Lowest-degree triple Massey products in moment-angle complexes

TL;DR

This paper provides a combinatorial classification of non-trivial triple Massey products in the cohomology of moment-angle complexes, advancing previous results by including cases with non-trivial indeterminacy.

Contribution

It extends prior work by Denham and Suciu to classify triple Massey products with non-trivial indeterminacy in moment-angle complexes.

Findings

Classified non-trivial triple Massey products combinatorially

Improved understanding of Massey products with indeterminacy

Enhanced the theoretical framework for cohomology of moment-angle complexes

Abstract

We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.

Equations

$${\mathcal{Z}}_{\mathcal{K}}=\bigcup _{I\in \mathcal{K}}{\left(D^2,S^1\right)}^I\subset (D^2{)}^m$$

$$\raisebox{2.0pt}{$\chi$}_{L}\otimes\raisebox{2.0pt}{$\chi$}_{M}\mapsto\left\{\begin{array}[]{lr}c_{L\cup M}\;\raisebox{2.0pt}{$\chi$}_{L\cup M}&\text{if }I\cap J=\varnothing,\\ 0&\text{otherwise}\end{array}\right.$$

$$a:3,3,2,2,2,2b:3,3,3,3,2,2c:3,3,3,3,2,2d:4,3,3,3,3,2$$

$$e:3,3,3,3,3,3f:4,4,3,3,3,3g:4,3,3,3,3,2h:3,3,3,3,2,2$$

Full text

Massey products are higher cohomology operations that refine cup products. The first examples of non-trivial Massey products in moment-angle complexes were by Baskakov [2], who gave an infinite family of moment-angle complexes with non-trivial triple Massey products. For moment-angle complexes we give a classification of non-trivial triple Massey products of classes in the lowest degree (degree three). Additionally this provides the smallest examples of Massey products in moment-angle complexes in which non-trivial indeterminacy can occur. This classification is important for recent results, see for example [5, Proposition 4.9].

Let $\mathcal{K}$ be a simplicial complex on $[m]$ vertices. Following [4], the moment-angle complex $\mathcal{Z}_{\mathcal{K}}$ is

[TABLE]

where $(D^{2},S^{1})^{I}=\prod_{i=1}^{m}Y_{i}$ for $Y_{i}=D^{2}$ if $i\in I$, and $Y_{i}=S^{1}$ if $i\notin I$.

As a subspace of the polydisc, $\mathcal{Z}_{\mathcal{K}}$ has a cellular decomposition that induces a multigrading on $C^{*}(\mathcal{Z}_{\mathcal{K}})$. The full subcomplex $\mathcal{K}_{J}$ is $\{\sigma\in\mathcal{K}\ |\ \sigma\subset J\}$ for $J\subset[m]$. Let $\widetilde{C}^{*}(\mathcal{K}_{J})$ be the augmented simplicial cochain complex. The following theorem is a combination of results by Hochster [7], Buchstaber-Panov [3], and Baskakov [1]. All coefficients are in $\mathbf{k}$, which is a field or $\mathbb{Z}$.

The cochain group $C^{p}(\mathcal{K}_{J})=\operatorname{Hom}(C_{p}(\mathcal{K}_{J}),\mathbf{k})$ has a basis of \raisebox{2.0pt}{\chi}_{L} for a $p$-simplex $L\in\mathcal{K}_{J}$, where \raisebox{2.0pt}{\chi}_{L} takes the value $1$ on $L$ and [math] otherwise. Let $\varepsilon(j,J)=(-1)^{r-1}$ for $j$ the $r$th element of $J$, and for $L\subset J$, let $\varepsilon(L,J)=\prod_{j\in L}\varepsilon(j,J)$. The product on $\bigoplus_{J\subset[m]}\widetilde{H}^{*}(\mathcal{K}_{J})$ is induced by $C^{p-1}(\mathcal{K}_{I})\otimes C^{q-1}(\mathcal{K}_{J})\to C^{p+q-1}(\mathcal{K}_{I\cup J})$,

[TABLE]

where $c_{L\cup M}=\varepsilon(L,I)\;\varepsilon(M,J)\;\zeta\;\varepsilon(L\cup M,I\cup J)$ and $\zeta=\prod_{k\in I\setminus L}\varepsilon(k,k\cup J\setminus M)$.

