# Models for the Cohomology of Certain Polyhedral Products

M. Bendersky  Department of Mathematics, Hunter College, CUNY, 695 Park Avenue
New York, NY 10065, U.S.A<EMAIL_ADDRESS>and  J. Grbić  School of
Mathematical Sciences, University of Southampton, SO17 1BJ Southampton, UK
<EMAIL_ADDRESS>
###### Abstract.

For a commutative ring $\mathbbm{k}$ with unit, we describe and study various
differential graded $\mathbbm{k}$-modules and $\mathbbm{k}$-algebras which are
models for the cohomology of polyhedral products
$(\underline{CX},\underline{X})^{K}$. Along the way, we prove that the
integral cohomology $H^{*}((D^{1},S^{0})^{K};\mathbb{Z})$ of the real moment-
angle complex is a Tor module, the one that does not come from a geometric
setting. We also reveal that the apriori different cup product structures in
$H^{*}((D^{1},S^{0})^{K};\mathbb{Z})$ and in
$H^{*}((D^{n},S^{n-1})^{K};\mathbb{Z})$ for $n\geq 2$ have the same origin. As
an application, this work sets the stage for studying the based loop space of
$(\underline{CX},\underline{X})^{K}$ in terms of the bar construction applied
to the differential graded $\mathbb{Z}$-algebras
$B(\mathcal{C}^{*}(\underline{X};\mathbb{Z}),K)$ quasi-isomorphic to the
singular cochain algebra
$\mathcal{C}^{*}((\underline{CX},\underline{X})^{K};\mathbb{Z})$.

###### Contents

  1. 1 Introduction
  2. 2 Additive models
    1. 2.1 Real moment-angle complexes
    2. 2.2 A generalisation to polyhedral products
  3. 3 Algebra models

##  1\. Introduction

One of the most studied algebraic invariants of topological spaces is their
cohomology ring. With its rich and relatively approachable structure,
cohomology is a homotopy theoretical flagship invariant harnessed by a host of
mathematical disciplines.

In the realm of toric topology, polyhedral products
$(\underline{X},\underline{A})^{K}$ are constructed as a functorial interplay
between topology and combinatorics in term of topological pairs
$(\underline{X},\underline{A})$ and a simplicial complex $K$. More precisely,
let $K$ be a simplicial complex on the vertex set $[m]=\\{1,\ldots,m\\}$ and
let $(\underline{X},\underline{A})=\\{(X_{i},A_{i})\\}_{i=1}^{m}$ be an
$m$-tuple of $CW$-pairs. The polyhedral product is defined by

$(\underline{X},\underline{A})^{K}=\bigcup_{\sigma\in
K}(\underline{X},\underline{A})^{\sigma}\subseteq\prod_{i=1}^{m}X_{i}$

where

$(\underline{X},\underline{A})^{\sigma}=\prod_{i=1}^{m}Y_{i},\quad
Y_{i}=\begin{cases}X_{i}&\text{for }i\in\sigma\\\ A_{i}&\text{for
}i\notin\sigma.\end{cases}$

With $CX_{i}$ being the cone on $X_{i}$, we restrict our attention to
polyhedral products where the topological pairs are $(CX_{i},X_{i})$ for all
$i$. Of particular importance are $X_{i}=S^{1}$ for all $i$ in which case the
polyhedral product is identified with the complex moment-angle complex, and is
denoted by $\mathcal{Z}_{K}$. If $X_{i}=S^{0}$ for all $i$, the polyhedral
product is the real moment-angle complex, denoted by
$\mathcal{Z}^{\mathbb{R}}_{K}$.

Before we start with the study of polyhedral products, let us lay down some
basic combinatorial notations. For an abstract simplicial complex $K$ on the
vertex set $[m]$, let $|K|$ denote its geometric realisation and let $K_{J}$
be the full subcomplex of $K$ consisting of all simplicies of $K$ with
vertices in $J\subset[m]$.

Moment-angle complexes are known as the bellwethers for the combinatorial
coding of topological and geometrical properties. For example, there is a
stable decomposition of a real moment-angle complex
$\mathcal{Z}^{\mathbb{R}}_{K}=(D^{1},S^{0})^{K}$ in terms of full subcomplexes
of the simplicial complex $K$ given by

$\Sigma(D^{1},S^{0})^{K}=\Sigma\bigvee_{J\subseteq[m]}|\Sigma K_{J}|.$

This decomposition was generalised in [1] to a stable decomposition of
polyhedral products, which specialises to the following stable decomposition
of $(\underline{CX},\underline{X})^{K}$,

$\Sigma(\underline{CX},\underline{X})^{K}\longrightarrow\Sigma\bigvee_{J\subseteq[m]}|\Sigma
K_{J}|\wedge Y^{\wedge J}$

where

$Y^{\wedge J}=\underset{i\in[m]}{\bigwedge}Y_{i}\mbox{ with }Y_{i}=X_{i}\mbox{
if }i\in J\mbox{ and }Y_{i}=S^{0}\mbox{ if }i\notin J.$

Directly from these stable decompositions, we can read off the underlying
$\mathbbm{k}$-module of the cohomology of the real moment-angle complex, of
the complex moment-angle complex and of the polyhedral product
$(\underline{CX},\underline{X})^{K}$, respectively as

$H^{*}((D^{1},S^{0})^{K};\mathbbm{k})\cong\bigoplus_{J\subseteq[m]}H^{*-1}(|K_{J}|;\mathbbm{k}),\qquad
H^{*}((D^{2},S^{1})^{K};\mathbbm{k})\cong\bigoplus_{J\subseteq[m]}H^{*-|J|-1}(|K_{J}|;\mathbbm{k})\qquad\text{and}$
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})\cong\bigoplus_{J\subseteq[m]}H^{\ast}(|\Sigma
K_{J}|;\mathbbm{k})\otimes H^{J}$

where $H^{J}=\underset{i\in[m]}{\otimes}H_{i}$ with
$H_{i}=\widetilde{H}^{\ast}(X_{i};\mathbbm{k})$ if $i\in J$ and
$Y_{i}=\mathbbm{k}$ if $i\notin J$ assuming that $H^{*}(X_{i};\mathbbm{k})$
are free $\mathbbm{k}$-modules for all $i$.

