–Ulam type theorem for arbitrary codomains and new Hopf-type results under injectivity-radius constraints. They further generalize to multivalued maps, using cycle maps and odd-degree pushforwards to obtain joint distance-coincidence conclusions, and discuss corollaries for ellipsoids and Lipschitz maps. Overall, the paper links topological coincidence phenomena with geometric constraints (width/waist) and broadens applicability to metric contexts and multivalued mappings, with potential ham-sandwich-type implications.

Abstract

In this paper we study the problems of the following kind: For a pair of topological spaces $X$ and $Y$ find sufficient conditions that under every continuous map $f : X\to Y$ a pair of sufficiently distant points is mapped to a single point.

Borsuk-Ulam type theorems for metric spaces

TL;DR

This work extends classical Borsuk–Ulam and Hopf-type results to metric and open-manifold settings by employing Gromov’s contraction in the space of cycles and short path maps. The authors construct a cycle-valued map and leverage its nontrivial homology class to force coincidences with controlled distance, yielding a broad Borsuk–Ulam type theorem for arbitrary codomains and new Hopf-type results under injectivity-radius constraints. They further generalize to multivalued maps, using cycle maps and odd-degree pushforwards to obtain joint distance-coincidence conclusions, and discuss corollaries for ellipsoids and Lipschitz maps. Overall, the paper links topological coincidence phenomena with geometric constraints (width/waist) and broadens applicability to metric contexts and multivalued mappings, with potential ham-sandwich-type implications.

Abstract

In this paper we study the problems of the following kind: For a pair of topological spaces and find sufficient conditions that under every continuous map a pair of sufficiently distant points is mapped to a single point.

Paper Structure

This paper contains 6 sections, 10 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. A Borsuk--Ulam type theorem for metric spaces
  3. Space of cycles and the proof of Theorem \ref{['theorem:bu-paths']}
  4. Classical Hopf type results
  5. New Hopf type results
  6. A Borsuk--Ulam--Hopf-type theorem for multivalued maps

Key Result

Theorem 1.1

Under any continuous map $f : S^n\to \mathbb R^n$ some two opposite points are mapped to a single point.

Theorems & Definitions (28)

  • Theorem 1.1: K. Borsuk, S. Ulam, 1933
  • Theorem 1.2: H. Hopf, 1944
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Remark 3.1
  • ...and 18 more