We compute Massey products $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle\subset H^{*}(\mathcal{Z}_{\mathcal{K}})$ in terms of $\mathcal{K}$ by using the isomorphism in Hochster’s formula. When indeterminacy is trivial, Denham and Suciu [6] showed that there is a non-trivial Massey product $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$ for $\alpha_{i}\in H^{3}(\mathcal{Z}_{\mathcal{K}})$ if and only if the one-skeleton $\mathcal{K}^{(1)}$ contains a full subcomplex isomorphic to one of the first $6$ graphs in Figure 1. Those six graphs do not capture Massey products with non-trivial indeterminacy.

For $a\in C^{p}(\mathcal{K}_{J})$, which corresponds to $a\in C^{p+|J|+1}(\mathcal{Z}_{\mathcal{K}})$, denote ${\overline{\overline{a}}}=(-1)^{p+|J|}a$. A cochain $a_{12}\in C^{0}(\mathcal{K}_{1234})$ such that d(a_{12})={\overline{\overline{\raisebox{2.0pt}{\chi}_{1}}}}\raisebox{2.0pt}{\chi}_{3}=0 is of the form a_{12}=c_{1}\raisebox{2.0pt}{\chi}_{3}+c_{2}(\raisebox{2.0pt}{\chi}_{1}+\raisebox{2.0pt}{\chi}_{4}+\raisebox{2.0pt}{\chi}_{2}) for any $c_{1},c_{2}\in\mathbf{k}$. A cochain $a_{23}\in C^{0}(\mathcal{K}_{3456})$ such that d(a_{23})={\overline{\overline{\raisebox{2.0pt}{\chi}_{3}}}}\cdot\raisebox{2.0pt}{\chi}_{5}=\raisebox{2.0pt}{\chi}_{35} is of the form a_{23}=c_{3}\raisebox{2.0pt}{\chi}_{4}+c_{4}(\raisebox{2.0pt}{\chi}_{6}+\raisebox{2.0pt}{\chi}_{3}+\raisebox{2.0pt}{\chi}_{5})+\raisebox{2.0pt}{\chi}_{5} for any $c_{3},c_{4}\in\mathbf{k}$, where $c_{3}=c_{4}$ if $\{4,6\}\in\mathcal{K}$. The associated cocycle $\omega\in C^{1}(\mathcal{K})$ is \omega={\overline{\overline{a}}}_{1}a_{23}+{\overline{\overline{a}}}_{12}a_{3}=c_{3}\raisebox{2.0pt}{\chi}_{14}+c_{4}(\raisebox{2.0pt}{\chi}_{16}+\raisebox{2.0pt}{\chi}_{15})+\raisebox{2.0pt}{\chi}_{15}+c_{1}\raisebox{2.0pt}{\chi}_{35}+c_{2}(\raisebox{2.0pt}{\chi}_{15}+\raisebox{2.0pt}{\chi}_{25}). Since d(\raisebox{2.0pt}{\chi}_{5})=\raisebox{2.0pt}{\chi}_{15}+\raisebox{2.0pt}{\chi}_{35}+\raisebox{2.0pt}{\chi}_{25} and d(\raisebox{2.0pt}{\chi}_{1})=-\raisebox{2.0pt}{\chi}_{16}-\raisebox{2.0pt}{\chi}_{14}-\raisebox{2.0pt}{\chi}_{15} for \raisebox{2.0pt}{\chi}_{1},\raisebox{2.0pt}{\chi}_{5}\in C^{0}(\mathcal{K}), \omega=(c_{3}-c_{4})\raisebox{2.0pt}{\chi}_{14}-c_{4}d(\raisebox{2.0pt}{\chi}_{1})+\raisebox{2.0pt}{\chi}_{15}+(c_{1}-c_{2})\raisebox{2.0pt}{\chi}_{35}+c_{2}d(\raisebox{2.0pt}{\chi}_{5}). Also [\omega]=[(c_{3}-c_{4})\raisebox{2.0pt}{\chi}_{14}+\raisebox{2.0pt}{\chi}_{15}+(c_{1}-c_{2})\raisebox{2.0pt}{\chi}_{35}]\neq 0 for any $c_{1},c_{2},c_{3},c_{4}\in\mathbf{k},c_{3}=c_{4}\text{ if }\{4,6\}\in\mathcal{K}$. Therefore $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle\subset H^{8}(\mathcal{Z}_{\mathcal{K}})$ is non-trivial with non-trivial indeterminacy, $\alpha_{1}\cdot\widetilde{H}^{0}(\mathcal{K}_{3456})+\alpha_{3}\cdot\widetilde{H}^{0}(\mathcal{K}_{1234}).$

