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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On Knaster’s conjecture
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by R. P. Jerrard PDF
Trans. Amer. Math. Soc. 170 (1972), 385-402 Request permission

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 ${F_m}$.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 385-402
  • MSC: Primary 55C20; Secondary 54H25
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0309101-7
  • MathSciNet review: 0309101