Neighboring mapping points theorem

Authors

Andrei V. Malyutin
Oleg R. Musin, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

9-26-2023

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 Sm→Rn with m

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.

Comments

© 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

Recommended Citation

Malyutin, Andrei V., and Oleg R. Musin. "Neighboring mapping points theorem." Algebraic & Geometric Topology 23, no. 7 (2023): 3043-3070. https://doi.org/10.2140/agt.2023.23.3043

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Algebraic & Geometric Topology

DOI

10.2140/agt.2023.23.3043