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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ  ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
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     {\flushleft\it Том ??, № ?. ??--?? (2022) \hspace{65mm}{\rm\small УДК \udcs}}
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     	{\rm\small DOI~10.????/semi.2022.??.??}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\title[Hochschild cohomology of the Manturov group $\Gamma ^4_5$]{Grobner Shirshov basis and Hochschild cohomology of the Manturov group $\Gamma ^4_5$}
\author{{Hassan AlHussein}}%
\address{AlHussein H.
\newline\hphantom{iii} Novosibirsk State University of Economics and Management, Russia,
\newline\hphantom{iii} St. Kamenskaya, 52,
\newline\hphantom{iii} 630099, Novosibirsk, Russia}%
\email{hassanalhussein2014@gmail.com}%

\thanks{\sc AlHussein, H.,
Grobner Shirshov basis and Hochschild cohomology of the Manturov group $\Gamma ^4_5$}
\thanks{\copyright \ 2022 AlHussein, H.}



\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. }In this paper, we construct a Gr\"obner---Shirshov basis for the group $\Gamma^4_5$ with respect to the 
 tower order on the words. By using this result, we apply the discrete algebraic Morse theory to find explicitly the differentials of the Anick resolution for $\Gamma^4_5$, and calculate the first and second Hochschild cohomology groups of the group algebra of $\Gamma^4_5$ 
with coefficients the trivial 1-dimensional bimodule over a field $\Bbbk$ of characteristic zero.\medskip

\noindent{\bf Keywords:} Gr\"obner---Shirshov basis, Anick resolution, Hochschild cohomology, Manturov group.
 \end{quote}
}


\section{Introduction}
V.~O.~Manturov \cite{manturov}--\cite{manturov25} introduced a family of groups $G^k_n$ depending on two positive integers $n>k$, and formulated the following principle: If a dynamical system describing a motion of $n$ particles possesses a nice codimension one property governed by $k$ particles, then it has topological invariants valued in the group $G^k_n$.
%which generalize classical  braid groups, virtual braid groups %and other gruops in a very broad sense. 
In the paper
\cite{manturov2}, V.O.~Manturov constructed maps from fundamental groups of configuration spaces to the groups $G^k_n$. 
Another group $\Gamma^4_n$ corresponding to the motion of points in $\mathbb{R}^3$
was proposed \cite{manturov19} from the point of view of Delaunay triangulations

In terms of generators and defining relations the group $\Gamma^4_n$ is defined as follows. It is generated by the set
\[
\{d_{(i,j,k,l)} \mid \{i,j,k,l\}|=4 \quad and \quad \{i,j,k,l\} \in \{1,2,\dots,n\}\}
\]
subject to the following relations:
\[
\begin{aligned}
&d^2_{(i,j,k,l)}=1\\
 &d_{(i,j,k,l)}d_{(s,t,u,v)}= d_{(s,t,u,v)} d_{(i,j,k,l)} ; |\{ i,j,k,l\}\cap\{ i,j,k,l\}|< 3\\
&d_{(i,j,k,l)} d_{(i,j,k,m)} d_{(i,j,l,m)} d_{(i,k,l,m)} d_{(j,k,l,m)}=1\\
&d_{(i,j,k,l)}= d_{(l,i,j,k)}= d_{(k,l,i,j)}= d_{(j,k,l,i)}=\\ 
&d_{(i,l,k,j)} = d_{(j,i,l,k)}= d_{(k,j,i,l)} = d_{(l,k,j,i)}.
\end{aligned}
\]
In the paper \cite{manturov19}, the authors studied homomorphisms from the group of pure braids on $n$ strands to the product of copies of $\Gamma^4_n$, and also studied the group of pure braids in $\mathbb{R}^3$, which is described by a fundamental
group of the restricted configuration space of $\mathbb{R}^3$.
They defined a group homomorphism from the group of pure braids in $\mathbb{R}^3$ to $\Gamma^4_n$, and gave some comments about relations between the restricted configuration
space of $\mathbb{R}^3$ and triangulations of the $3$-dimensional ball and Pachner moves.
Therefore, it is useful to study these groups from 
combinatorial point if view.

This paper is devoted to finding Gr\"obner---Shirshov basis and cohomology of $G^4_n$ for $n=5$ (the first infinite example in the series).
Gr\"obner basis and Gr\"obner---Shirshov basis were invented independently by A.I. Shirshov \cite{shir} for ideals of Lie algebras and, implicitly, associative algebras, by H.~Hironaka \cite{Hironaka} for ideals of the power series algebras (both formal and convergent), and by B.~Buchberger \cite{Buchberger} for ideals of the polynomial algebras. Gr\"obner basis and Gr\"obner---Shirshov basis theories have been proved to be very useful in different branches of mathematics, including commutative algebra and combinatorial algebra. It is a powerful tool to solve the following classical problems: normal form, word problem, conjugacy problem, rewriting system, embedding theorem, PBW theorem, growth function, (co)homology, etc., (see, e.g., \cite{Bokut} and references therein).
It is well known that the first cohomology group of an associative algebra is related to its derivations, and the elements of the second one are interpreted as its null extensions. Homological algebra also introduces important numerical invariants in the group theory, e.g., (co)homological dimension and Euler characteristic. In some way, the (co)homology groups carry in-depth information about the properties of a group or an algebra, e.g., the property of a finite group to be nilpotent or solvable can be characterized in homological terms. So we can say that homological methods allow us to get important information about the structure of an algebra.

The most important cohomology theory for associative algebras is the Hoch\-schild cohomology.
Given an associative algebra $A$ over a field $\Bbbk $ and an $A$-bimodule $W$, 
the set of Hochschild cochains $C^{m}(A,W)$ consists of all linear maps $A^{\otimes m}\to W$, 
$m\ge 0$, and the $m$th Hochschild cohomology group is defined as 
$H^m(A,W) = \Ker \Delta_m /\mathrm{Im}\,\Delta_{m-1}$, where $\Delta_m : C^{m}(A,W)\to C^{m+1}(A,W)$
is the Hochschild differential 
%
\begin{multline}\label{eq:Hoch}
\Delta_n f(r_1, \ldots, r_{n+1})=r_1 f(r_2, \ldots, r_{m+1}) \\
+ \sum\limits_{i=1}^{m}(-1)^i f(r_1,\ldots,r_i r_{i+1},\ldots,r_{m+1}) +
 (-1)^{m+1}f(r_1,\ldots,r_m)r_{m+1},
\end{multline}
%
One of the most important features of the Hochschild cohomology groups is that the first one, 
$H^1(A,W)$, describes outer derivations from $A$ to $W$ and the second one, 
$H^2(A,W)$, is in one-to-one correspondence with the equivalence classes of 
null extensions 
\[
0\to W\to E \to A \to 0,\quad W^2=0.
\]
It is often a difficult problem to find the groups $H^m(A,W)$ for a given 
algebra $A$ an $A$-bimodule~$W$.
The first important step of solution is to find a long exact sequence of $A$-bimodules
starting from~$A$, a resolution of~$A$. Next, one needs to calculate the derived 
functor to $\Hom (\cdot, W)$  to find the cohomology groups.
The most natural resolution known as the bar-resolution is easy to define 
but it is not convenient for the calculation of cohomologies.
A smaller resolution was proposed by D.J.~Anick \cite{Anick}:
he found a way to construct a free resolution for an associative algebra which is homotopy equivalent to the bar-resolution. The Anick resolution found numerous applications 
in combinatorial algebra, see \cite{ufn}.
This long exact sequence is more convenient for the calculation of cohomology groups, 
but the computation of differentials according to 
the original Anick algorithm described in \cite{Anick} is extremely hard. 
In order to visualize the computation of differentials it is possible to use the discrete algebraic
Morse theory\cite{jollwel, formancell, formanguide}  based on the concept of a Morse matching. 
In this paper, we choose a tower order on the free monoid generated by a well--ordered set $X$. By using this order, we find a Gr\"obner---Shirshov basis for the group $\Gamma^4_5$. Then we apply the discrete algebraic Morse theory to find explicitly the differentials $\delta_1$ and $\delta_2$ of the Anick resolution for $\Gamma^4_5$. As an application, we calculate the second Hochschild cohomology group of the group algebra of $\Gamma^4_5$ with coefficients in the trivial 1-dmensional 
$\Bbbk \Gamma^4_5$-bimodule~$M$. 
It turns out that 
 there exist five inequivalent 1-dimensional null extensions
 of $\Bbbk \Gamma^4_5$ by means of the trivial module.


