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Duality for Cohen--Macaulay Complexes through Combinatorial Sheaves

Richard D. Wade, Thomas A. Wasserman

TL;DR

The paper develops a comprehensive combinatorial sheaf-theoretic framework to prove a duality for Cohen--Macaulay complexes that generalizes Poincaré duality. By pairing local homology sheaves with local cohomology cosheaves through an explicit chain-level cap product, it constructs a fundamental class $[X]$ and proves a CM Duality Theorem relating compactly supported (co)homology to (locally finite) relative homology. The Mayer--Vietoris double complex and its associated spectral sequences are leveraged to produce explicit inverses to augmentation maps, connecting MV-homology to cap products with $[X]$; the results extend to coefficients in (co)sheaves and to relative, compactly supported, and natural settings. Functoriality and naturality are established via star-local homeomorphisms, ensuring the duality is robust under symmetries and group actions. The work situates CM duality within Verdier duality and duality groups, offering tools for broader applications in combinatorial topology and geometric group theory.

Abstract

We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincaré Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working knowledge of simplicial complexes and (co)homology. The main motivation is a link with Bieri-Eckmann duality for discrete groups, which is explored in a companion paper.

Duality for Cohen--Macaulay Complexes through Combinatorial Sheaves

TL;DR

The paper develops a comprehensive combinatorial sheaf-theoretic framework to prove a duality for Cohen--Macaulay complexes that generalizes Poincaré duality. By pairing local homology sheaves with local cohomology cosheaves through an explicit chain-level cap product, it constructs a fundamental class and proves a CM Duality Theorem relating compactly supported (co)homology to (locally finite) relative homology. The Mayer--Vietoris double complex and its associated spectral sequences are leveraged to produce explicit inverses to augmentation maps, connecting MV-homology to cap products with ; the results extend to coefficients in (co)sheaves and to relative, compactly supported, and natural settings. Functoriality and naturality are established via star-local homeomorphisms, ensuring the duality is robust under symmetries and group actions. The work situates CM duality within Verdier duality and duality groups, offering tools for broader applications in combinatorial topology and geometric group theory.

Abstract

We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincaré Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working knowledge of simplicial complexes and (co)homology. The main motivation is a link with Bieri-Eckmann duality for discrete groups, which is explored in a companion paper.
Paper Structure (70 sections, 49 theorems, 207 equations, 1 figure)

This paper contains 70 sections, 49 theorems, 207 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. CM Duality
  4. Cap product
  5. Naturality
  6. Cosheaf homology in degree 0
  7. A brief history: CM complexes, duality, and combinatorial sheaves
  8. Cohen--Macaulay complexes
  9. CM duality and combinatorial sheaves
  10. Verdier duality
  11. Duality groups
  12. A sketch of proof of CM duality
  13. Outline
  14. Setup
  15. CM complexes
  16. ...and 55 more sections

Key Result

Theorem 1

Let $X$ be a locally finite, locally Cohen--Macaulay simplicial complex of dimension $n$. Then there is a fundamental class $[X]\in H^{\textnormal{lf}}_n(X;h^*)$ in locally finite homology, such that the cap product induces isomorphisms and where $H^*_c$ denotes compactly supported cohomology.

Figures (1)

  • Figure 1: The MV double complex $D(L)$ of a $d$ dimensional subcomplex $L$ of an $n$-dimensional complex $X$ with the differentials $i^l$ and $d_k$, the last vertex lift $\lambda$, and the diagonal shift $\Lambda$.

Theorems & Definitions (111)

  • Theorem 1: CM Duality
  • Theorem 2
  • Theorem 3
  • Remark 2.3
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Definition 2.10: $\mathcal{G}$-transport and $\mathcal{F}$-dual
  • ...and 101 more