RELATIVE SECTIONAL NUMBER AND THE COINCIDENCE PROPERTY

Ipanaque Zapata C.A.; Torres Estrella F.A. · Topological Methods in Nonlinear Analysis (2025) · DOI: 10.12775/tmna.2025.016

Resumen

For a Hausdorff space Y, a topological space X and a map g: X → Y, we present a connection between the relative sectional number of the first coordinate projection π2,1Y: F (Y, 2) → 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 → 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 {Ui }1≤i≤n of X such that each Ui admits a local lifting for g with respect to π2,1Y 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.

Instituciones: Universidad Nacional Mayor de San Marcos