Relative Topological Complexity and Configuration Spaces

TL;DR

This paper introduces a relative version of topological complexity for subspaces of a space, applies it to configuration spaces, and establishes bounds relating it to the topological complexity of the underlying space.

Contribution

It develops a relative topological complexity framework and derives bounds for configuration spaces, extending known results to new relative settings.

Findings

Bounded the relative topological complexity of configuration spaces by the topological complexity of the base space.

Established conditions under which the relative topological complexity is bounded below by the topological complexity of the base space.

Provided general results that extend classical topological complexity results to relative and configuration space contexts.

Abstract

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 $TC(Y)$.