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Grapes and Alexander duality

Mario Marietti

TL;DR

The paper develops four variants of grape-like recursive decompositions for simplicial complexes and proves that the grape property is preserved under Alexander duality, enabling transfer of simple-homotopy information between a complex and its dual via Combinatorial Alexander Duality. A central result is that the strongest grape variant is simple-homotopy equivalent to either the void complex or the boundary of a cross-polytope, with a procedure to determine the corresponding dimension; this yields a powerful, unified framework to analyze dual complexes. The authors apply these results to graph-derived complexes, notably forest-associated complexes, and to the path-missing/path-free complexes, producing explicit simple-homotopy types and duality relations. Collectively, the work provides a cohesive combinatorial toolkit for transferring topological information across duals and for obtaining concrete homotopy classifications in graph- and digraph-related settings.

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.

Grapes and Alexander duality

TL;DR

The paper develops four variants of grape-like recursive decompositions for simplicial complexes and proves that the grape property is preserved under Alexander duality, enabling transfer of simple-homotopy information between a complex and its dual via Combinatorial Alexander Duality. A central result is that the strongest grape variant is simple-homotopy equivalent to either the void complex or the boundary of a cross-polytope, with a procedure to determine the corresponding dimension; this yields a powerful, unified framework to analyze dual complexes. The authors apply these results to graph-derived complexes, notably forest-associated complexes, and to the path-missing/path-free complexes, producing explicit simple-homotopy types and duality relations. Collectively, the work provides a cohesive combinatorial toolkit for transferring topological information across duals and for obtaining concrete homotopy classifications in graph- and digraph-related settings.

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.
Paper Structure (8 sections, 14 theorems, 11 equations, 3 figures)

This paper contains 8 sections, 14 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grapes
  4. Simple-homotopy
  5. Grapes and Alexander duals
  6. Applications
  7. Simplicial complexes associated with forests
  8. The path-missing and path-free complexes

Key Result

Proposition 2.2

For every simplicial complex $(\Delta, X)$ and every $x\in X$, we have

Figures (3)

  • Figure : The graph $G$
  • Figure : The graph $G$
  • Figure : $\mathcal{PF}(G,s,t)$

Theorems & Definitions (39)

  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.7
  • Lemma 3.8
  • ...and 29 more