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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Knaster's conjecture


Author: R. P. Jerrard
Journal: Trans. Amer. Math. Soc. 170 (1972), 385-402
MSC: Primary 55C20; Secondary 54H25
MathSciNet review: 0309101
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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

$\displaystyle g(r({q_1})) = g(r({q_2})) = \cdots = g(r({q_{n - m + 2}})).$

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}\vert{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

$\displaystyle {f_i}(r({q_1})) = \cdots = {f_i}(r({q_{n - i + 2}})),\quad i = 1,2, \ldots ,n,$

for each rotation. It follows that for $ m = 1,2, \ldots ,n$,

$\displaystyle {F_m}(r({q_1})) = {F_m}(r({q_2})) = \cdots = {F_m}(r({q_{n - m + 2}})),$

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

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0309101-7
PII: S 0002-9947(1972)0309101-7
Keywords: Knaster's conjecture, sphere maps, coincidence theorems, set-valued functions
Article copyright: © Copyright 1972 American Mathematical Society