Torsion in higher homology of unordered graph configuration spaces

Determine whether torsion can appear in homology groups of unordered configuration spaces of graphs beyond the first integral homology group.

Background

The paper contrasts anchored configuration spaces with standard graph configuration spaces. It notes that torsion is known to occur in the first integral homology groups of unordered graph configuration spaces and that this torsion detects planarity. The unresolved issue is whether torsion occurs in any other homological degree.

References

For example, while it has been known that the first integral homology groups of unordered configurations can have torsion (and this torsion actually detects planarity of the graph) for at least a decade , there has still not been confirmation whether torsion can appear in any other homology group.

— Universality in the algebra and topology of cographs  (2609.04554 - Mamun et al., 3 Sep 2026) in Section 5.1, subsection “Anchored configuration spaces of cographs”