Calculations are similar for $\mathcal{K}$ in Figure 2a, where a_{12}=\raisebox{2.0pt}{\chi}_{3}+c_{1}(\raisebox{2.0pt}{\chi}_{1}+\raisebox{2.0pt}{\chi}_{2}+\raisebox{2.0pt}{\chi}_{3}+\raisebox{2.0pt}{\chi}_{4}) and a_{23}=\raisebox{2.0pt}{\chi}_{5}+c_{2}(\raisebox{2.0pt}{\chi}_{3}+\raisebox{2.0pt}{\chi}_{4}+\raisebox{2.0pt}{\chi}_{5}+\raisebox{2.0pt}{\chi}_{6}), $c_{1},c_{2}\in\mathbf{k}$. In this case, \omega=-\raisebox{2.0pt}{\chi}_{25}+(c_{1}+1)d(\raisebox{2.0pt}{\chi}_{5})-c_{2}d(\raisebox{2.0pt}{\chi}_{1}) so $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$ only contains the non-zero class [\omega]=[-\raisebox{2.0pt}{\chi}_{25}].

Conversely, suppose $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$ is non-trivial for $\alpha_{1},\alpha_{2},\alpha_{3}\in H^{3}(\mathcal{Z}_{\mathcal{K}})$. Let $G$ on $[6]$ be the edge complement graph of $\mathcal{K}^{(1)}$, so $\{i,j\}\in G$ if and only if $\{i,j\}\notin\mathcal{K}$. We will show that $G$ is one of the graphs in Figure 2, where the dashed edges are optional. By Hochster’s formula, there are full subcomplexes $\mathcal{K}_{S_{i}}$ for $S_{i}\subset[m]$, $|S_{i}|=2$ such that $\alpha_{i}$ corresponds to $\alpha_{i}\in\widetilde{H}^{0}(\mathcal{K}_{S_{i}})$. Since $\mathcal{K}_{S_{i}}$ is a pair of disjoint vertices, $\{v_{i},v_{i}^{\prime}\}\notin\mathcal{K}$ for any $v_{i},v_{i}^{\prime}\in S_{i}$. Since $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$ is non-trivial, $S_{i}\cap S_{j}=\varnothing$ for $i\neq j$. Let $S_{1}=\{1,2\},S_{2}=\{3,4\},S_{3}=\{5,6\}$. Then the graph $G$ contains the edges $\{1,2\},\{3,4\},\{5,6\}$. Since $\alpha_{i}\alpha_{i+1}=0$, the full subcomplex $\mathcal{K}_{S_{i}\cup S_{i+1}}$ does not contain a cycle for $i=1,2$. Thus there exist edges $\{v_{1},v_{2}\},\{v_{2}^{\prime},v_{3}\}\in G$ for $v_{i},v_{i}^{\prime}\in S_{i}$.

Suppose $\{v_{1},v_{2}\},\{v_{2},v_{3}\}\in G$ for $v_{i}\in S_{i}$. Let a_{1}=\raisebox{2.0pt}{\chi}_{v_{1}}\in C^{0}(\mathcal{K}_{12}), a_{2}=\raisebox{2.0pt}{\chi}_{v_{2}}\in C^{0}(\mathcal{K}_{34}), a_{3}=\raisebox{2.0pt}{\chi}_{v_{3}}\in C^{0}(\mathcal{K}_{56}) be representing cocycles for $\alpha_{1},\alpha_{2},\alpha_{3}$. So a_{1}a_{2}=\raisebox{2.0pt}{\chi}_{v_{1}v_{2}}=0 and a_{2}a_{3}=\raisebox{2.0pt}{\chi}_{v_{2}v_{3}}=0. For $a_{12}=0=a_{23}$, $\omega=0$, which contradicts the non-triviality of $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$. Thus $\{v_{1},v_{2}\},\{v_{2},v_{3}\}\notin G$ for $v_{i}\in S_{i}$.

