Neighboring mapping points theorem

Malyutin, Andrei V¹ ; Musin, Oleg R²

¹ St Petersburg Department of Steklov Institute of Mathematics, St Petersburg, Russia
² School of Mathematical and Statistical Sciences, University of Texas – Rio Grande Valley, Brownsville, TX, United States

Algebraic and Geometric Topology, Tome 23 (2023) no. 7, pp. 3043-3070

Résumé

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 < 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.

Détail

DOI : 10.2140/agt.2023.23.3043

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