Grapes and Alexander duality

In this paper, we prove that the property of being a grape (in any of its variants) is invariant under Alexander duality. The explicitly determined (simple-)homotopy type of a grape can be transferred to its Alexander dual via Combinatorial Alexander Duality in (co)homology. We also provide several applications.

💡 Research Summary

The paper introduces and studies a family of simplicial complexes called “grapes,” which are defined by a recursive vertex‑based decomposition. Four variants are considered: topological grapes, combinatorial grapes, strong combinatorial grapes, and weak/strong‑weak combinatorial grapes. In each case there must exist a vertex a such that the link lkₐ(Δ) and the deletion dlₐ(Δ) are themselves grapes of the same type, together with an extra condition (e.g., the link is contained in a cone inside the deletion, or one of the two is a cone, or the link is simple‑homotopy equivalent to the void complex). The definitions depend only on the vertex set, not on the ambient ground set.

The authors first establish basic properties: removing a non‑vertex from the ground set does not affect grape status; cones and the full simplex minus its top face are strong combinatorial grapes; and any topological grape is either contractible or homotopy equivalent to a wedge of spheres. Lemma 3.8 and Proposition 3.7 formalize these observations.

A central result (Theorem 4.3) concerns strong combinatorial grapes. Using induction on the number of vertices, they show that any such grape is simple‑homotopy equivalent either to the void complex or to the boundary of an n‑dimensional cross‑polytope ∂βₙ. The proof hinges on the fact that if either the link or the deletion is a cone, the whole complex is either a suspension of the other piece (if the deletion is a cone) or collapses onto the other piece (if the link is a cone). Since the link and deletion are themselves strong combinatorial grapes, the induction proceeds, and suspensions of ∂βₙ give ∂βₙ₊₁.

The paper’s main novelty is the invariance of the grape property under Alexander duality. For a simplicial complex Δ on a ground set X, the Alexander dual Δ* consists of all subsets whose complements are not faces of Δ. Proposition 5.1 records elementary facts: deletion and link are swapped under duality, cones dualize to cones, and simple‑homotopy equivalence to the void complex is preserved. Using these facts and a double induction on |X|, Theorem 5.2 proves that if Δ is a grape of any of the four types, then Δ* is also a grape of the same type. The proof treats each variant separately, handling the base cases (void complex, irrelevant complex, single‑vertex complexes) and then using the duality relations for deletion and link to transfer the recursive structure.

Finally, Section 6 discusses applications. Many complexes previously studied in combinatorial topology—such as leaf complexes of forests, independence complexes of certain graphs, and boundaries of cross‑polytopes—fit naturally into the grape framework. Because the grape property is preserved under Alexander duality, results about contractibility, homotopy type, or simple‑homotopy type can be transferred between a complex and its dual, providing a unified approach to a variety of problems.

In summary, the paper establishes a robust class of recursively decomposable simplicial complexes whose defining property is stable under Alexander duality. Strong combinatorial grapes have a very restricted simple‑homotopy type (void or cross‑polytope boundary), and this information can be moved across duals. The work unifies several known results and opens the door to new applications where duality and recursive decomposition interact.