LS-category and sequential topological complexity of symmetric products
Ekansh Jauhari
TL;DR
The paper advances the understanding of sequential topological complexity $\mathsf{TC}_m$ and Lusternik–Schnirelmann category $\operatorname{cat}$ for symmetric products $SP^n(X)$, focusing on finite CW complexes and, in depth, closed orientable and non-orientable surfaces. It develops and employs a refined cohomological framework built around special zero-divisors $\mathsf{szcl}^m_Q(X)$ to derive sharp lower bounds that match classical upper bounds, yielding exact values such as $\mathsf{TC}_m(SP^n(M_g))=2m n$ for $n\le g$ and $\mathsf{TC}_m(SP^n(M_g))=m(n+g)$ for $n>g$, with corresponding $\operatorname{cat}$-values. The work confirms the Farber–Oprea rationality conjecture for the TC-generating function for symmetric products of orientable surfaces and their finite products, and establishes LS-logarithmic and TC_m-logarithmic behavior in these families, providing a robust toolkit for analyzing TC and LS-type invariants of polyhedral products. Additionally, it determines $\operatorname{cat}(SP^n(N_g))=2n$ for all $n,g$, explores homotopy groups and spin structures of universal covers, and develops cohomological descriptions (including the $\mathbb{Z}_2$-cohomology of $SP^n(N_g)$) that underlie these results. Overall, the paper combines cohomological machinery, diagonal embeddings, and distributional variants to yield new, sharp results for symmetric products of both orientable and non-orientable surfaces, with implications for motion-planning invariants in geometric topology.
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.