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Neighboring mapping points theorem

Andrei V Malyutin and Oleg R Musin

Algebraic & Geometric Topology 23 (2023) 3043–3070

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 𝕊m 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 Δn of the n–dimensional simplex Δn to a contractible metric space M. Then Δn contains a subset E such that E (is “large” in the sense that it) intersects all facets of Δn and the image f(E) (is “small” in the sense that it) is either a singleton or a subset of the boundary B of a metric ball B M whose interior does not meet f(Δn).

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.

Borsuk–Ulam theorem, topological Radon theorem, Hopf theorem, KKM lemma
Mathematical Subject Classification
Primary: 55M20, 55M25, 55P05
Received: 13 April 2021
Revised: 8 January 2022
Accepted: 3 February 2022
Published: 26 September 2023
Andrei V Malyutin
St Petersburg Department of Steklov Institute of Mathematics
St Petersburg
Oleg R Musin
School of Mathematical and Statistical Sciences
University of Texas – Rio Grande Valley
Brownsville, TX
United States

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