Sequential topological complexities (denoted $\text{TC}_m$ for each $m\ge 2$) are numerical homotopy invariants of topological spaces motivated by the motion planning problem in robotics. Given a space $X$, $\text{TC}_m(X)$ measures the discontinuity of planning a motion between any given sequence of $m$ points in $X$ for any robot whose configuration space is $X$. Usually, cohomological data of a space helps estimate its $\text{TC}_m$ values. In this talk, we focus on the case when our space $X$ is a symmetric product of a closed orientable surface. Using Macdonald’s description of the cohomology ring of these spaces, we completely determine all sequential topological complexities of all symmetric products of closed orientable surfaces. Our methods involve explicit computations of their Lusternik–Schnirelmann category and rational zero-divisor cup-lengths. Using our computations, we also verify the “TC-rationality conjecture” of Farber and Oprea for these spaces.

arXiv preprint: https://arxiv.org/abs/2503.04532