Let $K$ be a simplicial complex on the vertex set $[m]$. Denote by
$\mathbbm{k}[v_{1},\ldots,v_{m}]$ the polynomial ring on $m$ variables $v_{i}$
and for every set $I=\\{i_{1},\ldots,i_{r}\\}\subset[m]$, let
$v_{I}=v_{i_{1}}\cdots v_{i_{r}}$. Then the Stanley-Reisner ring is defined by

$SR[K]=\mathbbm{k}[v_{1},\ldots,v_{m}]/\mathcal{I}_{K}$

where $\mathcal{I}_{K}=(v_{I}\ |\ I\notin K)$ is the ideal generated by those
square-free monomials $v_{I}$ for which $I$ is not a simplex of $K$. If
$v_{1},\ldots,v_{m}$ are all of degree 2, then

$H^{*}(({\mathbb{C}}P^{\infty},*)^{K};\mathbbm{k})\cong SR[K].$

Using the language of toric varieties Franz [7] and independently using the
existence of the homotopy fibration

$T^{m}=(S^{1},S^{1})^{K}\longrightarrow(D^{2},S^{1})^{K}\longrightarrow({\mathbb{C}}P^{\infty},*)^{K}\longrightarrow({\mathbb{C}}P^{\infty},{\mathbb{C}}P^{\infty})^{K}=BT^{m}$

Buchstaber and Panov [4] showed that there is an algebra isomorphism

$H^{*}((D^{2},S^{1})^{K};\mathbb{Z})\cong\mathrm{Tor}_{\mathbb{Z}[v_{1},\ldots,v_{m}]}(SR[K],\mathbb{Z}).$

Similarly, Franz [6] showed that there is an additive isomorphism

$H^{*}((D^{1},S^{0})^{K};\mathbb{Z}_{2})\cong\mathrm{Tor}_{\mathbb{Z}_{2}[t_{1},\ldots,t_{m}]}(SR_{\mathbb{Z}_{2}}[K],\mathbb{Z}_{2})$

where $t_{i}$ are of degree 1 and
$SR_{\mathbb{Z}_{2}}[K]=\mathbb{Z}_{2}[t_{1},\ldots,t_{m}]/\mathcal{I}_{K}$
which cannot be extended to a multiplicative isomorphism for the canonical
product on the Tor. This additive description of the cohomology of the real
moment-angle complex can be related to the homotopy fibration

$\mathbb{Z}_{2}^{m}=(S^{0},S^{0})^{K}\longrightarrow(D^{1},S^{0})^{K}\longrightarrow(\mathbb{R}P^{\infty},*)^{K}\longrightarrow(\mathbb{R}P^{\infty},\mathbb{R}P^{\infty})^{K}=B\mathbb{Z}_{2}^{m}.$

In this paper we extend Franz’s result by showing, as Corollary 2.7, that
$H^{*}((D^{1},S^{0})^{K};\mathbbm{k})$ is additively isomorphic to a Tor
$\mathbbm{k}$-module. However this Tor $\mathbbm{k}$-module does not have a
geometric origin and the additive isomorphism cannot be extended to a one of
algebras.

Mimicking our approach for the cohomology of the real moment-angle complex,
that is, by considering the polynomial ring on suspended cohomological classes
of all $X_{i}$’s and introducing a generalised Stanley-Reisner ring
$SR(\underline{X},K)$, see Definitions 2.12 and 2.13, we next try to describe
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ as a Tor module. This
approach allow us only to identify
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ as a summand of such a
Tor module, see Proposition 2.16. To possibly get a description of
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ as a Tor module, in
future work we plan to replace the polynomial ring on suspended cohomology
classes of all $X_{i}$’s by an algebraic object that behaves as an algebraic
‘‘delooping" of $X_{i}$’s.

In Section 3 we give two integral algebraic models $B(\underline{X},K)$ and
$B(\mathcal{C}^{*}(\underline{X}),K)$ for the cohomology of
$(\underline{CX},\underline{X})^{K}$, see Propositions 3.2 and 3.4. The
results are obtained by combining Franz’s [6] and Cai’s [5] model of the
integral cohomology ring of the real moment-angle complex with the result of
Bahri, Bendersky, Cohen and Gitler [2] stating that the algebra structure in
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ depends only on the
cup product structure in $H^{*}((D^{1},S^{0})^{K};\mathbbm{k})$ and the cup
product structure in $H^{*}(X_{i};\mathbbm{k})$ for all $i$. The algebraic
model $B(C^{*}(X);\mathbbm{k})$ is particularly interesting as it is quasi-
isomorphic to the $\mathbbm{k}$-cochains on
$(\underline{CX},\underline{X})^{K}$ and therefor the bar construction can be
applied to it to deduce a model over $\mathbbm{k}$ for the based loop space of
$(\underline{CX},\underline{X})^{K}$ when $X_{i}$ have free
$\mathbbm{k}$-cohomology.

We finish the paper by explaining how seemingly different the cup product
structures in $H^{*}((D^{1},S^{0})^{K};\mathbb{Z})$ and in
$H^{*}((D^{n},S^{n-1})^{K};\mathbb{Z})$ for $n\geq 2$ have the same origin.

##  2\. Additive models

###  2.1. Real moment-angle complexes

Let $\mathbbm{k}$ be a commutative ring, which in this paper will be assumed
to be either the ring $\mathbb{Z}$ of integers or a field. Let
$y_{1},\ldots,y_{m}$ be of degree $1$. Define the graded algebra
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$ as

$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle=T(y_{1},\ldots,y_{m})/(y_{i}y_{j}=y_{j}y_{i})$

where $T$ is a free associative algebra. Notice that here we are assuming
commutativity not graded commutativity. Considered as a non-graded object
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$ is isomorphic to the
polynomial algebra $\mathbbm{k}[y_{1},\ldots,y_{m}]$. Given a subset
$I=\\{i_{1},\ldots,i_{r}\\}\subset[m]$, we denote by $y_{I}$ the square-free
monomial $y_{i_{1}}\cdots y_{i_{r}}$ in
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$.

Define an analogue of the Stanley-Reisner ring to be

$SR\langle
K\rangle=\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle/\mathcal{I}_{K}$

where $\mathcal{I}_{K}=(y_{I}\ |\ I\notin K)$ is the ideal generated by those
monomials $y_{I}$ for which $I$ is not a simplex of $K$.

For $\omega_{1},\ldots,\omega_{m}$ of degre $0$, define the graded algebra $L$
as

(1)
$L(\omega_{1},\ldots,\omega_{m})=T(\omega_{1},\ldots,\omega_{m})/(\omega_{i}^{2}=0,\
\omega_{i}\omega_{j}=-\omega_{j}\omega_{i}).$

As in the case of $\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$, the
multiplication is not graded commutative, and considered as a non-graded
object $L(\omega_{1},\ldots,\omega_{m})$ is isomorphic to the exterior algebra
$\Lambda(\omega_{1},\ldots,\omega_{m})$.

Define the bigraded differential algebra $(E,d)$ by

(2) $E=T(\omega_{1},\ldots,\omega_{m},y_{1},\ldots,y_{m})/(\omega_{i}^{2}=0,\
\omega_{i}\omega_{j}=-\omega_{j}\omega_{i},\ y_{i}y_{j}=y_{j}y_{i},\
\omega_{i}y_{j}=y_{j}\omega_{i})$

such that

$\begin{gathered}\mathop{\rm bideg}\omega_{i}=(-1,1),\quad\mathop{\rm
bideg}y_{i}=(0,1),\\\ d\omega_{i}=y_{i},\quad dy_{i}=0\end{gathered}$

and requiring the differential $d$ to satisfy the identity

(3) $d(a\cdot b)=d(a)\cdot b+(-1)^{\deg_{1}(a)}a\cdot d(b)$

where $\mathop{\rm bideg}(a)=(\deg_{1}(a),\deg_{2}(a))$.

######  Lemma 2.1.

The differential graded algebra $(E,d)$ with the differential given by
$d(\omega_{i})=y_{i}$ and $d(y_{i})=0$ is a free
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-resolution of $\mathbbm{k}$.

