Motion-planning complexity of symmetric products of non-orientable surfaces
Let $N_g = \#^g \mathbb{RP}^2$ be the closed non-orientable surface of genus $g \ge 2$, and for an integer $n \ge 2$ let $$X_{n,g} = \operatorname{SP}^n(N_g) = N_g^n / \mathfrak{S}_n$$ be its $n$-th symmetric product, the symmetric group permuting coordinates (coincident points allowed), with the quotient topology; it is a closed non-orientable $2n$-manifold. Use normalized topological complexity: $\operatorname{TC}(X)$ is the least $r$ such that $X \times X$ has an open cover $U_0, \dots, U_r$ with continuous maps $s_i : U_i \to C([0,1], X)$, the path space carrying the compact-open topology, satisfying $s_i(x, y)(0) = x$ and $s_i(x, y)(1) = y$.
Write $2^e \le n < 2^{e+1}$. González and Jauhari computed the mod-$2$ zero-divisor cup length of $X_{n,g}$, a lower bound for $\operatorname{TC}(X_{n,g})$, and deduced from a theorem of Cohen and Vandembroucq that $\operatorname{TC}(X_{n,g}) \le 4n - 1$. Together, $$L(n,g) \le \operatorname{TC}(X_{n,g}) \le 4n - 1, \qquad L(n,g) = \begin{cases} 2^{e+2} + g - 2, & g \le 2n - 2^{e+1} + 1, \\ 2^{e+1} + 2n - 1, & g \ge 2n - 2^{e+1} + 1, \end{cases}$$ the two expressions agreeing when $g = 2n - 2^{e+1} + 1$. In particular $\operatorname{TC}(X_{2^e,g}) = 2^{e+2} - 1$ for $e \ge 1$. Jauhari proved that the Lusternik–Schnirelmann category of $X_{n,g}$ is $2n$. For $g = 1$, $\operatorname{SP}^n(\mathbb{RP}^2) \cong \mathbb{RP}^{2n}$.
Problem. Determine $\operatorname{TC}(X_{n,g})$ for all $n, g \ge 2$. In particular, close the gap between the lower and upper bounds above when $n$ is not a power of two.
Source: Jauhari (2025), Section 6, who records that the topological complexity of these symmetric products is undetermined for n ≥ 2 (arXiv 2025). Stated for g ≥ 2 by AIM — Open Applied Problems, with the bounds of González and Jauhari (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- GT
- Keywords
- AIM Open Applied Problems, non-orientable surfaces, symmetric products, topological complexity
References
@article{jauhari2025symmetric,
author = {Jauhari, Ekansh},
title = {{LS}-category and sequential topological complexity of symmetric products},
journal = {J. Appl. Comput. Topol.},
volume = {9},
number = {4},
year = {2025},
pages = {25},
doi = {10.1007/s41468-025-00223-x},
eprint = {2503.04532},
archivePrefix = {arXiv},
}
@misc{gonzalez2026nonorientable,
author = {Gonz{\'a}lez, Jes{\'u}s and Jauhari, Ekansh},
title = {On the sequential topological complexity and the {LS}-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces},
year = {2026},
eprint = {2603.14788},
archivePrefix = {arXiv},
}
@article{cohen2021abelian,
author = {Cohen, Daniel C. and Vandembroucq, Lucile},
title = {On the topological complexity of manifolds with abelian fundamental group},
journal = {Forum Math.},
volume = {33},
number = {6},
year = {2021},
pages = {1403--1421},
doi = {10.1515/forum-2021-0094},
}
@misc{aim-nonorientable-symmetric-product-motion-planning,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Motion-planning complexity of symmetric products of non-orientable surfaces}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/533-nonorientable-symmetric-product-motion-planning.md},
note = {Entry 533 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00883, "Motion-planning complexity of symmetric products of non-orientable surfaces", clopen.solutions.