Relative sectional number and the coincidence property

TL;DR

This paper establishes a link between the relative sectional number of a projection in configuration spaces and the coincidence property in topology, introducing the concept of relative topological complexity of a map.

Contribution

It provides a new characterization of the coincidence property using sectional category, connecting coincidence theory with topological robotics and introducing relative topological complexity.

Findings

Characterizes the coincidence property via minimal open covers with local liftings.

Relates the coincidence property to the relative sectional number of a projection.

Introduces the notion of relative topological complexity of a map.

Abstract

For a Hausdorff space $Y$, a topological space $X$ and a map $g:X\to Y$, we present a connection between the relative sectional number of the first coordinate projection $\pi_{2,1}^Y:F(Y,2)\to Y$ 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 $f:X\to Y$, 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.