###### Proof.

Consider $\mathbbm{k}$ with the
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-module structure given by the
augmentation map sending each $y_{i}$ to zero. Rewrite $E$ as

$E=L(\omega_{1},\ldots,\omega_{m})\otimes_{\mathbbm{k}}\mathbbm{k}\langle
y_{1},\ldots,y_{m}\rangle$

emphasising that $\omega_{i}y_{j}=y_{j}\omega_{i}$. Then $(E,d)$ together with
the augmentation map $\varepsilon\colon E\longrightarrow\mathbbm{k}$ defines a
cochain complex of $\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-modules

(4) $0\longrightarrow
L^{m}(\omega_{1},\ldots,\omega_{m})\otimes\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\cdots\\\ \stackrel{{\scriptstyle
d}}{{\longrightarrow}}L^{1}(\omega_{1},\ldots,\omega_{m})\otimes\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle\stackrel{{\scriptstyle\varepsilon}}{{\longrightarrow}}\mathbbm{k}\longrightarrow
0$

where $L^{i}(\omega_{1},\ldots,\omega_{m})$ is the submodule of
$L(\omega_{1},\ldots,\omega_{m})$ generated by monomials of length $i$. We
shall show that $\varepsilon\colon(E,d)\longrightarrow(\mathbbm{k},0)$ is a
quasi-isomorphism. There is an obvious inclusion
$\eta\colon\mathbbm{k}\longrightarrow E$ such that
$\varepsilon\eta=\mathrm{id}$. To finish the proof we construct a cochain
homotopy between $\mathrm{id}$ and $\eta\varepsilon$, that is, a set of
$\mathbbm{k}$-linear maps $s=\\{s^{-i,j}\colon E^{-i,j}\longrightarrow
E^{-i-1,j}\\}$ satisfying the identity

(5) $ds+sd=\mathrm{id}-\eta\varepsilon.$

For $m=1$, we define the map $s_{1}\colon
E_{1}^{0,*}=\mathbbm{k}\langle{y}\rangle\longrightarrow E_{1}^{-1,*}$ by

$s_{1}(a_{0}+a_{1}y+\cdots+a_{j}y^{j})=\omega(a_{1}+a_{2}y+\cdots+a_{j}y^{j-1}).$

Then for $f=a_{0}+a_{1}y+\cdots+a_{j}y^{j}\in E_{1}^{0,*}$ we have
$ds_{1}f=f-a_{0}=f-\eta\varepsilon f$ and $s_{1}df=0$. On the other hand, for
$\omega f\in E_{1}^{-1,*}$ we have $s_{1}d(\omega f)=\omega f$ and
$ds_{1}(\omega f)=0$. In any case (5) holds. Now we assume by induction that
for $m=k-1$ the required cochain homotopy $s_{k-1}\colon
E_{k-1}\longrightarrow E_{k-1}$ is already constructed. Since
$E_{k}=E_{k-1}\otimes E_{1}$,
$\varepsilon_{k}=\varepsilon_{k-1}\otimes\varepsilon_{1}$ and
$\eta_{k}=\eta_{k-1}\otimes\eta_{1}$, a direct calculation shows that the map

$s_{k}=s_{k-1}\otimes\mathrm{id}+\eta_{k-1}\varepsilon_{k-1}\otimes s_{1}$

is a cochain homotopy between $\mathrm{id}$ and $\eta_{k}\varepsilon_{k}$.

Since
$L^{i}(\omega_{1},\ldots,\omega_{m})\otimes\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$
is a free $\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-module, (4) is a
free resolution for the $\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-module
$\mathbbm{k}$. This is an analog of the Koszul resolution. ∎

Since
$L(\omega_{1},\ldots,\omega_{m})\otimes\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$
is a resolution of $\mathbbm{k}$ by free
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$-modules, it follows that the
cohomology of the complex

$\Big{(}L(\omega_{1},\ldots,\omega_{m})\otimes_{\mathbbm{k}}\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle\Big{)}\otimes_{\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle}SR\langle
K\rangle=L(\omega_{1},\ldots,\omega_{m})\otimes_{\mathbbm{k}}SR\langle
K\rangle$

is isomorphic as a $\mathbbm{k}$-module to
$\mathrm{Tor}_{\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle}(SR\langle
K\rangle,\mathbbm{k})$.

We aim to show that there is an additive isomorphism between the cohomology
$H^{*}((D^{0},S^{1})^{K};\mathbbm{k})$ and the $\mathbbm{k}$-module
$\mathrm{Tor}_{\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle}(SR\langle
K\rangle,\mathbbm{k})$. Following Buchstaber-Panov’s work [4], the idea is to
first reduce the differential graded algebra $(E,d)$ to a finite dimensional
quotient $\bar{R}(K)$ without changing the cohomology. We then proceed by
showing that as a $\mathbbm{k}$-module $\bar{R}(K)$ is quasi-isomorphic to the
underlying $\mathbbm{k}$-module of a certain differential graded algebra
$B(K)$ which in turn is quasi-isomorphic to the singular cochains of
$(D^{1},S^{0})^{K}$.

######  Definition 2.2.

Let $K$ be a simplicial complex on $[m]$. Define the differential graded
algebra $(\bar{R}(K),d)$ by

$\bar{R}(K)=L(\omega_{1},\ldots,\omega_{m})\otimes SR\langle
K\rangle/(y_{i}^{2}=\omega_{i}y_{i}=0)$

with differential $d$ induced from $(E,d)$, that is, $d(w_{i})=y_{i}$ and
$d(y_{i})=0$.

######  Proposition 2.3.

The quotient map

$L(\omega_{1},\ldots,\omega_{m})\otimes SR\langle
K\rangle\longrightarrow\bar{R}(K)$

is an algebra quasi-isomorphism.

###### Proof.

There is a short exact sequence of differential graded algebras

$0\longrightarrow\mathcal{I}\longrightarrow\Lambda(\omega_{1},\ldots,\omega_{m})\otimes
SR\langle K\rangle\longrightarrow\Lambda(\omega_{1},\ldots,\omega_{m})\otimes
SR\langle K\rangle/\mathcal{I}\longrightarrow 0$

where $\mathcal{I}$ is the ideal $(y_{i}^{2}=\omega_{i}y_{i}=0)$.
Specifically, $\mathcal{I}$ is generated by monomials which are divisible by
$y_{i}^{2}$ or $\omega_{i}y_{i}$ for some $i$.

We show that $H^{*}\mathcal{I}=0$. For a monomial $x\in\mathcal{I}$, there is
a minimal index $i(x)$ such that either $y_{i(x)}^{2}$ or
$\omega_{i(x)}y_{i(x)}$ divides $x$. Define
$s\colon\mathcal{I}\longrightarrow\mathcal{I}$ on generating monomials by

$s(x)=\omega_{i(x)}\frac{x}{y_{i(x)}}.$

By showing that $s$ is a chain homotopy between the identity and zero, that
is, $ds(x)-sd(x)=x$ the proposition statement follows.

