Relative Topological Complexity and Configuration Spaces

The topological complexity of a space X, denoted by TC(X), can be viewed as the minimum number of “continuous rules” needed to describe how to move between any two points in X. Given subspaces Y1 and Y2 of X, there is a “relative” version of topological complexity, in which one only considers paths starting at a point y1 ∈ Y1 and ending at a point y2 ∈ Y2, but the path from y1 to y2 can pass through any point in X.We discuss general results that provide relative analogues of well-known results concerning TC(X) before focusing on configuration spaces.Our primary interest is the case inwhich configurations must start in some space Y1 and end in some space Y2, but the configurations have an extra degree of motion which allows them to move “above” Y1 ∪ Y2 throughout the intermediate stages. We show that in this case, the relative topological complexity is bounded above by TC(Y n) and with certain hypotheses is bounded below by TC(Y ), where Y = Y1 ∪ Y2.