Maximal motion-planning complexity from positive simplicial volume
Let $M$ be a closed, connected, oriented aspherical manifold of dimension $n \ge 3$, and let $$\|M\| = \inf\Bigl\{\sum_j |a_j| : \sum_j a_j \sigma_j \text{ is a rational singular cycle representing } [M] \in H_n(M;\mathbb{Q})\Bigr\}$$ be its simplicial volume. Let $\operatorname{TC}(M)$ be Farber's topological complexity in the unreduced convention: the least number of open sets covering $M \times M$ on each of which the endpoint map $\gamma \mapsto (\gamma(0), \gamma(1))$, from the space of paths $[0,1] \to M$ with the compact-open topology, has a continuous section.
Farber's dimensional bound gives $\operatorname{TC}(M) \le 2n + 1$. Dranishnikov proved that a finitely generated torsion-free hyperbolic group of cohomological dimension $n$ has topological complexity $2n + 1$ in this convention; as Neofytidis notes, every closed negatively curved $n$-manifold therefore has $\operatorname{TC}(M) = 2n + 1$. Closed negatively curved manifolds have positive simplicial volume (Gromov; Inoue and Yano).
Problem. Suppose $\|M\| > 0$. Must $\operatorname{TC}(M) = 2n + 1$? No curvature assumption is made.
Source: Neofytidis (2025), Question 5.1(b), for aspherical n-manifolds (arXiv 2022). Stated by AIM — Open Applied Problems for closed, connected, oriented manifolds of dimension n ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- GT
- Keywords
- AIM Open Applied Problems, aspherical manifolds, simplicial volume, topological complexity
References
@article{neofytidis2025topological,
author = {Neofytidis, Christoforos},
title = {Topological complexity, asphericity and connected sums},
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volume = {9},
number = {1},
year = {2025},
pages = {10},
doi = {10.1007/s41468-025-00205-z},
eprint = {2212.08962},
archivePrefix = {arXiv},
}
@article{farber2003topological,
author = {Farber, Michael},
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number = {2},
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pages = {211--221},
doi = {10.1007/s00454-002-0760-9},
eprint = {math/0111197},
archivePrefix = {arXiv},
}
@article{dranishnikov2020hyperbolic,
author = {Dranishnikov, Alexander},
title = {On topological complexity of hyperbolic groups},
journal = {Proc. Amer. Math. Soc.},
volume = {148},
number = {10},
year = {2020},
pages = {4547--4556},
doi = {10.1090/proc/15094},
eprint = {1904.06720},
archivePrefix = {arXiv},
}
@article{gromov1982volume,
author = {Gromov, Michael},
title = {Volume and bounded cohomology},
journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.},
volume = {56},
year = {1982},
pages = {5--99},
doi = {10.5802/pmihes.3},
}
@article{inoue1982gromov,
author = {Inoue, Hisao and Yano, Koichi},
title = {The {G}romov invariant of negatively curved manifolds},
journal = {Topology},
volume = {21},
number = {1},
year = {1982},
pages = {83--89},
doi = {10.1016/0040-9383(82)90043-X},
}
@misc{aim-positive-volume-motion-planning,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Maximal motion-planning complexity from positive simplicial volume}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/524-positive-volume-motion-planning.md},
note = {Entry 524 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-00892, "Maximal motion-planning complexity from positive simplicial volume", clopen.solutions.