Since $x=\frac{x}{y_{i(x)}}y_{i(x)}$, we have

$ds(x)-sd(x)=d(\omega_{i(x)}\frac{x}{y_{i(x)}})-s(d(\frac{x}{y_{i(x)}})y_{i(x)})=x+\omega_{i(x)}d(\frac{x}{y_{i(x)}})-s(d(\frac{x}{y_{i(x)}})y_{i(x)}).$

We first observe that $y^{2}_{i(x)}$ divides $d(\frac{x}{y_{i(x)}})y_{i(x)}$.
To see this, note that either $y_{i(x)}$ or $\omega_{i(x)}$ divide
$\frac{x}{y_{i(x)}}$. In either case $y_{i(x)}$ divides
$d(\frac{x}{y_{i(x)}})$.

We claim that for $j<i(x)$, neither $y_{j}^{2}$ nor $\omega_{j}y_{j}$ divide
$d(\frac{x}{y_{i(x)}})$.

Assuming the claim for the moment, we first finish the proof. Since
$y_{i(x)}^{2}$ divides $d(\frac{x}{y_{i(x)}})y_{i(x)}$ and for smaller $j$,
both $y_{j}^{2}$ and $\omega_{j}y_{j}$ do not divide
$d(\frac{x}{y_{i(x)}})y_{i(x)}$, it follows that
$s(d(\frac{x}{y_{i(x)}})y_{i(x)})=\omega_{i(x)}d(\frac{x}{y_{i(x)}})$.

We next prove the claim. When $j<i(x)$, for $y_{j}^{2}$ or $\omega_{j}y_{j}$
to divide $d(\frac{x}{y_{i(x)}})$ either $y_{j}$ or $\omega_{j}$ divides
$\frac{x}{y_{i(x)}}$. If $y_{j}$ divides $\frac{x}{y_{i(x)}}$, then
$y_{j}^{2}$ cannot divide $x$ because $j<i(x)$. So if either $y_{j}^{2}$ or
$\omega_{j}y_{j}$ divides $d(\frac{x}{y_{i(x)}})$, we must have $\omega_{j}$
devides $\frac{x}{y_{i(x)}}$. But then neither $\omega_{j}$ nor $y_{j}$ can
divide $(\frac{x}{y_{i(x)}})/\omega_{j}$. Thus

$d\left(\frac{x}{y_{i(x)}}\right)=d\omega_{j}\left[\left(\frac{x}{y_{i(x)}}\right)/\omega_{j}\right]=y_{j}\left[\left(\frac{x}{y_{i(x)}}\right)/\omega_{j}\right]+\omega_{j}d\left[\left(\frac{x}{y_{i(x)}}\right)/\omega_{j}\right].$

Neither summand is divisible by $y_{j}^{2}$ or $\omega_{j}y_{j}$. ∎

######  Corollary 2.4.

As $\mathbbm{k}$-modules,

$\mathrm{Tor}_{\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle}(SR\langle
K\rangle,\mathbbm{k})\cong H^{\ast}(\bar{R}(K)).$

∎

Recalling that there is the bigrading of $\omega_{i}$ and $y_{i}$ given by
$\mathop{\rm bideg}(w_{i})=(-1,1)$ and $\mathop{\rm bideg}(y_{i})=(0,1)$, we
can consider $\bar{R}(K)$ as a bigraded differential algebra. Recall that for
$I=(i_{1},\ldots,i_{p}),i_{1}<\ldots<i_{p},L=(y_{l_{1}},\ldots,y_{l_{s}}),l_{1}<\ldots<l_{s}$,
the monomial $\omega_{i_{1}}\cdots\omega_{i_{p}}\cdot y_{l_{1}}\cdots
y_{l_{s}}$ is denoted by $\omega_{I}y_{L}$.

For $0\leq p\leq m$, a $\mathbbm{k}$-module basis for $\bar{R}^{-p,*}(K)$ can
be given as

$\\{\omega_{I}y_{L}\,|L\in K,|I|=p,I\cap L=\emptyset\\}.$

To consider $\bar{R}(K)$ as a differential $\mathbbm{k}$-module, the
differential $d\colon\bar{R}^{*,*}(K)\longrightarrow\bar{R}^{*,*}(K)$, induced
from its differential algebra structure, is given on the $\mathbbm{k}$-module
generators by

$d(\omega_{I}y_{L})=\sum_{k=1}^{p}(-1)^{k+1}\omega_{i_{1}}\cdots\omega_{i_{k-1}}\hat{\omega}_{i_{k}}\omega_{i_{k+1}}\cdots\omega_{i_{p}}y_{l_{1}}\cdots
y_{l_{r}}y_{i_{k}}y_{l_{r+1}}\cdots y_{l_{s}}$

where $l_{r}<i_{k}<l_{r+1}$. Notice that the sign is induced by the derivation
identity (3) and if the set
$(l_{1},\ldots,l_{r},i_{k},l_{r+1},\ldots,l_{s})\notin K$, then
$\omega_{i_{1}}\cdots\omega_{i_{k-1}}\hat{\omega}_{i_{k}}\omega_{i_{k+1}}\cdots\omega_{i_{p}}y_{l_{1}}\cdots
y_{l_{r}}y_{i_{k}}y_{l_{r+1}}\cdots y_{l_{s}}=0$.

Next we relate the $\mathbbm{k}$-module $\bar{R}(K)$ to the
$\mathbbm{k}$-cohomology module of the real moment-angle complex
$(D^{1},S^{0})^{K}$ via the differential graded algebra model $B(K)$ of
$H^{*}((D^{1},S^{0})^{K};\mathbbm{k})$ given by Cai [5] and Franz [6].

The differential graded algebra $B(K)$ is presented with generators $s_{i}$
and $t_{i}$ such that $\deg(s_{i})=0$ and $\deg(t_{i})=1$ and the relations

(6) $s_{i}s_{i}=s_{i},\quad t_{i}s_{i}=t_{i},\quad s_{i}t_{i}=0,\quad
t_{i}t_{i}=0,\quad\prod_{j\in\sigma,\sigma\notin K}t_{j}=0$

for all $1\leq i\leq m$ and all variables with different indices are graded
commutative, that is,

$s_{i}s_{j}=s_{j}s_{i},\quad t_{i}t_{j}=-t_{j}t_{i},\quad
s_{i}t_{j}=t_{j}s_{i}\quad\text{for }i\neq j.$

The differential $d\colon B^{*}(K)\longrightarrow B^{*}(K)$ is given by

$ds_{i}=-t_{i},\quad dt_{i}=0$

and extended using the Leibniz rule $d(a\cdot b)=d(a)\cdot
b+(-1)^{\deg(a)}a\cdot d(b)$. Franz [6] proved that $B(K)$ is quasi-isomorphic
to $\mathcal{C}^{\ast}((D^{1},S^{0})^{K},\mathbbm{k}))$, the singular
$\mathbbm{k}$-cochains of the real moment-angle complex $(D^{1},S^{0})^{K}$.

