On Knaster’s conjecture

Abstract:

Knaster’s conjecture is: given a continuous $g:{S^n} \to {E^m}$ and a set $\Delta$ of $n - m + 2$ distinct points $({q_1}, \ldots ,{q_{n - m + 2}})$ in ${S^n}$ there exists a rotation $r:{S^n} \to {S^n}$ such that \[ g(r({q_1})) = g(r({q_2})) = \cdots = g(r({q_{n - m + 2}})).\] We prove a stronger statement about a smaller class of functions. If $f:{S^n} \to {E^n}$ we write $f = ({f_1},{f_2}, \ldots ,{f_n})$ where ${f_i}:{S^n} \to {E^1}$, and put ${F_i} = ({f_1}, \ldots ,{f_i}):{S^n} \to {E^i}$ so that ${F_n} = f$. The level surface of ${F_i}$ in ${S^n}$ containing $x$ is ${l_i}(x) = \{ y \in {S^n}|{F_i}(x) = {F_i}(y)\}$. Theorem. Given an $(n + 1)$-frame $\Delta \subset {S^n}$ and a real-analytic function $f:{S^n} \to {E^n}$ such that each ${l_i}(x)$ is either a point or a topological $(n - i)$-sphere, there exist at least ${2^{n - 1}}$ distinct rotations $r:{S^n} \to {S^n}$ such that \[ {f_i}(r({q_1})) = \cdots = {f_i}(r({q_{n - i + 2}})),\quad i = 1,2, \ldots ,n,\] for each rotation. It follows that for $m = 1,2, \ldots ,n$, \[ {F_m}(r({q_1})) = {F_m}(r({q_2})) = \cdots = {F_m}(r({q_{n - m + 2}})),\] so that the functions ${F_m}:{S^n} \to {E^m}$ satisfy Knaster’s conjecture simultaneously. Given ${F_i}$, the definition of $f$ can be completed in many ways by choosing ${f_{i + 1}}, \cdots ,{f_n}$, each way giving rise to different rotations satisfying the Theorem. A suitable homotopy of $f$ which changes ${f_n}$ slightly will give locally a continuum of rotations $r$ each of which satisfies Knaster’s conjecture for ${F_{n - 1}}$. In general there exists an $(n - m)$-dimensional family of rotations satisfying Knaster’s conjecture for ${F_m}$.