Relative Topological Complexity and Configuration Spaces

Given a space $X$, the topological complexity of $X$, 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 $Y_1$ and $Y_2$ of $X$, there is a "relative" version of topological complexity, denoted by $TC_X(Y_1\times Y_2)$, 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 $TC(X)$ before focusing on the case in which we have $Y_1=Y_2=C^n(Y)$, the configuration space of $n$ points in some space $Y$, and $X=C^n(Y\times I)$, the configuration space of $n$ points in $Y\times I$, where $I$ denotes the interval $[0,1]$. Our main result shows $TC_{C^n(Y\times I)}(C^n(Y)\times C^n(Y))$ is bounded above by $TC(Y^n)$ and under certain hypotheses is bounded below by $TC(Y)$.