[Paper Review] The homology of simpicial complements and the cohomology of moment-angle complexes

Qibing Zheng, Xiangjun Wang|arXiv (Cornell University)|Sep 29, 2011

Advanced Combinatorial Mathematics11 references3 citations

TL;DR

This paper introduces the simplicial complement of a simplicial complex $K$ and uses the Taylor resolution of its associated monomial ideal to construct a differential graded algebra $(\overline{\Lambda}(\mathbb{P}), d)$ that computes the $\mathrm{Tor}$ algebra $\mathrm{Tor}_*^{\mathbf{k}[\mathbf{x}]}(\mathbf{k}(K), \mathbf{k})$ and determines its product structure. The key contribution is a new algebraic model for the cohomology of generalized moment-angle complexes $\mathcal{Z}_K(X,A)$, with explicit computations for cases like $({X},{A}) = (S^2, S^1)$, yielding a total Betti number of 216.

ABSTRACT

We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.

Motivation & Objective

• To develop a new method for computing the $\mathrm{Tor}$ algebra $\mathrm{Tor}_*^{\mathbf{k}[\mathbf{x}]}(\mathbf{k}(K), \mathbf{k})$ associated with the Stanley–Reisner face ring of a simplicial complex $K$.
• To define the simplicial complement $\mathbb{P}$ of $K$ as a system of generators for a monomial ideal and use its Taylor resolution to construct a DGA model.
• To determine the algebra structure of the $\mathrm{Tor}$ algebra by realizing it as the cohomology of a chain complex $\overline{\Lambda}(\mathbb{P})$.
• To relate the cohomology of generalized moment-angle complexes $\mathcal{Z}_K(X,A)$ to the homology of the simplicial complement $\mathbb{P}$, particularly when $A_i \hookrightarrow X_i$ are null-homotopic.
• To compute explicit Betti numbers and Poincaré series for $\mathcal{Z}_K(X,A)$ in special cases, such as $({X},{A}) = (D^2, S^1)$ and $({X},{A}) = (S^2, S^1)$.

Proposed method

• The simplicial complement $\mathbb{P} = \{\sigma_1, \dots, \sigma_s\}$ is defined as a sequence of subsets of $[m]$ generating a square-free monomial ideal $\mathcal{I}_{\mathbb{P}}$, with $K$ being the associated simplicial complex.
• The Taylor resolution of $\mathcal{I}_{\mathbb{P}}$ is used to construct a differential graded algebra $(\overline{\Lambda}(\mathbb{P}), d)$ over $\mathbf{k}$, which computes $\mathrm{Tor}_*^{\mathbf{k}[\mathbf{x}]}(\mathbf{k}(K), \mathbf{k})$ as its cohomology.
• The chain complex $\overline{\Lambda}(\mathbb{P})$ is built from exterior algebra generators corresponding to the simplices in $\mathbb{P}$, with a differential $d$ defined via the least common multiple of monomials.
• The cohomology of generalized moment-angle complexes $\mathcal{Z}_K(X,A)$ is computed via a decomposition involving suspensions and the homology of $E_\omega\mathbb{P}$, the simplicial complement relative to $\omega \in K$, using the isomorphism $H^*(\mathcal{Z}_K(X,A), \mathbf{k}) \cong \bigoplus_{\omega \in K} \bigoplus_{\sigma} H_{*,\sigma}(\overline{\Lambda}^{*,\sigma}(E_\omega\mathbb{P}), d)$.
• For $({X},{A}) = (S^2, S^1)$, the cohomology is computed by summing contributions from all $\omega \in K$, with each $\omega$ contributing a graded vector space whose Poincaré series is derived from $H_{*,\sigma}(\overline{\Lambda}^{*,\sigma}(E_\omega\mathbb{P}), d)$.
• The method leverages the fact that $\overline{\Lambda}^{*,\sigma}(E_\omega\mathbb{P})$ is a free $\mathbf{k}$-module, enabling explicit computation of Betti numbers via Poincaré series decomposition.

Experimental results

Research questions

• RQ1 How can the $\mathrm{Tor}$ algebra $\mathrm{Tor}_*^{\mathbf{k}[\mathbf{x}]}(\mathbf{k}(K), \mathbf{k})$ be computed with its full algebra structure, including the product, beyond Hochster's formula?
• RQ2 What is the role of the simplicial complement $\mathbb{P}$ of a simplicial complex $K$ in modeling the cohomology of moment-angle complexes?
• RQ3 How does the cohomology of generalized moment-angle complexes $\mathcal{Z}_K(X,A)$ decompose in terms of the homology of simplicial complements $E_\omega\mathbb{P}$ for $\omega \in K$?
• RQ4 What are the Betti numbers and Poincaré series of $\mathcal{Z}_K(X,A)$ when $X_i$ are contractible or $A_i$ are contractible, or $({X}_i, {A}_i) = (S^2, S^1)$?
• RQ5 Can the cohomology of $\mathcal{Z}_K(D^2, S^1)$ be fully described using the homology of the simplicial complement $\mathbb{P}$, and what is its total Betti number?

Key findings

• The cohomology ring $H^*(\mathcal{Z}_K(D^2, S^1), \mathbf{k})$ is isomorphic to the exterior algebra $\Lambda(\sigma_1, \sigma_2, \sigma_3)$, with Poincaré series $1 + 3x^3 + 3x^6 + x^9$ and total Betti number 8.
• For $({X}, {A}) = (S^2, S^1)$, the cohomology $H^*(\mathcal{Z}_K(S^2, S^1), \mathbf{k})$ has Poincaré series $1 + 6x^2 + 9x^3 + 12x^4 + 36x^5 + 35x^6 + 36x^7 + 54x^8 + 27x^9$, yielding a total Betti number of 216.
• The cohomology of $\mathcal{Z}_K(X,A)$ is isomorphic to the direct sum over $\omega \in K$ of the homology groups $H_{*,\sigma}(\overline{\Lambda}^{*,\sigma}(E_\omega\mathbb{P}), d)$, with each $\omega$ contributing a graded vector space whose structure depends on the simplicial complement $E_\omega\mathbb{P}$.
• When $\mathbb{P} = \{\{1,2\}, \{3,4\}, \{5,6\}\}$, the homology of $E_\phi\mathbb{P}$ is isomorphic to $\Lambda(\sigma_1, \sigma_2, \sigma_3)$, confirming the exterior algebra structure of $H^*(\mathcal{Z}_K(D^2, S^1), \mathbf{k})$.
• For $\omega = \{i\}$, the contribution to the cohomology has Poincaré series $x^2(1 + x + 2x^3 + 2x^4 + x^6 + x^7)$, and for $\omega = \{i,j\}$, it is $x^4(1 + 2x + x^2 + x^3 + 2x^4 + x^5)$, with $\omega = \{i,j,k\}$ contributing $x^6(1 + 3x + 3x^2 + x^3)$.
• The method provides a new proof of the Hochster formula and explicitly determines the product structure of the $\mathrm{Tor}$ algebra via the DGA $(\overline{\Lambda}(\mathbb{P}), d)$, which is a vector space over $\mathbf{k}$ and respects the algebra multiplication.