\section{Composition--Diamond Lemma for Associate Algebras}
Let $\Bbbk$ be a field, $X$ be a nonempty set, $\Bbbk\langle X\rangle$ be the free associative algebra generated by $X$.
Denote by $X^*$ the free monoid generated by $X$, where the empty word is denoted by~1. 
Let $\deg (u)$ stand for the length (number of letters) of a word $u=x_1x_2\dots x_n \in X^*$, 
$x_i \in X$.
In particular, $\deg(u)=0$ means $u=1$. 

A well order $<$ on $X^*$ is said to be monomial if
$u>v$ implies $wu>wv$ and $uw>vw$ for all $u,v,w\in X^*$.
Let us fix a monomial order $<$ on $X^*$. An arbitrary 
$f\in \Bbbk\langle X\rangle$ may be written as
\[
f=\sum^n_{i=1}\alpha_i u_i \in ,
\]
where $\alpha_i \in \Bbbk \setminus \{0\}$, 
$u_i\in X^*$, $n\in \mathbb{N}$, and 
$u_1>u_2>\dots>u_n$. 
Define the leading monomial of $f$, denoted by $\overline{f}$, to be $u_1$. 
If $\alpha_1=1$, then we say $f$ is a monic polynomial.

Let us recall the main statements of the Gr\"obner---Shirshov 
basis theory for associative algebras according to \cite{Bokut,Bok72,shir}.
% можно сослаться на книгу Бокуть-Чен-Калоркоти-Колесников-Лопаткин

\begin{definition} 
Let $f,g \in \Bbbk\langle X\rangle$ be monic polynomials. Define the following two kinds of {\em compositions}:
\begin{itemize}
    \item 
If $w=\overline{f} b=a\overline{g}\in X^*$, where $a,b\in X^*$, $\deg(\overline{f})+\deg(\overline{g})>\deg(w)$, 
then $(f,g)_w=fb-ag$ is called an intersection composition of $f$ and $g$ with respect to~$w$;\\
\item 
 If $w=\overline{f}=a\overline{g}b \in X^*$, $a,b\in X^*$,
 then $(f,g)_w=f-agb$ is called an inclusion composition of $f$ and $g$ with respect to~$w$.
\end{itemize}
%The word $w$ in the above is called an ambiguity.
\end{definition}


Let $S\subseteq \Bbbk\langle X\rangle$ be a nonempty set of polynomials.
%and $<$ be a monomial order on $X^*$. 
Given $f \in \Bbbk\langle X\rangle$, $w\in X^*$, if $f=\sum \alpha_i a_i s_i b_i$, where $\alpha_i \in \Bbbk$, $a_i, b_i \in X^*$, $s_i\in S$, and $a_i\overline{s_i}b_i< w$ with respect to the 
fixed monomial order, then $f$ is said to be trivial modulo $(S,w)$, and is denoted by
\[
f\equiv 0 \pmod{(S,w)}
\]

\begin{definition}
A nonempty set $S\subset \Bbbk\langle X\rangle$ of monic polynomials
is called a {\em Gr\"obner---Shirshov basis}
in $\Bbbk\langle X\rangle$ if for every $f,g\in S$ having a composition we have $(f,g)_w \equiv 0$ modulo $(S,w)$.
\end{definition}

\begin{Theorem}[Composition--Diamond Lemma]\label{thm:GSB}
Let $<$ be a monomial order on $X^*$, 
$S\subseteq \Bbbk\langle X\rangle$ be a nonempty set of monic polynomials,  and let $\mathrm{Id}\,(S)$ 
be the ideal of $\Bbbk\langle X\rangle$ generated by~$S$. 
Then the following statements are equivalent.
\begin{enumerate}
    \item[(i)] 
$S$ is a Gr\"obner---Shirshov basis in $\Bbbk\langle X\rangle$.
    \item[(ii)] If $0\ne f \in \mathrm{Id}\,(S)$ then
$\overline{f}=a\overline{s}b$ for some $s\in S$, $a,b\in X^*$.
\item [(iii)] 
The set $\mathrm{Irr}\,(S)=
\{z\in X^* \mid  z\ne a \overline{s}b,\ a,b \in X^*,\ s\in S\}$ 
is a linear basis of the algebra $\Bbbk\langle X \mid S\rangle:= \Bbbk\langle X\rangle/\mathrm{Id}\,(S)$.
\end{enumerate}
\end{Theorem}

If a subset $S$ of $\Bbbk\langle X\rangle$ is not 
a Gr\"obner---Shirshov basis then we can add to $S$ all nontrivial compositions of polynomials of $S$, and by continuing this process (maybe infinitely) many times, we eventually obtain a Gr\"obner---Shirshov basis $S^{\mathrm{comp}}$. 
Such a process is known as the Shirshov algorithm since \cite{shir}. 
If $S$ is a set of semigroup (or binomial) relations (that is, polynomails of the form $u-v$, where $u,v \in X^*)$, then every their composition has the same form. 
As a result, the set $S^{\mathrm{comp}}$ 
also consists of semigroup relations. 

Let $M=\mathrm{sgp}\,\langle X \mid S\rangle:=X^*/\rho(S)$ be a semigroup presentation, where $S\subset X^*\times X^*$ and 
$\rho (S)$ is the congruence on $X^*$ generated by~$S$. 
Then $S$ may be considered is a set of semigroup relations in $\Bbbk\langle X\rangle$,
and hence one can find the corresponding Gr\"obner---Shirshov basis $S^{\mathrm{comp}}$. 
The latter does not depend on $\Bbbk$ and, as mentioned above, it consists of semigroup relations. 
We will call $S^{\mathrm{comp}}$ a Gr\"obner---Shirshov basis for~$M$. This is the same as a Gr\"obner---Shirshov basis for the semigroup algebra $\Bbbk M=\Bbbk\langle X\mid S\rangle$.

The Gr\"obner---Shirshov basis of a given algebra 
highly depends on the set $X$ of generators and on the chosen 
monomial order on $X^*$. In what follows, we will use the following one known as the {\em tower order}.
Let $X$ be a well-ordered set. Given a word $u\in X^*$, $u\ne 1$, 
denote by $\max(u)$ the largest letter occurring in $u$, and let 
$n_u$ be the number of occurrences of $a=\max(u)$ in $u$. 
Then $u$ may be uniquely presented as
\[
u=u_0 a u_1 a u_2\dots u_{n_u-1} a u_{n_u} ,\quad u_i \in X^*,\
0\le i\le n_u.
\]
Here $\max(u_i)<\max (u)$ for every nonempty $u_i$.

For $u\in X^*$, $u\ne 1$, define the sequence
\[
\mathrm{wt}\,(u)=(\max(u), n_u, u_0, \dots, u_{n_u})
\]
If $u,v\in X^*$ are two nonempty words then set
$u>v$ if only and only if $\mathrm{wt}\,(u)>\mathrm{wt}\,(v)$ lexicographically, where $u_0, u_1, \dots, u_{n_u}$ are compared by induction. Such an order is called a {\em tower order} on $X^*$ which is a monomial order. 


\section{Grobner-Shirshov basis for $\Gamma^4_5$}

\begin{definition}[\cite{manturov19}]
The group $\Gamma^4_5$ is generated (as a semigroup) 
by the set of elements
\[
\begin{aligned}
A_1=\{a=d_{(1,2,3,4)} , b=d_{(1,2,4,3)} , c=d_{(1,3,2,4)}, d=d_{(1,2,3,5)}, e=d_{(1,2,5,3)}\\
f=d_{(1,3,2,5)}, h=d_{(1,2,4,5)}, g=d_{(1,2,5,4)}, k=d_{(1,4,2,5)}, l=d_{(1,3,4,5)}\\
m=d_{(1,3,5,4)}, n=d_{(1,4,3,5)}, q=d_{(2,3,4,5)}, r=d_{(2,3,5,4)}, s=d_{(2,4,3,5)}\}
\end{aligned}
\]
subject to the following relations:
\[
\begin{aligned}
S_1{}&=\{x^2=1,\ x\in A_1, \\
&adhlq=qadhl=lqadh=hlqad=dhlqa=1,\\
&bhdns=sbhdn=nsbhd=dnsbh=hdnsb=1,\\
&cflhr=rcflh=hrcfl=lhrcf=flhrc=1,\\
&dagmr=rdagm=mrdag=gmrda=agmrd=1,\\
&egans=segan=nsega=anseg=ganse=1,\\
&fcmgq=qfcmg=gqfcm=mgqfc=cmgqf=1,\\
&hbemr=rhbem=mrhbe=emrhb=bemrh=1,\\
&geblq=qgebl=lqgeb=blqge=eblqg=1,\\
&kcmes=skcme=eskcm=meskc=cmesk=1\}.
\end{aligned}
\]
\end{definition}

