2024/08/14 by Zapata, Cesar A. Ipanaque, Estrella, Felipe A. Torres
#Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2408.07316
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 (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\ of X such that each Ui admits a local lifting for g with respect to π2,1Y. This characterisation 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.