LS-category and sequential topological complexity of symmetric products

Abstract

The $n$-th symmetric product of a topological space $X$ is the orbit space of the natural action of the symmetric group $S_n$ on the product space $X^n$. In this paper, we compute the sequential topological complexities of (finite products of) the symmetric products of closed orientable surfaces, thereby verifying the rationality conjecture of Farber and Oprea for these spaces. Additionally, we determine the Lusternik-Schnirelmann category of (finite products of) the symmetric products of closed non-orientable surfaces. More generally, we provide lower bounds to the LS-category and the sequential topological complexities of the symmetric products of finite CW complexes $X$ in terms of the cohomology of $X$ and its products. On the way, we also obtain new lower bounds to the sequential distributional complexities of continuous maps and study the homotopy groups of the symmetric products of closed surfaces.