Homotopy types for arbitrary relative contact complexes

Generalise the homotopy-type description of the relative contact complex of a 2-connected quasi-median graph from prism-covering star-covering collections of gated subgraphs to arbitrary prism-covering collections of gated subgraphs.

Background

The paper determines the homotopy type of relative contact complexes when the collection of gated subgraphs consists of prisms or is star-covering. It does not resolve the intermediate cases, including relative contact complexes lying between crossing and contiguity complexes. The authors explain that fibres can create intersections not represented by simplices in the relative contact complex, and then formulate the extension to arbitrary prism-covering collections as a problem.

References

However, in full generality, three hyperplanes may have a fibre in common while only two of them are adjacent in the relative contact complex. As a consequence, one is tempted to both keep and remove the fibre from our subspaces. As a concrete example, our results do not allow us to identify the homotopy type of the coset intersection complex of the right-angled Artin group illustrated on the left.

— Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups  (2503.08411 - Abbott et al., 11 Mar 2025) in Section 6, paragraph “Other relative contact complexes,” Problem