Relative Topological Complexity and Configuration Spaces

Abstract

The topological complexity of a space X,  denoted by ${{\,\mathrm{\mathrm {TC}}\,}}(X)$, can be viewed as the minimum number of “continuous rules” needed to describe how to move between any two points in X. Given subspaces $Y_1$ and $Y_2$ of X,  there is a “relative” version of topological complexity, in which one only considers paths starting at a point $y_1\in Y_1$ and ending at a point $y_2\in Y_2,$ but the path from $y_1$ to $y_2$ can pass through any point in X. We discuss general results that provide relative analogues of well-known results concerning ${{\,\mathrm{\mathrm {TC}}\,}}(X)$ before focusing on configuration spaces. Our primary interest is the case in which configurations must start in some space $Y_1$ and end in some space $Y_2,$ but the configurations have an extra degree of motion which allows them to move “above” $Y_1\cup Y_2$ throughout the intermediate stages. We show that in this case, the relative topological complexity is bounded above by ${{\,\mathrm{\mathrm {TC}}\,}}(Y^n)$ and with certain hypotheses is bounded below by ${{\,\mathrm{\mathrm {TC}}\,}}(Y),$ where $Y=Y_1\cup Y_2.$