\begin{Lemma}\label{lem:Lemma1}
Let 
\[
A_2=\{ e,g,k,l,m,n,q,r,s\}
\] and 
\[
\begin{aligned}
 S_2{}&=\{x^2=1,\ x\in A_2, \ 
 ge=nsegns,\\
 &gnse=esng, le=rnsemrnkmgqksrmgqlrm, lrnse=egqlskqgmknrm,\\ &kse=rnsemrnkm,krnse=semknrm, me=rnsemrns, mrnse=esnrm,\\ &qrnse=nsegnkqgmknrm, qnse=rnsemrnkmgqkng,lqg=rmgqlrm,\\ &lrmg=qgmrlq, gqlnse=mregqlsnrm, gmre=qlnsemrnsrmgqlrm \}.
 \end{aligned}
 \]
Then $\Gamma^4_5= \mathrm{sgp}\,\langle A_2 \mid S_2 \rangle $. 
\end{Lemma}

\begin{proof}
The group $\Gamma_5^4$ is indeed generated by $A_2$ since 
$a,b,c,d,f,h$ may be expressed via $A_2$:
\[
a=nseg,\ 
b=emrh,\ 
c=mesk,\ 
d=nsemr,\ 
f=cmgq,\ 
h=rmgql 
\]
It remains to show that $S_2$  hold in $\Gamma_5^4$ and, conversely, 
that $S_1$ hold in $\mathrm{sgp}\,\langle A_2 \mid S_2 \rangle $.
For example, 
 



\[
1=adhlq=anseg=agmrd\] Since $a^2=d^2=h^2=1$, then\\
\[
\begin{aligned}
&a=dhlq=nseg =gmrd,\quad   nseg=gens\\
&d=nsemr, \quad   nsemr= rmesn \\
&h= rmgql ,\quad   rmgql=lqgmr
\end{aligned}
\]
So \[
\begin{aligned}
&ge=nsegns ,\quad   gnse=esng\\
&me=rnsemrns ,\quad  nsemr= rmesn\\
&lqg=rmgqlrm \quad   lrmg=gmrl
\end{aligned}\] 
Conversely, let
\[
a=gnse\ ,d=nsemr,\ h= rmgql
\]
then $a^2=d^2=h^2=1$ and
\[
\begin{aligned}
drm=nse\quad  \&\quad hlq=rmg\\
drmg=nseg\quad  \&\quad dhlq=drmg\\
\end{aligned}
\]
So
\[
\begin{aligned}
a=dhlq=nseg=drmg\\
1=adhlq=anseg=agmrd
\end{aligned}
\]
Similarly, we can get rest of the relations. 
\end{proof}

\begin{Theorem}
Let $<$ be the tower order on $A^*_2$ based on $e>g>k>l>m>n>q>r>s$. Then 
\[
\begin{aligned}
S{}&=S_2\cup \{
%\{e^2-1,g^2-1,k^2-1,l^2-1,m^2-1,n^2-1,q^2-1,r^2-1,s^2-1, ge-nsegns\\
%&gnse-esng, le-rnsemrnkmgqksrmgqlrm, lrnse-egqlskqgmknrm,\\ &kse-rnsemrnkm,
%krnse-semknrm, me-rnsemrns, mrnse-esnrm,\\ &qrnse-nsegnkqgmknrm,
%qnse-rnsemrnkmgqkng, lqg-rmgqlrm,\\ &lrmg-qgmrlq, gqlnse-mregqlsnrm,
%gmre-qlnsemrnsrmgqlrm,\\ 
g(qlrm)^jqlnse-(mrlq)^jmregqlsnrm,g(qlrm)^jnse-(mrlq)^jesng,\\ &g(mrlq)^jmre-(qlrm)^jqlnsemrnsrmgqlrm,  g(mrlq)^je-(qlrm)^jnsegns  ; j\ge 1\}  
\end{aligned}
\]
is a Grobner-Shirshov basis for $\Gamma^4_5$.
\end{Theorem}
\begin{proof}
Let us show that the last four series of relations in $S$ hold in $\Gamma^4_5$.
For example,
\[
\begin{aligned}
lrmg&{}=qgmrlq \Rightarrow 
gmrlq =qlrmg \Rightarrow 
gmrlqmre=qlrm(gmre)\\
\Rightarrow\ & gmrlqmre=qlrmqlnsemrnsrmgqlrm
\end{aligned}
\]
By induction, assume that
\[
\begin{aligned}
g(mrlq)^{j-1}mre=(qlrm)^{j-1}qlnsemrnsrmgqlrm
\end{aligned}
\]
then
\[
\begin{aligned}
g(mrlq)^jmre&{}=g(mrlq)^{j-1}mrlqmre\\
&=g(mrlq)^{j-1}mreelqmre\\
&=(qlrm)^{j-1}qlnsemrnsrmgqlr(me)lqmre\\
&=(qlrm)^{j-1}qlnsemrnsrmgqlr(rnsemrns)lqmre\\
&=(qlrm)^{j-1}qlnsemrnsrm(gqlnse)mrnslqmre\\
&=(qlrm)^{j-1}qlnsemrnsrm(mregqlsnrm)mrnslqmre\\
&=(qlrm)^{j-1}qlnse(mrnse)gmre\\
&=(qlrm)^{j-1}qlnse(esnrm)gmre\\
&=(qlrm)^{j-1}qlrm(gmre)\\
&=(qlrm)^{j-1}qlrm(gmre)\\
&=(qlrm)^j(gmre)\\
&=(qlrm)^jqlnsemrnsrmgqlrm.
\end{aligned}
\]
Similarly, we can get another relations. Let's check some compositions: 
\[
\begin{aligned}
(lrmg)mre-lrm(gmre)&{}=q(gmrlqmre)-lrmqlnsemrnsrmglrm\\
&=(qq)lrmqlnsemrnsrmglrm-lrmqlnsemrnsrmglrm=0;
\end{aligned}
\]
\[
\begin{aligned}
(gg)mrlqmre-g(gmrlqmre)&{}=mrlqmre-(gqlrmqlnse)mrnsrmgqlrm\\
&=mrlqmre-mrlqmregqlsnrmmrnsrmgqlrm\\
&=mrlqmre-mrlqmregq(lrmg)qlrm\\
&=mrlqmre-mrlqmregqqgmrlqqlrm\\
&=mrlqmre-mrlqmre=0;
\end{aligned}
\]
\[
\begin{aligned}
(lqg)qlnse-lq(gqlnse)&{}=rm(gqlrmqlnse)-lqmregqlsnrm\\
&=rmmrlqmregqlsnrm-lqmregqlsnrm=0;
\end{aligned}
\]
\[
\begin{aligned}
(gg)qlrmqlnse-g(gqlrmqlnse)&{}=qlrmqlnse-(gmrlqmre)gqlsnrm\\
&=qlrmqlnse-qlrmqlnsemrnsrmgq(lrmg)qlsnrm\\
&=qlrmqlnse-qlrmqlnsemrnsrmgqqgmrlqqlsnrm\\
&=qlrmqlnse-qlrmqlnse=0.
\end{aligned}
\]
This proves triviality of compositions for $j=1$.
Proceed by induction for $j>1$.
For example, we have
\[
\begin{aligned}
&lrm(g(mrlq)^jmre)- (lrmg)(mrlq)^jmre=\\ &=lrm(qlrm)^jqlnsemrnsrmgqlrm-qgmrlq(mrlq)^jmre\\
&= lrm(qlrm)^jqlnsemrnsrmgqlrm-q(g(mrlq)^{j+1}mre)\\
&= lrm(qlrm)^jqlnsemrnsrmgqlrm-q(g(mrlq)^{j+1}mre)\\
&= lrm(qlrm)^jqlnsemrnsrmgqlrm-q((qlrm)^{j+1}qlnsemrnsrmgqlrm)\\
&= lrm(qlrm)^jqlnsemrnsrmgqlrm-qqlrm (qlrm)^jqlnsemrnsrmgqlrm)\\
&= lrm(qlrm)^jqlnsemrnsrmgqlrm-lrm (qlrm)^jqlnsemrnsrmgqlrm=0;
\end{aligned}
\]
\[
\begin{aligned}
g(g(mrlq)^jmre)- (gg)(mrlq)^jmre &{}=(g(qlrm)^jqlnse)mrnsrmgqlrm-(mrlq)^jmre\\
&=(mrlq)^jmregq(lrmg)qlrm-(mrlq)^jmre\\
&=(mrlq)^jmregqqgmrlqqlrm-(mrlq)^jmre\\
&=(mrlq)^jmre-(mrlq)^jmre=0;
\end{aligned}
\]
\[
\begin{aligned}
lq(g(mrlq)^je)- (lqg)(mrlq)^je{}&=lq(qlrm)^jnsegns-rmgqlrm(mrlq)^je\\
&= lqqlrm(qlrm)^{j-1}nsegns- rm(g(mrlq)^{j-1}e) \\
&= rm(qlrm)^{j-1}nsegns- rm(qlrm)^{j-1}nsegns=0;
\end{aligned}
\]
\[
\begin{aligned}
g(g(mrlq)^je)- (gg)(mrlq)^je{}&=(g(qlrm)^jnse)gns-(mrlq)^je\\
&=(mrlq)^jesnggns-(mrlq)^je \\
&=(mrlq)^je-(mrlq)^je=0.
\end{aligned}
\]
Similarly, we can check that any compostion in $S$ is trivial.
\end{proof}


