Torsion in the directed-forest complex for digraphs with oriented cycles

Determine whether the complex of directed forests associated to a finite directed graph that contains oriented cycles has torsion in its integral homology; equivalently, decide if there exists a digraph with at least one oriented cycle whose directed-forest complex has a homology group with nontrivial torsion.

Background

The complex of directed forests is a monotone complex associated to digraphs. For digraphs without oriented cycles, this complex is shellable, implying torsion-free homology.

The authors point out that the status for digraphs with oriented cycles is unknown to them and pose an explicit question about the existence of torsion, highlighting a gap between acyclic and cyclic cases.

References

This observation leads to the following question, whose answer is not known to the authors: Does the complex of directed forests of graphs with oriented cycles contain torsion?

— The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology (2401.01248 - Caputi et al., 2 Jan 2024) in Section 4.1 (Multipath complexes and their torsion)