Neighboring mapping points theorem

Malyutin, Andrei V; Musin, Oleg R

doi:10.2140/agt.2023.23.3043

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Neighboring mapping points theorem

Andrei V Malyutin and Oleg R Musin

Algebraic & Geometric Topology 23 (2023) 3043–3070

DOI: 10.2140/agt.2023.23.3043

Abstract

We introduce and study a new family of theorems extending the class of Borsuk–Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form “the image of a subset that is large in some sense is a singleton” with requirements of the milder form “the image of a subset that is large in some sense is a subset that is small in some sense”. This approach covers the case of mappings $𝕊^{m} \to ℝ^{n}$ with $m < n$ and extends to wider classes of spaces.

An example of a statement from this new family is the following theorem. Let $f$ be a continuous map of the boundary $\partial Δ^{n}$ of the $n$ –dimensional simplex $Δ^{n}$ to a contractible metric space $M$ . Then $\partial Δ^{n}$ contains a subset $E$ such that $E$ (is “large” in the sense that it) intersects all facets of $Δ^{n}$ and the image $f (E)$ (is “small” in the sense that it) is either a singleton or a subset of the boundary $\partial B$ of a metric ball $B \subset M$ whose interior does not meet $f (\partial Δ^{n})$ .

We generalize this theorem to noncontractible normal spaces via covers and deduce a series of its corollaries. Several of these corollaries are similar to the topological Radon theorem.

Keywords

Borsuk–Ulam theorem, topological Radon theorem, Hopf theorem, KKM lemma

Mathematical Subject Classification

Primary: 55M20, 55M25, 55P05

References

Publication

Received: 13 April 2021

Revised: 8 January 2022

Accepted: 3 February 2022

Published: 26 September 2023

Authors

Andrei V Malyutin
St Petersburg Department of Steklov Institute of Mathematics St Petersburg Russia

Oleg R Musin
School of Mathematical and Statistical Sciences University of Texas – Rio Grande Valley Brownsville, TX United States

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Volume 23, issue 7 (2023)