Sectional category and the fixed point property

Abstract

In this work we exhibit an unexpected connection between sectional category theory and the fixed point property. On the one hand, a topological space $X$ is said to have \textit{the fixed point property} (FPP) if, for every continuous self-map $f$ of $X$, there is a point $x$ of $X$ such that $f(x)=x$. On the other hand, for a continuous surjection $p:E\to B$, the \textit{standard sectional number} $sec_{\text{op}}(p)$ is the minimal cardinality of open covers $\{U_i\}$ of $B$ such that each $U_i$ admits a continuous local section for $p$. Let $F(X,k)$ denote the configuration space of $k$ ordered distinct points in $X$ and consider the natural projection $\pi_{k,1}:F(X,k)\to X$. We demonstrate that a space $X$ has the FPP if and only if $sec_{\text{op}}(\pi_{2,1})=2$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.