We treat $B(K)$ as a bigraded differential algebra by setting that
$\mathop{\rm bideg}(s_{i})=(-1,1)$ and $\mathop{\rm bideg}(t_{i})=(0,1)$ for
all $1\leq i\leq m$. Then the $\mathbbm{k}$-module basis for
$B^{-p,*}(K),0\leq p\leq m$ is similar to the one for $\bar{R}^{-p,*}(K)$ and
is given by

$\\{s_{I}t_{L}\,|\,L\in K,\,I\cap L=\emptyset\\}$

where $s_{I}=s_{i_{1}}\cdots s_{i_{p}}$ for $I=(i_{1},\ldots,i_{p})$ and
$t_{L}=t_{l_{1}}\cdots t_{l_{s}}$ for $L=(l_{1},\ldots,l_{s})$. The
$\mathbbm{k}$-module differential $d\colon B^{*,*}(K)\longrightarrow
B^{*,*}(K)$ is given by

$d(s_{I}t_{J})=\sum_{k=1}^{p}(-1)^{r+1}s_{i_{1}}\cdots
s_{i_{k-1}}s_{i_{k+1}}\cdots s_{i_{p}}t_{l_{1}}\cdots
t_{l_{r}}t_{i_{k}}t_{l_{r+1}}\cdots t_{l_{s}}$

where $l_{r}<i_{k}<l_{r+1}$. Notice that in $B(K)$ the summand
$s_{i_{1}}\cdots s_{i_{k-1}}s_{i_{k+1}}\cdots s_{i_{p}}t_{l_{1}}\cdots
t_{l_{r}}t_{i_{k}}t_{l_{r+1}}\cdots t_{l_{s}}=0$ if
$(l_{1},\ldots,l_{r},i_{k},l_{r+1},\ldots,l_{s})\notin K$. The graded-
commutative Leibniz contributes $-1$ as $\deg(s_{i})=0$ and
$d(s_{i_{k}})=-t_{i_{k}}$. The additional $(-1)^{r}$ in the differential
formula comes about from $t_{l_{k}}$ passing $r$ many $t_{j}$’s.

######  Definition 2.5.

Let an additive isomorphism $f\colon\bar{R}^{-p,*}(K)\longrightarrow
B^{-p,*}(K)$ be defined by

$f(\omega_{I}y_{L})=\epsilon(I,L)s_{I}t_{L}$

where $\epsilon(I,L)$ is the sign of the permutation that converts $IL$, the
concatenation of $I$ followed by $L$, into an increasing sequence.

######  Proposition 2.6.

The $\mathbbm{k}$-isomorphism $f$ commutes up to sign with the differentials.
Specifically, if $\alpha\in\bar{R}^{-p,*}(K)$ then

$fd_{\bar{R}}(\alpha)=(-1)^{p}d_{B}f(\alpha).$

###### Proof.

For $\alpha=\omega_{I}y_{L}$,

$d(\alpha)-\Sigma(-1)^{k+1}\omega_{i_{1}}\cdots\omega_{i_{k-1}}\omega_{i_{k+1}}\cdots\omega_{i_{p}}y_{l_{1}}\cdots
y_{l_{r}}y_{i_{k}}y_{l_{r+1}}\cdots y_{l_{s}}.$

We compute the sign of the permutation that converts

(7)
$i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{p},l_{1},\ldots,l_{r},i_{k},l_{r+1},\ldots,l_{s}$

into an increasing sequence.

To start, note that the sequence of $i^{\prime}s$ and the sequence of
$l^{\prime}s$ are increasing. Therefore it suffices to permute the
$l_{j}^{\prime}s$ to the left starting with $l_{1}$ followed by increasing
$l_{j}^{\prime}s$.

The sign of the permutation that converts (7) into an increasing sequences
differs from $\epsilon(I,L)$ in two ways.

First, we move the indices $l_{1},\ldots,l_{r}$ to the left. Note that
$l_{j}<i_{k}$ for $j\leq r$. These $l_{j}^{\prime}s$ are meshed into
$i_{1},\ldots,i_{k-1}$. The number of permutations needed to place $l_{j}$
into its final position differs by $-1$ from the number needed to order $IL$
because $i_{k}$ is missing. So moving $l_{1},\ldots,l_{r}$ differs from
$\epsilon(I,L)$ by $(-1)^{r}$.

Second, we have to move $i_{k}$. Moving this index contributes $(-1)^{p-k}$.

Thus the sign of the permutation that converts (7) into an increasing sequence
differs from $\epsilon(I,L)$ by $(-1)^{r+k+p}$. Therefore the map $f$
multiplies the $k$th term in $d_{\bar{R}}(\alpha)$ by
$(-1)^{r+k+p}\epsilon(I,L)$ which is $(-1)^{p}\epsilon(I,L)$ times the
coefficient of the $k$-th term in $d_{B}$. ∎

######  Corollary 2.7.

The chain map $f$ induces a $\mathbbm{k}$-module isomorphism

$\mathrm{Tor}_{\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle}(SR\langle
K\rangle,\mathbbm{k})\longrightarrow H^{*}(B(K))\cong
H^{*}((D^{1},S^{0})^{K};\mathbbm{k}).$

∎

###  2.2. A generalisation to polyhedral products

In many ways, a combinatorial contribution of the simplicial complex $K$ to
homotopy theoretical properties of the real and complex moment-angle complexes
is comparable and usually suggests a formulation of those properties for
polyhedral products $(\underline{CX},\underline{X})^{K}$, see for example [8,
3]. Having that both cohomology $\mathbbm{k}$-modules
$H^{*}((D^{2},S^{1});\mathbbm{k})$ and $H^{*}(D^{1},S^{0});\mathbbm{k})$ can
be expressed in terms of appropriate Tor $\mathbbm{k}$-modules, we generalise
the previous discussion to give an additive description of
$H^{\ast}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ as a direct summand
of $\mathrm{Tor}_{A}(M,\mathbbm{k})$, where $A$ and $M$ are defined below.

Throughout the remainder of the paper, when the cohomology
$H^{\ast}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ of the polyhedral
product $(\underline{CX},\underline{X})^{K}$ is considered we assume that
$H^{\ast}(X_{i};\mathbbm{k})$ are free $\mathbbm{k}$-modules for all $i$. If
not explicitly stated all cohomology groups of topological spaces are taken
with $\mathbbm{k}$ coefficients.

Bahri, Bendersky, Cohen and Gitler [1] gave a stable homotopy decomposition of
$(\underline{CX},\underline{X})^{K}$ by establishing the homotopy equivalence

(8)
$\Sigma(\underline{CX},\underline{X})^{K}\longrightarrow\Sigma\bigvee_{J\subseteq[m]}|\Sigma
K_{J}|\wedge Y^{\wedge J}$

where

$Y^{\wedge J}=\underset{i\in[m]}{\bigwedge}Y_{i}\mbox{ with }Y_{i}=X_{i}\mbox{
if }i\in J\mbox{ and }Y_{i}=S^{0}\mbox{ if }i\notin J.$

They further showed [2] that the map

(9)
$H^{\ast}((\underline{CX},\underline{X})^{K};\mathbbm{k})\longrightarrow\bigoplus_{J\subseteq[m]}H^{\ast}(|\Sigma
K_{J}|;\mathbbm{k})\otimes H^{J}$

where $H^{J}=\underset{i\in[m]}{\otimes}H_{i}$ with
$H_{i}=\widetilde{H}^{\ast}(X_{i};\mathbbm{k})$ if $i\in J$ and
$H_{i}=\mathbbm{k}$ if $i\notin J$ is an isomorphism of rings. This
specialises to an isomorphism

(10)
$H^{\ast}((D^{1},S^{0})^{K};k)\cong\bigoplus_{J\subseteq[m]}H^{\ast}(|\Sigma
K_{J}|;\mathbbm{k}).$

It is proven in [2] that the cup product on
$H^{\ast}((D^{1},S^{0})^{K};\mathbbm{k})$ restricts to a pairing

$H^{\ast}(|\Sigma K_{J}|;\mathbbm{k})\otimes H^{\ast}(|\Sigma
K_{L}|;\mathbbm{k})\to H^{\ast}(|\Sigma K_{J\cup L}|;\mathbbm{k}).$

This in turn induces a product on

(11) $\bigoplus_{J\subseteq[m]}H^{\ast}(|\Sigma K_{J}|;\mathbbm{k})\otimes
H^{J}$

which is called the $\ast-$product in [2].

