Compact orbit covers of aspherical manifolds
Let $M$ be a closed connected smooth aspherical manifold, so that its universal cover $\widetilde M$ is contractible, and let $G = \pi_1(M)$ act on $\widetilde M$ by deck transformations. When $M$ has constant curvature, Hua, Manin, Queer and Wang observed that the Voronoi cells of a $G$-orbit in $\widetilde M$ are compact, convex translates of one another that tile $\widetilde M$, so that all their nonempty intersections are convex and hence contractible. Since these cells and their intersections are convex, the union of any set of cells is homotopy equivalent to the nerve of the cover the cells form.
Problem. Is there always a compact set $K \subseteq \widetilde M$ such that $$\widetilde M = \bigcup_{g \in G} gK$$ and, for every finite nonempty $F \subseteq G$, the intersection $\bigcap_{g \in F} gK$ is either empty or contractible? (Taking $F$ a singleton, $K$ itself must be contractible.)
Source: Hua, Manin, Queer and Wang (2024), Conjecture 1.4; their Conjecture 6.8 asks, for closed aspherical manifolds, for a closed fundamental domain with this property. Stated by AIM — Open Applied Problems for closed connected smooth aspherical manifolds.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GT — Geometric Topology
- Secondary
- AT, GR
- Keywords
- AIM Open Applied Problems, aspherical manifolds, universal covers
References
@article{hua2024,
author = {Hua, Dongming Merrick and Manin, Fedor and Queer, Tahda and Wang, Tianyi},
title = {Local behavior of the {E}den model on graphs and tessellations of manifolds},
journal = {J. Appl. Comput. Topol.},
volume = {8},
number = {6},
year = {2024},
pages = {1607--1647},
doi = {10.1007/s41468-023-00153-6},
eprint = {2212.14146},
archivePrefix = {arXiv},
}
@misc{aim-aspherical-compact-orbit-cover,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Compact orbit covers of aspherical manifolds}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/646-aspherical-compact-orbit-cover.md},
note = {Entry 646 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 It will not change if the problem is solved, reclassified or withdrawn.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-00848, "Compact orbit covers of aspherical manifolds", clopen.solutions.