The centre-rank formula for aspherical motion planning
Let $M$ be a closed, connected, oriented aspherical manifold of dimension $n \ge 3$, let $G = \pi_1(M)$, and let $$z = \dim_{\mathbb{Q}}\bigl(Z(G) \otimes_{\mathbb{Z}} \mathbb{Q}\bigr)$$ be the rank of its centre $Z(G)$. 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 proved $\operatorname{TC}(X) \le 2\dim X + 1$ for path-connected paracompact $X$ and $\operatorname{TC}(T^n) = n + 1$. Dranishnikov proved that a finitely generated torsion-free hyperbolic group of cohomological dimension $n \ge 2$ has topological complexity $2n + 1$ in this convention, as do therefore closed hyperbolic $n$-manifolds. Grant proved $\operatorname{TC}(G) \le 2\operatorname{rank}(G) + 1 - \operatorname{rank}(Z(G))$ for every finitely generated torsion-free nilpotent group $G$, with $\operatorname{rank}$ the Hirsch rank.
Problem. Does every such manifold satisfy $$\operatorname{TC}(M) = 2n + 1 - z?$$ No curvature assumption is made.
Source: Neofytidis (2025), Question 5.8, for aspherical n-manifolds. 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
- GR, GT
- Keywords
- AIM Open Applied Problems, aspherical manifolds, centre of the fundamental group, topological complexity
References
@article{neofytidis2025topological,
author = {Neofytidis, Christoforos},
title = {Topological complexity, asphericity and connected sums},
journal = {J. Appl. Comput. Topol.},
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},
title = {Topological complexity of motion planning},
journal = {Discrete Comput. Geom.},
volume = {29},
number = {2},
year = {2003},
pages = {211--221},
doi = {10.1007/s00454-002-0760-9},
eprint = {math/0111197},
archivePrefix = {arXiv},
}
@article{dranishnikov2020topological,
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{grant2012topological,
author = {Grant, Mark},
title = {Topological complexity, fibrations and symmetry},
journal = {Topology Appl.},
volume = {159},
number = {1},
year = {2012},
pages = {88--97},
doi = {10.1016/j.topol.2011.07.025},
eprint = {1104.2755},
archivePrefix = {arXiv},
}
@misc{aim-aspherical-motion-planning-center,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The centre-rank formula for aspherical motion planning}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/513-aspherical-motion-planning-center.md},
note = {Entry 513 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct a closed connected oriented flat five-manifold whose fundamental group has centre of rank two and whose unreduced topological complexity is at least ten. Since its universal cover is Euclidean space, this gives a negative answer to the proposed formula $\TC(M)=2\dim M+1-\rank Z(\pi_1M)$ for closed oriented aspherical manifolds. The lower bound follows from a nonzero product of nine degree-one …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00780, "The centre-rank formula for aspherical motion planning", clopen.solutions.