Relative sectional number and the coincidence property

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 ᵢ\ 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.