Grapes and Alexander duality

TL;DR

This paper proves that the property of being a grape is preserved under Alexander duality and demonstrates how homotopy types transfer via combinatorial duality, with several applications.

Contribution

It establishes the invariance of the grape property under Alexander duality and links homotopy types through combinatorial duality, expanding understanding of topological invariants.

Findings

Grape property is invariant under Alexander duality

Homotopy types transfer via combinatorial Alexander duality

Provides applications of these duality properties

Abstract

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.