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JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2023 Volume 20, Issue 1, Pages 165–182 (Mi semr1579)

Geometry and topology

Hopf-type theorems for $f$-neighbors

A. V. Malyutinab, I. M. Shirokova

a St. Petersburg Department of Steklov Mathematical Institute, Fontanka, 27, 191023, St. Petersburg, Russia
b St. Petersburg State University, Universitetskaya Emb., 13B, 199034, St. Petersburg, Russia

Abstract: We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk–Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this paper we study variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. First, we generalize the Hopf theorem in a quantitative sense. Then we investigate the case of maps $f\colon M \to \mathbb{R}^m$ with $n < m$ and introduce several notions of varied types of $f$-neighbors, which is a pair of distinct points in $M$ such that $f$ takes it to a ‘small’ set of some type. Next for each type, we ask what distances on $M$ are realized as distances between $f$-neighbors of this type and study various characteristics of this set of distances. One of our main results is as follows. Let $f\colon M \to \mathbb{R}^{m}$ be a continuous map. We say that two distinct points $a$ and $b$ in $M$ are visual $f$-neighbors if the segment in $\mathbb{R}^{m}$ with endpoints $f(a)$ and $f(b)$ intersects $f(M)$ only at $f(a)$ and $f(b)$. Then the set of distances that are realized as distances between visual $f$-neighbors is infinite.

Keywords: Borsuk–Ulam type theorems, the Hopf theorem, winding number, locally injective.

UDC: 514.17, 514.172, 515.14

MSC: 47H10, 54H25, 55M20

Received August 29, 2022, published March 1, 2023

Language: English

DOI: 10.33048/semi.2023.20.015



© Steklov Math. Inst. of RAS, 2024