Relative sectional number and the coincidence property - Zapata, Cesar A. Ipanaque; Estrella, Felipe A. Torres (Topological Methods in Nonlinear Analysis, 2025)
For a Hausdorff space Y, a topological space X and a map gX→ Y, we present a connection between the relative sectional number of the first coordinate projection π_2,1^YF(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 fX→ 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≤ i≤ 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.