\section{Morse matching}
In this section, we state main definitions and known results related to the construction of the Anick chain complex via algebraic Morse theory following 
\cite{lopatkin, skold, jollwel}. 
The approach is based on the following idea. Suppose $\Bbbk $ is a field and $\Lambda $ is 
a unital associative  $\Bbbk$-algebra.
Let $B=(B_n,d_n)_{n\ge 0}$ be a chain complex of free left $\Lambda $-modules, and let $X_n$ be a basis 
of $B_n$ over $\Lambda $. One may represent the complex with a weighted oriented graph
$\Gamma (B)$
whose vertices are $\bigcup\limits_{n}X_n$ and there is an edge from $x\in X_n$ to $y\in X_{n-1}$ 
if the distribution of $d_n(x)\in B_{n-1}$ contains $\lambda y$ for $0\ne \lambda \in \Lambda $
(the weight of the edge is set to be this $\lambda $).

Assume $M$ is a particular matching in the graph $\Gamma (B)$, i.e., a subset of edges such that 
neither of vertices belongs to more than one edge from $M$. 
In addition, assume the weights of edges from $M$ are invertible central elements of $\Lambda $
Then transform the graph $\Gamma(B)$ in the following way: invert the direction of all edges from $M$
and replace their weights $\lambda $ with $-\lambda^{-1}$.
If the resulting graph $\Gamma_M(B)$ has no directed cycles then $M$ is called a {\em Morse matching}. The vertices that do not belong to  edge from $M$ are said to be {\em critical cells}.

For a Morse matching graph $\Gamma_M(B)$, one may construct another chain complex 
of free $\Lambda $-modules $(B_{(m)},\delta_m)_{m\ge 0}$, where $B_{(m)}$ is spanned over 
$\Lambda $ by the set $X_{(m)}$ of all critical cells from $X_m$ and the calculation of $\delta_m$ 
is based on the consideration of all paths in $\Gamma_M(B)$ (see, e.g., \cite{lopatkin}).
Namely,
$$
\delta_m(x)=\sum\limits_{y\in X_{(m-1)}}\Gamma(x,y)y,\quad  y\in X_{(m)}
$$
where $\Gamma(w,w')$ is the sum of path weights in the Morse matching graph $\Gamma_M(B)$.
The new complex $(B_{(m)},\delta_m)_{m\ge 0}$ is homotopy equivalent to the initial one, 
but it is smaller since we choose only critical cells as generators of the modules $B_{(m)}$.

For the bar-resolution $B$, there is a natural way to choose a Morse matching $M$ in $\Gamma(B)$
in such a way that the resulting complex constructed from $\Gamma_M(B)$ coincides with 
the Anick resolution. We will state the construction in bimodule setting which is easy to 
get by replacing $\Lambda$ with $\Lambda\otimes \Lambda^{op}$.

Suppose $\Lambda $ has an an augmentation $\varepsilon: \Lambda \to \Bbbk $, 
$A$ is a set of generators for $\Lambda$. 
Then $\Lambda $ is a homomorphic image of the free associative algebra 
$\Bbbk \langle A\rangle$ generated by~$A$.
Assume  $\le$ is a monomial order on the free monoid $A^{*}$,
and $\GSB_\Lambda$ is a Gr\"obner---Shirshov basis of $\Lambda $.
The latter may be considered as a confluent set of defining relations 
for the algebra~$\Lambda $, each of relations is of the form $u-f$, where $u\in A^*$, 
$f\in \Bbbk \langle A\rangle$, $u\ge \bar f$, $\bar f$ is the leading monomial of $f$
relative to $\le $.
Denote by $V$ the set of all leading terms $u$ of relations from $\GSB_\Lambda$. 
Following Anick \cite{Anick}, $V$ is called the set of {\em obstructions}.

The cokernel of  $\varepsilon: \Bbbk \to \Lambda$ is denoted by $\Lambda\slash \Bbbk$.
The set of all non-trivial words in $A^*$ that do not contain a word from $V$
as a subword (i.e., the set of $V$-reduced words \cite{Bok72})
forms a linear basis of $\Lambda/\Bbbk $. 
This is one of the equivalent 
conditions in the Composition-Diamond Lemma about Gr\"obner---Shirshov bases 
for associative algebras (see Theorem \eqref{thm:GSB}).
 
The two-sided bar resolution $(B_m(\Lambda,\Lambda),d_m)_{m\ge 0}$ is an 
exact sequence of $\Lambda$-bimodules 
$$
0\leftarrow \Bbbk \leftarrow \Lambda \leftarrow B_0(\Lambda,\Lambda) \leftarrow B_1(\Lambda , \Lambda) \leftarrow \dots,
$$
where
$$
B_m(\Lambda,\Lambda):=\Lambda\otimes (\Lambda\slash \Bbbk )^{\otimes (m+1)}\otimes \Lambda.
$$
We will use the standard convention denoting $1\otimes \lambda_1\otimes \dots \lambda_{m+1}\otimes 1\in B_m(\Lambda,\Lambda )$
 by $[\lambda_1|\ldots |\lambda_{m+1}]$ for $\lambda_i\in \Lambda/\Bbbk$.
The differential $d_m$ is defined as follows:
\begin{multline}\nonumber
 d_m([\lambda_1|\ldots|\lambda_{m+1}])=\lambda_1[\lambda_2|\ldots|\lambda_{m+1}]+(-1)^{m+1}[\lambda_1|\ldots|\lambda_{m}]\lambda_{m+1}+
 \\
 +\sum\limits_{i=1}^{m}(-1)^i[\lambda_1|\ldots|\lambda_i\lambda_{i+1}|
 \ldots|\lambda_{m+1}]
\end{multline} 

The role of critical cells $X_{(m)}$ is played by so called $m$-chains.
By definition, $0$-chains are the elements of $A$, $1$-chains are the words from $V$.
For $m\ge 1$, a word $v=x_{i_1}\ldots x_{i_t} \in A^{*}$ 
is said to be an $m$-{\em prechain} if and only if there exist indices 
$a_j,b_j \in \mathbb{Z}$, $1 \le j \le m$, satisfying the following conditions:
\begin{enumerate}
    \item $1=a_1<a_2 \le b_1<\ldots<a_m \le b_{m-1}<b_m=t$;
    \item $ x_{i_{a_j}}\ldots x_{i_{b_j}} \in V$ for $1 \le j \le m$.
\end{enumerate}
That is, $v$ is obtained by $m$ obstructions consecutively intersecting with each other.
An $m$-prechain $x_{i_1}\ldots x_{i_t}$ is an $m$-{\em chain} 
if the integers $a_j,b_j$ can be chosen in such a way that for every $1 \le l \le m$
and for every $s<b_l$ the prefix
$x_{i_1}\ldots x_{i_s}$  is not an $l$-prechain.
Denote the set of all $m$-chains by $V^{(m)}$.


