OP-00779 Solution claimed

Maximal motion-planning complexity of aspherical connected sums

Let $M$ and $N$ be closed, connected, oriented aspherical manifolds of the same dimension $n \ge 3$, and let $M \# N$ be their oriented connected sum; the connected sum itself need not be aspherical. For a space $X$ let $\operatorname{TC}(X)$ be Farber's topological complexity in the unreduced convention: the least number of open sets covering $X \times X$ on each of which the endpoint map $\gamma \mapsto (\gamma(0), \gamma(1))$, from the space of paths $[0,1] \to X$ with the compact-open topology, has a continuous section.

Farber proved $\operatorname{TC}(X) \le 2\dim X + 1$ for path-connected paracompact $X$. Neofytidis proved that $\operatorname{TC}(M \# L) = 9$ whenever $M$ is a closed oriented negatively curved $4$-manifold with nonzero second Betti number and $L$ is any closed oriented $4$-manifold.

Problem. Is $\operatorname{TC}(M \# N) = 2n + 1$ for every such pair $M, N$?

Source: Neofytidis (2025), Question 5.5(b). Stated by AIM — Open Applied Problems for closed, connected, oriented summands of dimension n ≥ 3.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AT — Algebraic Topology
Secondary: GT
Keywords: AIM Open Applied Problems, aspherical manifolds, connected sums, 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},
}
@misc{aim-aspherical-connected-sum-complexity,
  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 of aspherical connected sums}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/514-aspherical-connected-sum-complexity.md},
  note         = {Entry 514 at commit 24edaad, 2026-10-15},
}

Resolution claims

Maximal topological complexity of connected sums of aspherical manifolds v1 GitHub ↗

Kaiwen Jin

Let $M$ and $N$ be closed, connected, oriented aspherical manifolds of the same dimension $n\geq 3$. We prove that their connected sum has ordinary unreduced topological complexity $2n+1$. Thus joining two aspherical summands always attains the dimensional upper bound, although the connected sum is not required to be aspherical. The proof detects the two ordered cross products of the summands' …

Full resolution · not verified · fidelity not established · posted on GitHub 13 Oct 2026

Status history

When	Change	By	Reason
2026-10-18	Needs verification → Solution claimed	—	Recorded here: AIM @ 24edaad, paper/aspherical-connected-sums.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00779, "Maximal motion-planning complexity of aspherical connected sums", clopen.solutions.