Label the vertices of $G$ so that there is a path $1,\ldots,6$. Consider the case when $\{1,3\},\{4,6\}\notin G$. Since $\{v_{1},v_{2}\},\{v_{2},v_{3}\}\notin G$ for $v_{i}\in S_{i}$, the vertices $3$ and $4$ have valency two. Suppose $\{2,5\}\in G$. Let \raisebox{2.0pt}{\chi}_{2}\in C^{0}(\mathcal{K}_{12}),\raisebox{2.0pt}{\chi}_{3}\in C^{0}(\mathcal{K}_{34}),\raisebox{2.0pt}{\chi}_{5}\in C^{0}(\mathcal{K}_{56}) represent $\alpha_{1},\alpha_{2},\alpha_{3}$, respectively. Since $a_{1}a_{2}=0$, let $a_{12}=0$ and let a_{23}=\raisebox{2.0pt}{\chi}_{5}. Then \omega=\raisebox{2.0pt}{\chi}_{25} is zero, contradicting the non-triviality of $\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle$. Hence $\{2,5\}\notin G$ and $G$ is the graph in Figure 3a, where $\{1,5\}$, $\{1,6\}$ and $\{2,6\}$ are optional.

There are three more cases. When $\{1,3\}\in G$ and $\{4,6\}\notin G$, it is necessary that $\{1,5\}\notin G$ otherwise in the Example, \omega=\raisebox{2.0pt}{\chi}_{15} is zero. Also $\{2,5\}\notin G$ as in the previous case. The edges $\{1,6\},\{2,6\}$ in $G$ are optional as they do not change the calculations in the Example. If $\{1,6\},\{2,6\}\notin G$, then $G$ is the graph in Figure 3b with $\{4,6\}\notin G$. For other selections of $\{1,6\},\{2,6\}$, $G$ is isomorphic to a graph in Figure 3a.

When $\{4,6\}\in G$ and $\{1,3\}\notin G$, $G$ is symmetric to the case when $\{1,3\}\in G$ and $\{4,6\}\notin G$, so up to isomorphism we obtain the same graphs. Finally when $\{1,3\},\{4,6\}\in G$, then $\{1,5\},\{2,5\},\{2,6\}\notin G$ for the same reasons as in the last two cases. Let $\alpha_{1},\alpha_{2},\alpha_{3}$ be represented by \raisebox{2.0pt}{\chi}_{1}\in C^{0}(\mathcal{K}_{12}), \raisebox{2.0pt}{\chi}_{3}\in C^{0}(\mathcal{K}_{34}), \raisebox{2.0pt}{\chi}_{6}\in C^{0}(\mathcal{K}_{56}), respectively. Let $a_{12}=0$, a_{23}=\raisebox{2.0pt}{\chi}_{6}. Then \omega=\raisebox{2.0pt}{\chi}_{16} and therefore $\{1,6\}\notin G$. So $G$ is the graph in Figure 3b with $\{4,6\}\in G$.

The graphs in Figure 3a and Figure 3b are edge complement graphs to the graphs in Figure 2a and Figure 2b, respectively. Up to graph isomorphism, all graphs in Figure 2 are exactly those in Figure 1. Thus Figure 1 contains all $1$-skeletons of simplicial complexes $\mathcal{K}$ on six vertices with a non-trivial triple Massey product in $H^{8}(\mathcal{Z}_{\mathcal{K}})$. ∎

[TABLE]

Thus graphs $a,e,f$ are not isomorphic to any of the other graphs. Also the graphs $d$ and $g$ are not isomorphic because the vertices of valency $2$ and $4$ are adjacent in $g$ but not adjacent in $d$. The graph $c$ is different to $b,h$ because the two vertices of valency $2$ are adjacent in graph $c$ but not adjacent in $b$ or $h$. The graph $b$ is different to $h$ because the two vertices of valency $2$ are at a minimal distance of $2$ from each other, that is, there is one vertex in between them in $b$. In $h$, these two vertices are at a minimal distance of $3$ from each other. Therefore each of these graphs is not isomorphic to another. ∎