According \cite{lopatkin,jollwel,skold}, there is a way to choose 
a Morse matching in the graph $\Gamma(B)$ corresponding to the bar-resolution 
in such a way that $X_{(m)}=V^{(m)}$ and the new complex is exactly the Anick resolution
$$
 0\leftarrow \Bbbk \overset{\varepsilon}{\leftarrow} \Lambda \overset{\delta_0}{\leftarrow} B_{(0)}\overset{\delta_1}{\leftarrow} B_{(1)} \overset{\delta_2}{\leftarrow} B_{(2)}
 \leftarrow \dots ,
$$
where
$$
B_{(m)} =\Lambda\otimes \Bbbk V^{(m)}  \otimes \Lambda, \quad m\ge 0.
$$

We suppress the construction of graphs in the algorithm sketched above in order to replace graphical manipulations with 
algebraic expressions. 
Our aim is to evaluate 
$\delta_n(w) \in \Lambda \otimes \Bbbk V^{(n-1)}\otimes \Lambda $ for a given $w\in V^{(n)}$. 
For 
 $w=w_1w_2\dots w_{n}w_{n+1}\in V^{(n)}$, where all $w_i$ are $V$-reduced words, 
denote $[w] = [w_1|w_2|\ldots |w_n|w_{n+1}]\in (\Lambda/\Bbbk)^{\otimes (n+1)}$. 
In the sequel we will often omit symbols $|$ in 
the elements of the Anick resolution.

Define two  $\Lambda $-bilinear operators 
$\delta_n'$ and $\delta_n''$ via a recurrent procedure 
which includes the following steps:
\begin{enumerate}
\item 
\begin{multline*}
    \delta_n'[w] 
= w_1 [w_2|\ldots |w_n|w_{n+1}]
+ (-1)^n [w_1|w_2|\dots | w_n]w_{n+1} \\
 +\sum\limits_{j=1}^n [w_1|w_2|\ldots |s_j |\ldots | w_{n+1}] \in \Lambda\otimes (\Lambda /\Bbbk)^{\otimes n} \otimes \Lambda,
\end{multline*}
where 
 $s_j$ are the $V$-reduced forms of the products $w_jw_{j+1}$, 
   $j=1,\dots, n$;
%
 \item If $[v]\in (\Lambda/\Bbbk)^{\otimes n}$ belongs to $V^{(n-1)}$, then
\[
 \delta_n''[v] = [v],
\]
otherwise, proceed to the following step.
%
 \item If $[v]=[v_1|\dots |v_n]$ does not belong to $V^{(n-1)} $ 
 (here all $v_k$s are $V$-reduced)
 then there exists the largest integer 
 $i\ge 0$ such that 
 $v_1\dots v_i \in V^{(i-1)}$,
 $v_{i+1}$ may be presented as $v_{i+1}=v_{i+1}'v_{i+1}''$,
 and
   $[v_1|\ldots |v'_{i+1}]$ belongs to $V^{(i)}$ \cite{jollwel}.
Then set
\[
 \delta_n''[v] = (-1)^{i-1}[v_1|\ldots|v'_{i+1}|v''_{i+1}|\ldots|v_{n}].
\]
\end{enumerate}
%
The recurrent application of $\delta_n''\delta_n'$ leads to the result: after finitely many steps, 
the computation of $\delta_n''$ does not involve step~(3). Therefore, 
\[
 \delta_n(w) = (\delta_n''\delta_n')^{k} [w], 
\]
where $k=k(w)\ge 1$ is a sufficiently large integer.

Let $\Lambda $ be a $\Bbbk$-algebra as above, and let $W$ be a unital $\Lambda $-bimodule. 
Consider the Hochschild cochain complex $(C^m(\Lambda , W), \Delta_m)_{m\ge 0}$
where $\Delta_m $ is given by \eqref{eq:Hoch}.
We may use the Anick resolution to calculate
$H^m(\Lambda ,W)={\Ker \Delta_m}/{\mathrm{Im}\,\Delta_{m-1}}$
via
\[
 \Delta_m (f) = f\delta _{m},\quad f:B_{(m-1)}\to W.
\]
Indeed, $H^m(\Lambda,W)=Z^m \slash \beta^m$, where 
\[
\begin{gathered}
\beta^m:=\mathrm{Im}\,\Delta_{m-1}
=\{f:B_{(m-1)} \rightarrow W \mid f=g \delta _{m-1}, \, g: B_{(m-2)} \rightarrow W\},\\ 
Z^m:=\Ker\Delta_{m}=\{f:B_{(m-1)} \rightarrow W \mid f \delta_{m}=0\}.
\end{gathered}
\]

\section{The Anick complex for $\Gamma^4_5$}\label{sec:Complex}

To find the Anick complex, we need two steps.
First, we have to find the set of obstructions for $\Gamma^4_5$
relative to the given Gr\"obner---Shirshov basis (the set of leading terms in $S=\GSB_{\Gamma^4_5}$) 
and the set of $2$-chains. 
Next, build a Morse graph and calculate the path weights 
$\Gamma(w,w')$ for every $m$-chain $w$ and $(m-1)$-chain $w'$ for $m=1,2$.
Alternatively, we may 
apply the algorithm described in the previous section to find 
the Anick differentials $\delta_m$, $m=1,2$.


For $m=1$ we have 
\[
\begin{aligned}
V^{(1)}&{}=\{e^2, g^2, k^2,l^2,m^2,n^2,q^2,r^2,s^2,ge,gnse,le,lrnse,kse,krnse,me,mrnse,qrnse\\
&qnse,lqg,lrmg, gqlnse, gmre, g(qlrm)^jqlnse,g(mrlq)^jmre, g(qlrm)^jnse, g(mrlq)^je
\mid
j\ge 1\}
\end{aligned}
\]

