Representation of d-Leray complexes by good covers

Construct, for every d-Leray complex K, a d-dimensional simplicial complex Y with vanishing d-dimensional homology and a good cover F in Y whose nerve is isomorphic to K, or determine that such a representation need not exist.

Background

The paper observes that a good cover in a d-dimensional simplicial complex with vanishing homology in dimension d has a d-Leray nerve. It notes that the converse is known for d=1 through representations by subtrees of a tree, and asks whether an analogous representation exists in every dimension.

References

Does there exist a $d$-dimensional simplicial complex $Y$ with vanishing homology in dimension $d$, and good cover $F$ in $Y$ such that the nerve of $F$ is isomorphic to $K$?

— Overlap-Helly theorems  (2609.10023 - Holmsen et al., 9 Sep 2026) in Section 4, first problem in the concluding remarks