To give a $\mathbbm{k}$-module model for
$H^{\ast}((\underline{CX},\underline{X})^{K};\mathbbm{k})$, we shall only use
the underlying additive isomorphism in (9). In Section 3 we shall invoke the
multiplicative structure.

By Corollaries 2.4 and 2.7, the cohomology $\mathbbm{k}$-module structure of
$(D^{1},S^{0})^{K}$ is also modeled by the differential graded algebra
$\bar{R}(K)$. We next identify the summands in (10) with sub dga’s of
$\bar{R}(K)$.

######  Definition 2.8.

For an element $\omega_{I}y_{L}\in\bar{R}(K)$, define its support as

$\mathrm{supp}(\omega_{I}y_{L})=I\cup L.$

The differential sub-module of $\bar{R}(K)$ generated by monomials with
support $J$ is denoted by $\bar{R}_{J}(K)$.

The following lemma is a direct consequence of [6] and [4, Lemma 4.5.1].

######  Lemma 2.9.

There is an algebra isomorphism

$H^{\ast}(\bar{R}_{J}(K);\mathbbm{k})\cong
H^{\ast}(\Sigma|K_{J}|);\mathbbm{k}).$

∎

######  Definition 2.10.

Let $(C(\underline{X},K),d)$ be a differential graded $\mathbbm{k}$-module
defined by

$C(\underline{X},K)=\bigoplus_{J\subset[m]}\bar{R}_{J}(K)\otimes H^{J}$

where $H^{J}=\underset{i\in[m]}{\bigotimes}H_{i}$ and
$H_{i}=\widetilde{H}^{\ast}(X_{i};\mathbbm{k})$ if $i\in J$ and
$H_{i}=\mathbbm{k}$ if $i\notin J$.

The differential $d\colon C^{*}(\underline{X},K)\longrightarrow
C^{*}(\underline{X},K)$ is given as the tensor product of the differential on
$\bar{R}_{J}(K)$, induced from the dga $\bar{R}(K)$, and the trivial
differential on $H^{J}$, induced by the trivial differential on
$H^{*}(X_{i})$.

Combining Lemma 2.9 with (9), the following statement holds.

######  Lemma 2.11.

There is a $\mathbbm{k}$-module isomorphism

$H^{*}((\underline{CX},\underline{X})^{K});\mathbbm{k})\longrightarrow
H^{*}(C(\underline{X},K)).$

∎

To integrate the topological structure of the polyhedral product
$(\underline{CX},\underline{X})^{K}$ into our cohomology model, we start by
generalising the polynomial algebra
$\mathbbm{k}\langle{y_{1},\ldots,y_{m}}\rangle$ to a free commutative algebra
on the cohomology of $\Sigma\underline{X}$.

######  Definition 2.12.

For $CW$-complexes $X_{i},1\leq i\leq m$, let

(12) $\mathbbm{k}\langle{\bigoplus_{1\leq i\leq m}\widetilde{H}^{*}(\Sigma
X_{i})}\rangle=T(\bigoplus_{1\leq i\leq m}\widetilde{H}^{*}(\Sigma
X_{i}))/(\alpha\beta=\beta\alpha\,|\,\alpha,\beta\in\bigoplus_{1\leq i\leq
m}\widetilde{H}^{*}(\Sigma X_{i}))$

where $T(M)$ denotes the free associative algebra generated by a free
$\mathbbm{k}$-module $M$.

To algebra 12 we associate a generalised Stanley-Reisner ring.

######  Definition 2.13.

For $CW$-complexes $X_{i},1\leq i\leq m$, define the generalised Stanley-
Reisner ring $SR(\underline{X},K)$ as

$SR(\underline{X},K)=\mathbbm{k}\langle{\underset{1\leq i\leq
m}{\bigoplus}\widetilde{H}^{*}(\Sigma X_{i})}\rangle/\mathcal{I}_{K}$

where $\mathcal{I}_{K}$ is the ideal generated by square free monomial

$\alpha_{i_{1}}\cdots\alpha_{i_{t}},\text{ where
}\alpha_{i_{j}}\in\widetilde{H}^{\ast}(X_{i_{j}})\text{ and
}\\{i_{1},\cdots,i_{t}\\}\notin K.$

We prove that $H^{*}(C(\underline{X},K))$ additively splits off
$\mathrm{Tor}_{\mathbbm{k}\langle{\bigoplus\widetilde{H}^{*}(\Sigma
X_{i})}\rangle}(SR(\underline{X},K),\mathbbm{k})$. To this end we choose an
ordered bases $B_{i}$ for $\widetilde{H}^{\ast}(\Sigma X_{i})$,

$B_{i}=\\{b_{i,1},\cdots,b_{i,k_{i}}\\}$

which induces an ordering on
$\underset{i}{\bigoplus}\widetilde{H}^{\ast}(\Sigma X_{i})$ by saying that
$b<b^{\prime}$ if $b\in H^{\ast}(\Sigma X_{i}),b^{\prime}\in H^{\ast}(\Sigma
X_{j})$ and $i<j$. To make $\underset{i}{\bigoplus}\widetilde{H}^{\ast}(\Sigma
X_{i})$ into a bigraded object, let $\mathop{\rm bideg}(b)=(0,|b|)$.

We define $(L(\widetilde{H}^{*}(\underline{X})),d)$ and
$(E(\widetilde{H}^{*}(\underline{X})),d)$, the natural generalisation of (1)
and (2), as

$L(\widetilde{H}^{*}(\underline{X}))=L(\underset{i}{\oplus}uB_{i})=T(\oplus
uB_{i})/(ub_{i}^{2}=0,\ ub_{i}ub_{j}=-ub_{i}ub_{j})$
$E((\widetilde{H}^{*}(\underline{X}))=L(\widetilde{H}^{*}(\underline{X}))\otimes\mathbbm{k}\langle{\bigoplus_{1\leq
i\leq m}\widetilde{H}^{*}(\Sigma X_{i})}\rangle$

where $uB_{i}$ is generated by classes $ub$ of bidegree $(-1,|b|)$
corresponding to $b\in B_{i}$. Define the differential $d$ on
$E(\widetilde{H}^{*}(\underline{X}))$ by $d(ub)=b$, $d(b)=0$ and by requiring
that $d$ satisfies the Leibniz identity $d(a\cdot b)=d(a)\cdot
b+(-1)^{|\deg_{1}(a)|}a\cdot d(b)$, where $\mathop{\rm
bideg}(a)=(\deg_{1}(a),\deg_{2}(a))$. As in Lemma 2.1,
$E(\widetilde{H}^{*}(\underline{X}))$ is a resolution of $\mathbbm{k}$.
Consequently the cohomology of the Koszul complex

$L(\underset{i}{\bigoplus}uB_{i})\otimes SR(\underline{X},K)$

is $\mathrm{Tor}_{\mathbbm{k}\langle{\bigoplus\widetilde{H}^{*}(\Sigma
X_{i})}\rangle}(SR(\underline{X},K),\mathbbm{k})$.