Let us calculate $\delta_1$ as stated in the previous section:
\[
 \delta_1[x^2]=x[x]+[x]x,\quad x\in \{e,g,k,l,m,n,q,r,s\}.
\]
A more complicated construction is needed for (See Fig.~1)
\[
 \delta_1[ge]=g[e]+[g]e-[n]segns-n[s]egns-ns[e]gns-nse[g]ns-nseg[n]s-nsegn[s].
\]
\[
\delta_1[gnse]=[g]nse+g[n]se+gn[s]e+gns[e]-[e]sng-e[s]ng-es[n]g-esn[g]
\]
\[
\begin{aligned}
 \delta_1[mrnse]&{}=[m]rnse+m[r]nse+mr[n]se+mrn[s]e+mrns[e]-[e]snrm-e[s]nrm\\
 &-es[n]rm-esn[r]m-esnr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[me]&{}=[m]e+m[e]-[r]nsemrns-r[n]semrns-rn[s]emrns-rns[e]mrns\\
 &-rnse[m]rns-rnsem[r]ns-rnsemr[n]s-rnsemrn[s].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[le]&{}=[l]e+l[e]-[r]nsemrnkmgqksrmgqlrm-r[n]semrnkmgqksrmgqlrm\\
 &-rn[s]emrnkmgqksrmgqlrm-rns[e]mrnkmgqksrmgqlrm\\
 &-rnse[m]rnkmgqksrmgqlrm-rnsem[r]nkmgqksrmgqlrm\\
 &-rnsemr[n]kmgqksrmgqlrm-rnsemrn[k]mgqksrmgqlrm\\
 &-rnsemrnk[m]gqksrmgqlrm-rnsemrnkm[g]qksrmgqlrm\\
 &-rnsemrnkmg[q]ksrmgqlrm-rnsemrnkmgq[k]srmgqlrm\\
 &-rnsemrnkmgqk[s]rmgqlrm-rnsemrnkmgqks[r]mgqlrm\\
 &-rnsemrnkmgqksr[m]gqlrm-rnsemrnkmgqksrm[g]qlrm\\
 &-rnsemrnkmgqksrmg[q]lrm-rnsemrnkmgqksrmgq[l]rm\\
 &-rnsemrnkmgqksrmgql[r]m-rnsemrnkmgqksrmgqlr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[lrnse]&{}=[l]rnse+l[r]nse+lr[n]se+lrn[s]e+lrns[e]-[e]gqlskqgmknrm\\
 &-e[g]qlskqgmknrm-eg[q]lskqgmknrm-egq[l]skqgmknrm-egql[s]kqgmknrm\\&-egqls[k]qgmknrm-egqlsk[q]gmknrm-egqlskq[g]mknrm-egqlskqg[m]knrm\\&-egqlskqgm[k]nrm-egqlskqgmk[n]rm-egqlskqgmkn[r]m-egqlskqgmknr[m]
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[kse]&{}=[k]se+k[s]e+ks[e]-[r]nsemrnkm-r[n]semrnkm-rn[s]emrnkm\\
 &-rns[e]mrnkm-rnse[m]rnkm-rnsem[r]nkm-rnsemr[n]km-rnsemrn[k]m\\
 &-rnsemrnk[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[krnse]&{}=[k]rnse+k[r]nse+kr[n]se+krn[s]e+krns[e]-[s]emknrm-s[e]mknrm\\
 &-se[m]knrm-sem[k]nrm-semk[n]rm-semkn[r]m-semknr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[qrnse]&{}=[q]rnse+q[r]nse+qr[n]se+qrn[s]e+qrns[e]-[n]segnkqgmknrm\\
 &-n[s]egnkqgmknrm-ns[e]gnkqgmknrm-nse[g]nkqgmknrm\\
 &-nseg[n]kqgmknrm-nsegn[k]qgmknrm-nsegnk[q]gmknrm\\
 &-nsegnkq[g]mknrm-nsegnkqg[m]knrm-nsegnkqgm[k]nrm\\
 &-nsegnkqgmk[n]rm-nsegnkqgmkn[r]m-nsegnkqgmknr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[qnse]&{}=[q]nse+q[n]se+qn[s]e+qns[e]-[r]nsemrnkmgqkng-r[n]semrnkmgqkng\\
 &-rn[s]emrnkmgqkng-rns[e]mrnkmgqkng-rnse[m]rnkmgqkng\\
 &-rnsem[r]nkmgqkng-rnsemr[n]kmgqkng-rnsemrn[k]mgqkng\\
 &-rnsemrnk[m]gqkng-rnsemrnkm[g]qkng-rnsemrnkmg[q]kng\\
 &-rnsemrnkmgq[k]ng-rnsemrnkmgqk[n]g--rnsemrnkmgqkn[g].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[gqlnse]&{}=[g]qlnse+g[q]lnse+gq[l]nse+gql[n]se+gqln[s]e+gqlns[e]\\
 &-m[r]egqlsnrm-mr[e]gqlsnrm-mre[g]qlsnrm-mreg[q]lsnrm\\
 &-mregq[l]snrm-mregql[s]nrm-mregqls[n]rm-mregqlsn[r]m\\
 &-mregqlsnr[m]-[m]regqlsnrm.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[gmre]&{}=[g]mre+g[m]re+gm[r]e+gmr[e]-[q]lnsemrnsrmgqlrm\\
 &-q[l]nsemrnsrmgqlrm-ql[n]semrnsrmgqlrm-qln[s]emrnsrmgqlrm\\
 &-qlns[e]mrnsrmgqlrm-qlnse[m]rnsrmgqlrm-qlnsem[r]nsrmgqlrm\\
 &-qlnsemr[n]srmgqlrm-qlnsemrn[s]rmgqlrm-qlnsemrns[r]mgqlrm\\
 &-qlnsemrnsr[m]gqlrm-qlnsemrnsrm[g]qlrm-qlnsemrnsrmg[q]lrm\\
 &-qlnsemrnsrmgq[l]rm-qlnsemrnsrmgql[r]m-qlnsemrnsrmgqlr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[lqg]&{}=[l]qg+l[q]g+lq[g]-[r]mgqlrm-r[m]gqlrm-rm[g]qlrm-rmg[q]lrm\\
 &-rmgq[l]rm-rmgql[r]m-rmgqlr[m].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[lrmg]&{}=[l]rmg+l[r]mg+lr[m]g+lrm[g]-[q]gmrlq-q[g]mrlq-qg[m]rlq\\
 &-qgm[r]lq-qgmr[l]q-qgmrl[q]
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_1[g(qlrm)^jqlnse]&{}=[g](qlrm)^jqlnse+\sum_{i=0}^{j-1} g(qlrm)^{i}[q]lrm(qlrm)^{j-i-1}qlnse\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}q[l]rm(qlrm)^{j-i-1}qlnse+g(qlrm)^j[q]lnse\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}ql[r]m(qlrm)^{j-i-1}qlnse+g(qlrm)^jq[l]nse\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}qlr[m](qlrm)^{j-i-1}qlnse+g(qlrm)^jql[n]se\\
 &-\sum_{i=0}^{j-1}(mrlq)^i[m]rlq(mrlq)^{j-i-1}mregqlsnrm+g(qlrm)^jqlns[e]\\
 &-\sum_{i=0}^{j-1}(mrlq)^im[r]lq(mrlq)^{j-i-1}mregqlsnrm+g(qlrm)^jqln[s]e\\
 &-\sum_{i=0}^{j-1}(mrlq)^imr[l]q(mrlq)^{j-i-1}mregqlsnrm-(mrlq)^jmreg[q]lsnrm\\
 &-\sum_{i=0}^{j-1}(mrlq)^imrl[q](mrlq)^{j-i-1}mregqlsnrm-(mrlq)^jmregq[l]snrm\\
 &-(mrlq)^j[m]regqlsnrm-(mrlq)^jm[r]egqlsnrm-(mrlq)^jmr[e]gqlsnrm\\
 &-(mrlq)^jmre[g]qlsnrm-(mrlq)^jmregql[s]nrm-(mrlq)^jmregqls[n]rm\\
 &-(mrlq)^jmregqlsn[r]m-(mrlq)^jmregqlsnr[m].\\
 \\
 \delta_1[g(mrlq)^jmre]&{}=[g](mrlq)^jmre+\sum_{i=0}^{j-1}g(mrlq)^i[m]rlq(mrlq)^{j-i-1}mre\\
 &+\sum_{i=0}^{j-1}g(mrlq)^im[r]lq(mrlq)^{j-i-1}mre+\sum_{i=0}^{j-1}g(mrlq)^imr[l]q(mrlq)^{j-i-1}mre\\
 &+\sum_{i=0}^{j-1}g(mrlq)^imrl[q](mrlq)^{j-i-1}mre+g(mrlq)^j[m]re+g(mrlq)^jm[r]e\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}[q]lrm(qlrm)^{j-i-1}qlnsemrnsrmgqlrm+g(mrlq)^jmr[e]\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}q[l]rm(qlrm)^{j-i-1}qlnsemrnsrmgqlrm
 \end{aligned}
\]
\[
\begin{aligned}
 &-\sum_{i=0}^{j-1} (qlrm)^{i}ql[r]m(qlrm)^{j-i-1}qlnsemrnsrmgqlrm\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}[q]lr[m](qlrm)^{j-i-1}qlnsemrnsrmgqlrm\\
 &-(qlrm)^{j}[q]lnsemrnsrmgqlrm-(qlrm)^{j}q[l]nsemrnsrmgqlrm\\
 &-(qlrm)^{j}ql[n]semrnsrmgqlrm-(qlrm)^{j}qln[s]emrnsrmgqlrm\\
 &-(qlrm)^{j}qlns[e]mrnsrmgqlrm-(qlrm)^{j}qlnse[m]rnsrmgqlrm\\
 &-(qlrm)^{j}qlnsem[r]nsrmgqlrm-(qlrm)^{j}qlnsemr[n]srmgqlrm\\
 &-(qlrm)^{j}qlnsemrn[s]rmgqlrm-(qlrm)^{j}qlnsemrns[r]mgqlrm\\
 &-(qlrm)^{j}qlnsemrnsr[m]gqlrm-(qlrm)^{j}qlnsemrnsrm[g]qlrm\\
 &-(qlrm)^{j}qlnsemrnsrmg[q]lrm-(qlrm)^{j}qlnsemrnsrmgq[l]rm\\
 &-(qlrm)^{j}qlnsemrnsrmgql[r]m-(qlrm)^{j}qlnsemrnsrmgqlr[m].
 \end{aligned}
\]

\[
\begin{aligned}
 \delta_1[g(qlrm)^jnse]&{}=[g](qlrm)^jnse+\sum_{i=0}^{j-1} g(qlrm)^{i}[q]lrm(qlrm)^{j-i-1}nse\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}q[l]rm(qlrm)^{j-i-1}nse-(mrlq)^jesn[g]\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}ql[r]m(qlrm)^{j-i-1}nse-(mrlq)^jes[n]g\\
 &+\sum_{i=0}^{j-1} g(qlrm)^{i}qlr[m](qlrm)^{j-i-1}nse+g(qlrm)^j[n]se\\
 &-\sum_{i=0}^{j-1}(mrlq)^i[m]rlq(mrlq)^{j-i-1}esng+g(qlrm)^jns[e]\\
 &-\sum_{i=0}^{j-1}(mrlq)^im[r]lq(mrlq)^{j-i-1}esng+g(qlrm)^jn[s]e\\
 &-\sum_{i=0}^{j-1}(mrlq)^imr[l]q(mrlq)^{j-i-1}esng-(mrlq)^je[s]ng\\
 &-\sum_{i=0}^{j-1}(mrlq)^imrl[q](mrlq)^{j-i-1}esng-(mrlq)^j[e]sng.
 \end{aligned}
\]

