Counterexamples to Knaster's conjecture
type: Item
instance of: scholarly article
title: Counterexamples to Knaster's conjecture
zbMATH Open document ID: 0887.55004
DOI: 10.1016/S0040-9383(97)00029-3
author: William Chen
published in: Topology
publication date: 1998-02-05T00:00:00Z
review text: In 1944, Hopf proved that if \(f:S^n \to \mathbb{R}^n\) is a map and \(s_1\), \(s_2\in S^n\) are any points, then there is a rotation \(\rho \in \text{SO} (n+1)\) such that \(f\rho (s_1)= f\rho (s_2)\). This result generalizes the Borsuk-Ulam Theorem, which is the case that \(s_1\) and \(s_2\) are antipodal points. In 1947, Knaster conjectured that Hopf's result would extend to a map \(f:S^n\to \mathbb{R}^m\) and \(k=n- m+2\) points \(s_1, \dots, s_k\), that is, there would exist a rotation \(\rho \in \text{SO} (n+1)\) such that \(f\rho (s_1)= \cdots =f\rho (s_k)\). Knaster's Conjecture was disproved in 1986 by Makeev, who showed that it fails whenever \(\dim V_{n+ 1,d} <m (k-1)\), where \(V_{i,j}\) is the Stiefel manifold of \(i\)-dimensional orthonormal frames in \(\mathbb{R}^j\) and \(d\) is the dimension of the configuration of \(k\) points. Makeev conjectured that the conclusion of Knaster's Conjecture would hold otherwise, that is, whenever \(\dim V_{n+1,d} \geq m(k-1)\). In this paper, the author constructs counterexamples to demonstrate that Knaster's Conjecture fails for all \(n>m>2\) and it follows that Makeev's Conjecture is also incorrect. The paper is well-written, the counterexamples are established with only the use of elementary methods so the arguments are self-contained, and it concludes with an interesting discussion of further questions in this area.
Mathematics Subject Classification ID: 55M99
zbMATH DE Number: 1113442
zbMATH Keywords: Borsuk-Ulam theorem
zbMATH Keywords: Stiefel manifold
Wikidata QID: Q122957880
reviewed by: Robert F. Brown
MaRDI profile type: Publication
full work available at URL: s0040-9383(97)00029-3
OpenAlex ID: W1976400744
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description: scientific article; zbMATH DE number 1113442