######  Definition 2.14.

Let $K$ be a simplicial complex on $[m]$ and let
$\underline{X}=\\{X_{i}\\}_{i=1}^{m}$ be $CW$-complexes. Define the
differential graded algebra $(R(\underline{X},K),d)$ by

$R(\underline{X},K)=L(\underset{i}{\oplus}uB_{i})\otimes
SR(\underline{X},K)/(b_{i}^{2}=(ub_{i})b_{i}=0)$

with differential $d$ induced from the differential graded algebra
$(E(\widetilde{H}^{*}(\underline{X})),d)$.

Following the lines of the proof of Proposition 2.3, we established that the
finite differential graded algebra $R(\underline{X},K)$ is a model for the Tor
$\mathbbm{k}$-module
$\mathrm{Tor}_{\mathbbm{k}\langle{\bigoplus\widetilde{H}^{*}(\Sigma
X_{i})}\rangle}(SR(\underline{X},K),\mathbbm{k})$.

######  Lemma 2.15.

The quotient map

$L(\underset{i}{\oplus}uB_{i})\otimes SR(\underline{X},K)\longrightarrow
R(\underline{X},K)$

is an algebra quasi-isomorphism. ∎

To describe an additive basis of $R(\underline{X},K)$ we note that $\oplus
uB_{i}$ inherits an ordering from the ordering of $\oplus B_{i}$. A basis for
$R(\underline{X},K)$ is given by

(13) $\\{ub_{i_{1},k_{1}}\cdots ub_{i_{s},k_{s}}b_{l_{1},j_{1}}\cdots
b_{l_{t},j_{t}}\\}$

where

  1. (i)

if $ub$ is a factor, then $b$ is not a factor;

  2. (ii)

$ub_{i_{1},k_{1}}<\cdots<ub_{i_{s},k_{s}}$,
$b_{l_{1},j_{1}}<\cdots<b_{l_{t},j_{t}}$;

  3. (iii)

$\\{l_{1},\cdots,l_{t}\\}\in K$.

There is a similar basis for $C(\underline{X},K)$,

(14) $\left\\{\omega_{i_{1}}\cdots\omega_{i_{s}}y_{l_{1}}\cdots
y_{l_{t}}\otimes\Big{[}s^{-1}b_{i_{1},k_{1}}\cdots
s^{-1}b_{i_{s},k_{s}}\Big{]}\otimes\Big{[}s^{-1}b_{l_{1},j_{1}}\cdots
s^{-1}b_{l_{t},j_{t}}\Big{]}\right\\}.$

The difference between the basis (13) and basis (14) is that the integers
$\\{i_{1},\cdots i_{s},l_{1},\cdots l_{t}\\}$ are all distinct in (14).

We now compare the differential graded algebras $R(\underline{X},K)$ and
$C(\underline{X},K)$.

######  Proposition 2.16.

The cohomology $H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$, seen
as a $\mathbbm{k}$-module, is a direct summand of the $\mathbbm{k}$-module
$\mathrm{Tor}_{\mathbbm{k}\langle{\bigoplus\widetilde{H}^{*}(\Sigma
X_{i})}\rangle}(SR(\underline{X},K),\mathbbm{k})$.

###### Proof.

There are maps of differential $\mathbbm{k}$-modules:

$h\colon C(\underline{X},K)\longrightarrow R(\underline{X},K)$

and

$g\colon R(\underline{X},K)\longrightarrow C(\underline{X},K)$

given by

$h\Big{(}\omega_{i_{1}}\cdots\omega_{i_{s}}y_{l_{1}}\cdots
y_{l_{t}}\otimes\Big{[}s^{-1}b_{i_{1},k_{1}}\cdots
s^{-1}b_{i_{s},k_{s}}\Big{]}\otimes\Big{[}s^{-1}b_{l_{1},j_{1}}\cdots
s^{-1}b_{l_{t},j_{t}}\Big{]}\Big{)}=(ub_{i_{1},k_{1}}\cdots
ub_{i_{s},k_{s}})(b_{l_{1},j_{1}}\cdots b_{l_{t},j_{t}})$

and

$g\Big{(}(ub_{i_{1},k_{1}}\cdots ub_{i_{s},k_{s}})(b_{l_{1},j_{1}}\cdots
b_{l_{t},j_{t}})\Big{)}=\omega_{i_{1}}\cdots\omega_{i_{s}}y_{l_{1}}\cdots
y_{l_{t}}\otimes\Big{[}s^{-1}b_{i_{1},k_{1}}\cdots
s^{-1}b_{i_{s},k_{s}}\Big{]}\otimes\Big{[}s^{-1}b_{l_{1},j_{1}}\cdots
s^{-1}b_{l_{t},j_{t}}\Big{]}.$

The homomorphisms $h$ and $g$ commute with differentials and $g\circ h=id$. ∎

We note that
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})\cong\mathrm{Tor}_{\mathbbm{k}\langle{\bigoplus\widetilde{H}^{*}(\Sigma
X_{i})}\rangle}(SR(\underline{X},K),\mathbbm{k})$ if $X_{i}=S^{n_{i}}$ for
each $i$. For $\widetilde{H}^{*}(X_{i})$ with more than one generator, the map
$g$ is zero on generators, $(ub_{i_{1},k_{1}}\cdots
ub_{i_{s},k_{s}})(b_{l_{1},j_{1}}\cdots b_{l_{t},j_{t}})$ whenever a
repetition occurs in the list

$i_{1},\ldots,i_{s},l_{1},\ldots,l_{t}$

obtained by dropping the second subscripts in the bi-indexing.

##  3\. Algebra models

In this section we coalesce the Bahri-Bendersky-Cohen-Gitler $*-$product (11)
on $H^{*}((\underline{CX},\underline{X})^{K})$ with the Cai [5] and Franz [6]
differential algebra $B(K)$ to give a natural differential algebra
$B(\underline{X},K)$ whose cohomology is isomorphic the the cohomology of
$(\underline{CX},\underline{X})^{K}$.

######  Definition 3.1.

Let $K$ be a simplicial complex on $[m]$ and let
$\underline{X}=\\{X_{i}\\}_{i=1}^{m}$ be $CW$-complexes. Define a differential
bigraded non-commutative algebra $(B(\underline{X},K),d)$ as

$B(\underline{X},K)=\bigoplus_{J\subset[m]}B_{J}(K)\otimes H^{J}$

where $B_{J}(K)$ is a subalgebra of $B(K)$ consisting of elements with support
$J$ and $H^{J}=\underset{i\in[m]}{\bigotimes}H_{i}$ with
$H_{i}=\widetilde{H}^{\ast}(X_{i};\mathbbm{k})$ if $i\in J$ and
$H_{i}=\mathbbm{k}$ if $i\notin J$.