\[
\begin{aligned}
 \delta_1[g(mrlq)^je]&{}=[g](mrlq)^je+g(mrlq)^j[e]+\sum_{i=0}^{j-1}g(mrlq)^i[m]rlq(mrlq)^{j-i-1}e\\
 &+\sum_{i=0}^{j-1}g(mrlq)^im[r]lq(mrlq)^{j-i-1}e+\sum_{i=0}^{j-1}g(mrlq)^imr[l]q(mrlq)^{j-i-1}e\\
 &+\sum_{i=0}^{j-1}g(mrlq)^imrl[q](mrlq)^{j-i-1}e-(qlrm)^{j}[n]segns-(qlrm)^{j}nsegn[s]\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}[q]lrm(qlrm)^{j-i-1}nsegns-(qlrm)^{j}ns[e]gns\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}q[l]rm(qlrm)^{j-i-1}nsegns-(qlrm)^{j}n[s]egns\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}ql[r]m(qlrm)^{j-i-1}nsegns-(qlrm)^{j}nse[g]ns\\
 &-\sum_{i=0}^{j-1} (qlrm)^{i}[q]lr[m](qlrm)^{j-i-1}nsegns-(qlrm)^{j}nseg[n]s.
 \end{aligned}
\]

For $m=2$ we have
\[
\begin{aligned}
V^{(2)}&{}=\{e^3,g^3,k^3,l^3,m^3,n^3,q^3,r^3,s^3,g^2e,ge^2, g^2nse, l^2e, l^2rnse, k^2se, k^2rnse, m^2e, m^2rnse,\\
&q^2rnse,q^2nse,l^2qg,l^2rmg, g^2qlnse, g^2mre, gnse^2, le^2, lrnse^2, kse^2, krnse^2, me^2, mrnse^2,\\
&qrnse^2,qnse^2,lqg^2,lrmg^2, gqlnse^2, gmre^2,lqge, lrmge,lqgnse, lrmgnse, lqgqlnse,\\
&g^2(qlrm)^jqlnse, g(qlrm)^jqlnse^2, g^2(mrlq)^jmre, g(mrlq)^jmre^2,g^2(qlrm)^jnse,g(qlrm)^jnse^2\\
&g^2(mrlq)^je, g(mrlq)^je^2,lrmgqlnse,lqgmre,lrmgmre,lqg(qlrm)^jqlnse,lrmg(qlrm)^jqlnse\\
&lqg(mrlq)^jmre,lrmg(mrlq)^jmre,lqg(qlrm)^jnse,lrmg(qlrm)^jnse, lqg(mrlq)^je\\
&lrmg(mrlq)^je
\mid 
j\ge1
\}
\end{aligned}
\]
Let us construct  a Morse matching
and evaluate the differential $\delta_2: B_{(3)}\to B_{(2)}$ (See Fig.~2). In the simplest case $w=x^3$, $x\in \{e,g,k,l,m,n,q,r,s\}$, we have
$\delta_2[x^3]=x[x^2]-[x^2]x$ for $x\in \{e,g,k,l,m,n,q,r,s\}$.
In a similar way, one may calculate
\[
\begin{aligned}
 \delta_2[g^2e]&{}=g[ge]-[g^2]e+[gnse]gns+esn[g^2]ns+es[n^2]s+e[s^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[ge^2]&{}=g[e^2]-[ge]e-nse[gnse]-ns[e^2]sng-n[s^2]ng-[n^2]g.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[g^2nse]&{}=g[gnse]-[g^2]nse+[ge]sng+nsegn[s^2]ng+nseg[n^2]g+nse[g^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[gnse^2]&{}=gns[e^2]-[gnse]e-esn[ge]-es[n^2]segns-e[s^2]egns-[e^2]gns.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[m^2rnse]&{}=m[mrnse]-[m^2]rnse+[me]snrm+rnsemrn[s^2]nrm+rnsemr[n^2]rm\\
 &+rnsem[r^2]m+rnse[m^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[mrnse^2]&{}=mrns[e^2]-[mrnse]e-esnr[me]-esn[r^2]nsemrns-es[n^2]semrns\\
 &-e[s^2]emrns-[e^2]mrns.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[m^2e]&{}=m[me]-[m^2]e+[mrnse]mrns+esnr[m^2]rns+esn[r^2]ns+es[n^2]s+e[s^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[me^2]&{}=m[e^2]-[me]e-rnse[mrnse]-rns[e^2]snrm-rn[s^2]nrm-r[n^2]rm-[r^2]m.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[k^2se]&{}=k[kse]-[k^2]se+[krnse]mrnkm+semknr[m^2]rnkm+semkn[r^2]nkm\\
 &+semk[n^2]km+sem[k^2]m+se[m^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[kse^2]&{}=ks[e^2]-[kse]e-rnsemrnk[me]-rnsemrn[krnse]mrns-rnse[mrnse]mks\\
 &-rns[e^2]snrks-rn[s^2]nrks-r[n^2]rks-[r^2]ks-knr[m^2]rns-kn[r^2]ns\\
 &-k[n^2]s-[m^2]ks.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[k^2rnse]&{}=k[krnse]-[k^2]rnse+[kse]mknrm+rnsemrnk[m^2]knrm+rnsemrn[k^2]nrm\\
 &+rnsemr[n^2]rm+rnsem[r^2]m+rnse[m^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[krnse^2]&{}=krns[e^2]-[krnse]e-semknr[me]-sem[kse]mrns-k[m^2]rns\\
 &-se[mrnse]mrnkrns-nr[m^2]rnkrns-n[r^2]nkrns-[n^2]krns-s[e^2]skrns\\
 &-[s^2]krns-semkn[r^2]nsemrns-semk[n^2]semrns.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[q^2rnse]&{}=q[qrnse]-[q^2]rnse+[qnse]gnkqgmknrm+rnsemrnkmgqkn[g^2]nkqgmknrm\\
 &+rnsemrnkmgqk[n^2]kqgmknrm+rnsemr[n^2]rm+rnsem[r^2]m\\
 &+rnsemrnkmgq[k^2]qgmknrm+rnsemrnkmg[q^2]gmknrm+rnse[m^2]\\
 &+rnsemrnkm[g^2]mknrm+rnsemrnk[m^2]knrm+rnsemrn[k^2]nrm.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[qrnse^2]&{}=qrns[e^2]-[qrnse]e-nsegnkqgmknr[me]-nsegnkqgmkn[r^2]nsemrns\\
 &-nsegnkqgmk[n^2]semrns-nsegnkqgm[kse]mrns-nsegnkqg[mrnse]mrnkrns\\
 &-nsegnkq[ge]skrns-nsegnk[qnse]gnkrns-nsegn[krnse]mrnkmgqrns\\
 &-nse[gnse]gqrns-[g^2]qrns-ns[e^2]snqrns-n[s^2]nqrns-[n^2]qrns-q[m^2]rns\\
 &-knr[m^2]rnkrns-kn[r^2]nkrns-k[n^2]krns-qk[s^2]krns-qkn[g^2]nkrns\\
 &-qk[n^2]krns-q[k^2]rns-gmknr[m^2]rnkmgqrns-gmkn[r^2]nkmgqrns\\
 &-gmk[n^2]kmgqrns-gm[k^2]mgqrns-g[m^2]gqrns.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[q^2nse]&{}=q[qnse]-[q^2]nse+[qrnse]mrnkmgqkng+nsegnkqgmknr[m^2]rnkmgqkng\\
 &+nsegnkqgmkn[r^2]nkmgqkng+nsegnkqgmk[n^2]kmgqkng\\
 &+nsegnkqgm[k^2]mgqkng+nsegnkqg[m^2]gqkng+nsegnkq[g^2]qkng\\
 &+nsegnk[q^2]kng+nsegn[k^2]ng+nseg[n^2]g+nse[g^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[qnse^2]&{}=qns[e^2]-[qnse]e-rnsemrnkmgqkn[ge]-rnsemrnkmgqk[n^2]segns\\
