Counterexamples to Knaster's conjecture
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.
scientific article; zbMATH DE number 1113442
scientific article