Relative sectional number and the coincidence property

Abstract

For a Hausdorff space $Y$, a topological space $X$ and a map $g\colon X\to Y$, we present a connection between the relative sectional number of the first coordinate projection $\pi_{2,1}^Y\colon F(Y,2)\to Y$ with respect to $g$, and the coincidence property (CP) for $(X,Y;g)$, where $F(Y,2)$ stands for the ordered configuration space of $2$ distinct points on $Y$, and $(X,Y;g)$ has the coincidence property CP if, for every map $f\colon X\to Y$, there is a point $x$ of $X$ such that $f(x)=g(x)$. Explicitly, we demonstrate that $(X,Y;g)$ has the CP if and only if 2 is the minimal cardinality of open covers $\{U_i\}_{1\leq i\leq n}$ of $X$ such that each $U_i$ admits a local lifting for $g$ with respect to $\pi_{2,1}^Y$ This characterization connects a standard problem in coincidence theory to current research trends in sectional category and topological robotics. Motivated by this connection, we introduce the notion of relative topological complexity of a map.