The differential $d$ on $B(\underline{X},K)$ is given as the tensor product of
the differential on $B_{J}(K)$, induced from the dga $B(K)$, and the trivial
differential on $H^{J}$, induced by the trivial differential on
$H^{*}(X_{i})$.

We now recognise $B(\underline{X},K)$ as a dga model for the cohomology of
$(\underline{CX},\underline{X})^{K}$.

######  Proposition 3.2.

Let $K$ be a simplicial complex on $[m]$ and let
$\underline{X}=\\{X_{i}\\}_{i=1}^{m}$ be $CW$-complexes with $H^{*}(X_{i})$
being free $\mathbbm{k}$-modules. Then the dga $B(\underline{X},K)$ is quasi-
isomorphic to $H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$.

###### Proof.

The statement follows as a direct consequence of the Bhari-Bendersky-Cohen-
Gitler description of the cup product structure on
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$ given by the
$*-$product 11, the Künneth theorem and the result of Franz [6] stating that
the dga $B(K)$ is quasi-isomorphic to $\mathcal{C}^{\ast}((D^{1},S^{0})^{K})$.
∎

We next enhance the construction to give a dga which is quasi-isomorphic to
$\mathcal{C}^{\ast}((\underline{CX},\underline{X})^{K},\mathbbm{k})$, the
singular cochains of $(\underline{CX},\underline{X})^{K}$.

######  Definition 3.3.

Let $K$ be a simplicial complex on vertex set $[m]$ and let
$\underline{X}=\\{X_{i}\\}_{i=1}^{m}$ be $CW$-complexes. Define a differential
bigraded non-commutative algebra $(B(\mathcal{C}^{*}(\underline{X}),K),d)$ as

$B(\mathcal{C}^{*}(\underline{X}),K)=\bigoplus_{J\subset[m]}B_{J}(K)\otimes\mathcal{C}^{J}$

where $B_{J}(K)$ is a subalgebra of $B(K)$ consisting of elements with support
$J$ and $\mathcal{C}^{J}=\underset{i\in[m]}{\bigotimes}\mathcal{C}_{i}$ with
$\mathcal{C}_{i}=\mathcal{C}^{\ast}(X_{i},\mathbbm{k})$ if $i\in J$ and
$\mathcal{C}_{i}=\mathbbm{k}$ if $i\notin J$.

The differential $d$ on $B(\mathcal{C}^{*}(\underline{X}),K)$ is given as the
tensor product of the differential on $B_{J}(K)$, induced from the dga $B(K)$,
and the differential on $\mathcal{C}^{J}$, induced by the differential on
$\mathcal{C}^{*}(X_{i},\mathbbm{k})$.

######  Proposition 3.4.

Let $K$ be a simplicial complex on $[m]$ and let
$\underline{X}=\\{X_{i}\\}_{i=1}^{m}$ be $CW$-complexes with $H^{*}(X_{i})$
being free $\mathbbm{k}$-modules. Then the dga
$\mathcal{C}^{*}((C\underline{X},\underline{X})^{K})$ is quasi-isomorphic to
the dga $B(\mathcal{C}^{\ast}(\underline{X}),K)$.∎

###### Proof.

The statement is a straightforward consequence of Proposition 3.2 and the
Künneth formula. ∎

The bar construction applied to $B(\mathcal{C}^{\ast}(\underline{X}),K)$ gives
a model for the loops space of $(\underline{CX},\underline{X})^{K}$ when
spaces $X_{i}$ have torsion free $\mathbbm{k}$-cohomology. We will return to
this point in a future paper.

We finish the paper by compering the dga $C(\underline{X},K)$ with the dga
$B(\underline{X},K)$. Restricting to the $\mathbbm{k}$-module structures, we
observe that $B(\underline{X},K)$ is a differential bigraded
$\mathbbm{k}$-module model for
$H^{*}((\underline{CX},\underline{X})^{K};\mathbbm{k})$.

By extending the map $f\colon\bar{R}(K)\longrightarrow B(K)$ in Definition
2.5, we define an isomorphism of differential $\mathbbm{k}$-modules
$f_{\underline{X}}\colon C(\underline{X},K)\longrightarrow B(\underline{X},K)$
by

$f_{\underline{X}}(\omega_{I}y_{L}\otimes h)=\epsilon(I,L)s_{I}t_{L}\otimes h$

where $\epsilon(I,L)$ is the sign of the permutation that converts $IL$, the
concatenation of $I$ followed by $L$, into an increasing sequence.

Straightforwardly, following the proof of Proposition 2.6 we have the
following statement.

######  Lemma 3.5.

The additive isomorphism $f_{\underline{X}}$ commutes up to sign with the
differentials. Specifically,

$f_{\underline{X}}d_{C(\underline{X},K)}(\omega_{I}y_{L}\otimes
h)=(-1)^{|I|}d_{B(\underline{X},K)}f_{\underline{X}}(\omega_{I}y_{L}\otimes
h).$

∎

There is a natural algebra structure on
$C(\underline{X},K)=\bigoplus_{J\subset[m]}\bar{R}_{J}(K)\otimes H^{J}$
induced by the algebra structures on $\bar{R}(K)$ and $H^{*}(X_{i})$ for all
$i$. Notice that $\bar{R}(K)$ is a commutative algebra while $B(K)$ is not
commutative. Therefore, although $B(\underline{X},K)$ and $C(\underline{X},K)$
are isomorphic as $\mathbbm{k}$-modules, the isomorphism cannot be extended to
the one of algebras. However, in the case when all $X_{i}$’s are suspension
spaces the algebra structure of $B(\underline{X},K)$ reduces to the one of
$C(\underline{X},K)$.

######  Proposition 3.6.

The algebra structures on $B(\underline{X},K)$ and $C(\underline{X},K)$
coincide up to sign on the product of classes with disjoint support. In
particular, the algebra structures are isomorphic up to sign if $X_{i}$ is a
suspension space for all $i$.

###### Proof.

It is enough to see that dgas $\bar{R}(K)$ and $B(K)$ differ only on elements
with repeated indices. Notice that in Definition 2.2 of $\bar{R}(K)$, we
quotient out $y_{i}^{2}$ and $\omega_{i}y_{i}$. Therefore elements in which
indices are repeated are trivial. On the other hand, the defining relations
(6) of $B(K)$ state that the products with repeated indices, such as
$s_{i}s_{i}=s_{i}$ and $t_{i}s_{i}=t_{i}$, are not trivial. Moreover, together
with $s_{i}t_{i}=0$, these relations imply that $B(K)$ is a non-commutative
dga.

If we assume that $X_{i}$’s are suspension spaces, then since the diagonal map
on suspension spaces is null-homotopic, the cup product on $H^{J}$ is trivial
if indices are repeated. This trivialises products with repeated indices in
$B(\underline{X},K)$ as well despite the product being non-trivial in $B(K)$.
∎

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