Relative sectional number and the coincidence property

TL;DR

This work characterizes the coincidence property (CP) for a map $g:X\to Y$ in terms of a relative sectional number: CP holds if and only if $\mathrm{sec}_g(\pi_{2,1}^Y)=2$, linking CP to sectional category theory and configuration spaces. It introduces relative sectional theory, including $\mathrm{sec}_g(p)$, and extends these ideas to a relative notion of topological complexity $\mathrm{TC}_g(f)$, establishing bounds and relations with classical invariants. The results unify CP with topological robotics concepts and provide cohomological and geometric criteria (e.g., for targets like $D^m$) as well as implications for the contractibility of configuration spaces $F(Y,2)$. Applications include deducing when CP holds for spheres, projective spaces, and groups, and showing how CP constrains or is constrained by the (relative) motion planning problems encoded by $\mathrm{TC}_g$.

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 $π_{2,1}^Y:F(Y,2)\to Y$ 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 $π_{2,1}^Y$. 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.

Theorems & Definitions (49)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Definition 2.7
• Remark 2.8