 &-rnsemrnkmgq[kse]gns-rnsemrnkmg[qrnse]mrnkmgns\\
 &-rnsemrnkm[gnse]gnkqns-rnsemrnk[me]skqns-rnsemrn[krnse]mrnkqns\\
 &-rnse[mrnse]mqns-rns[e^2]snrqns-rn[s^2]nrqns-r[n^2]rqns-[r^2]qns\\&-[m^2]qns-[k^2]qns-k[n^2]kqns-kn[r^2]nkqns-knr[m^2]rnkqns\\
 &-k[s^2]kqns-k[n^2]kqns-kn[g^2]nkqns-q[g^2]ns-qg[m^2]gns-qgm[k^2]mgns\\
 &-qgmk[n^2]kmgns-qgmkn[r^2]nkmgns-qgmknr[m^2]rnkmgns.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[l^2e]&{}=l[le]-[l^2]e+[lrnse]mrnkmgqksrmgqlrm+egqlskqgmknr[m^2]rnkmgqksrmgqlrm\\
 &+egqlskqgmkn[r^2]nkmgqksrmgqlrm+egqlskqgmk[n^2]kmgqksrmgqlrm\\
 &+egqlskqgm[k^2]mgqksrmgqlrm+egqlskqg[m^2]gqksrmgqlrm\\
 &+egqlskq[g^2]qksrmgqlrm+egqlsk[q^2]ksrmgqlrm\\
 &+egqls[k^2]srmgqlrm+egql[s^2]rmgqlrm+egq[lrmg]qlrm\\
 &+e[g^2]+eg[q^2]g+e[m^2]+em[r^2]m+emr[l^2]rm+emrl[q^2]lrm
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[le^2]&{}=l[e^2]-[le]e-rnsemrnkmgqksrmgqlr[me]-rnsemrnkmgqksrmgql[r^2]nsemrns\\
 &-rnsemrnkmgqksrm[gqlnse]mrns-rnsemrnkmgq[kse]gql\\
 &-rnsemrnkmg[qrnse]mrnkmgql-rnsemrnkm[gnse]gnkl-rnsemrnk[me]skl\\
 &-rnsemrn[krnse]mrnkl-rnse[mrnse]ml
 -rns[e^2]snrl-rn[s^2]nrl-r[n^2]rl\\
 &-[r^2]l-[k^2]l-k[n^2]kl-kn[r^2]nkl-knr[m^2]rnkl-[q^2]l-q[g^2]ql-qg[m^2]gql\\
 &-qgm[k^2]mgql-qgmk[n^2]kmgql-qgmkn[r^2]nkmgql-qgmknr[m^2]rnkmgql\\
 & -rnsemrnkmgqksr[m^2]regql-rnsemrnkmgqks[r^2]egql-kn[g^2]nkl-k[n^2]kl\\
 &-k[s^2]kl-lsnr[m^2]rns-lsn[r^2]ns-ls[n^2]s-l[s^2]-[m^2]l.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[l^2rnse]&{}=l[lrnse]-[l^2]rnse+[le]gqlskqgmknrm\\
 &+rnsemrnkmgqksrmgq[lrmg]qlskqgmknrm\\
 &+rnsemrnkmgqksrmg[q^2]gmrskqgmknrm\\
 &+rnsemrnkmgqksrm[g^2]mrskqgmknrm\\
 &+rnsemrnkmgqksr[m^2]rskqgmknrm+rnsemrnkmgqks[r^2]skqgmknrm\\
 &+rnsemrnkmgqk[s^2]kqgmknrm+rnsemrnkmgq[k^2]qgmknrm\\
 &+rnsemrnkmg[q^2]gmknrm+rnsemrnkm[g^2]mknrm+rnsemrnk[m^2]knrm\\
 &+rnsemrn[k^2]nrm+rnsemr[n^2]rm+rnsem[r^2]m+rnse[m^2]\\
 &+rnsemrnkmgqksl[q^2]lskqgmknrm+rnsemrnkmgqks[l^2]skqgmknrm
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[lrnse^2]&{}=lrns[e^2]-[lrnse]e-egqlskqgmknr[me]-egqlskqgmkn[r^2]nsemrns\\
 &-egqlskqgmk[n^2]semrns-egqlskqgm[kse]mrns-egqlskqg[mrnse]mrnkrns\\
 &-l[m^2]rns-egqlskq[ge]skrns-lk[n^2]krns-lkn[r^2]nkrns-lknr[m^2]rnkrns\\
 &-egqlsk[qnse]gnkrns-lk[s^2]krns-egqls[krnse]mrnkmgqrns-l[k^2]rns\\&-lk[n^2]krns-lkn[g^2]nkrns-egql[s^2]egqrns-rmgqlrm[m^2]gqrns\\
 &-rmgqlr[k^2]mgqrns-rmgqlrk[n^2]kmgqrns-rmgqlrkn[r^2]nkmgqrns\\
 &-rmgqlrknr[m^2]rnkmgqrns-egq[le]gqrns-skqgmknr[m^2]rnkmgqkslrns\\
 &-skqgmkn[r^2]nkmgqkslrns-skqgmk[n^2]kmgqkslrns-skqgm[k^2]mgqkslrns\\
 &-skqg[m^2]gqkslrns-skq[g^2]qkslrns-sk[q^2]kslrns-s[k^2]slrns\\
 &-eg[qrnse]mrnkmgqkslrns-e[gnse]gnslrns-rmgq[lrmg]qrns\\
 &-[e^2]lrns-[s^2]lrns-s[n^2]slrns-sn[g^2]nslrns-[r^2]lrns-r[m^2]rlrns\\
 &-rm[g^2]mrlrns-rmg[q^2]gmrlrns-l[q^2]rns.
 \end{aligned}
\]
\[
\begin{aligned}
\delta_2[l^2qg]&{}=l[lqg]-[l^2]qg+[lrmg]qlrm+qgmrl[q^2]lrm\\
 &+qgmr[l^2]rm+qgm[r^2]m+qg[m^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[lqg^2]&{}=lq[g^2]-[lqg]g-rmgq[lrmg]-rmg[q^2]gmrlq\\
 &-rm[g^2]mrlq-r[m^2]rlq-[r^2]lq.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[l^2rmg]&{}=l[lrmg]-[l^2]rmg+[lqg]mrlq+rmgqlr[m^2]rlq\\
 &+rmgql[r^2]lq+rmgq[l^2]q+rmg[q^2].
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[lrmg^2]&{}=lrm[g^2]-[lrmg]g-qgmr[lqg]-qgm[r^2]mgqlrm\\
 &-qg[m^2]gqlrm-q[g^2]qlrm-[q^2]lrm.
 \end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2qlnse]&{}=g[gqlnse]-[g^2]qlnse+[gmre]gqlsnrm+qlnsemrnsrmgq[lrmg]qlsnrm\\
&+qlnsemrns[q^2]lsnrm+qlnsemrns[l^2]snrm+qlnsemrnsrmg[q^2]gmrsnrm\\
&+qlnsemrnsrm[g^2]mrsnrm+qlnsemrnsr[m^2]rsnrm+qlnsemrns[r^2]snrm\\
&+qlnsemrn[s^2]nrm+qlnsemr[n^2]rm+qlnsem[r^2]m+qlnse[m^2].
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[gqlnse^2]&{}=gqlns[e^2]-[gqlnse]e-mregqlsnr[me]-mregqlsn[r^2]nsemrns\\
&-mregqls[n^2]semrns-mregql[s^2]emrns-mregq[le]mrns-gqlr[m^2]rns\\
&-mreg[qrnse]mrnkmgqksrmgqlns-mre[gnse]gnsrmgqlns-mr[e^2]rmgqlns\\
&-m[r^2]mgqlns-[m^2]gqlns-mrsn[g^2]nsrmgqlns-mrs[n^2]srmgqlns\\
&-mr[s^2]rmgqlns-gql[r^2]ns-mrskqgmknr[m^2]rnkmgqksrmgqlns\\
&-mrskqgmkn[r^2]nkmgqksrmgqlns-mrskqgmk[n^2]kmgqksrmgqlns\\
&-mrskqgm[k^2]mgqksrmgqlns-mrskqg[m^2]gqksrmgqlns\\
&-mrskq[g^2]qksrmgqlns-mrsk[q^2]ksrmgqlns-mrs[k^2]srmgqlns.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2mre]&{}=g[gmre]-[g^2]mre+[gqlnse]mrnsrmgqlrm+mregq[lrmg]qlrm+mreg[q^2]g\\
&+mre[g^2]+mremrl[q^2]lrm+mremr[l^2]rm+mrem[r^2]m+mre[m^2]\\
&+mregqlsnr[m^2]rnsmgqlrm+mregqlsn[r^2]nsmgqlrm+mregqls[n^2]smgqlrm\\
&+mregql[s^2]mgqlrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[gmre^2]&{}=gmr[e^2]-[gmre]e-qlnsemrnsrmgqlr[me] - qlnsemrnsrmgql[r^2]nsemrns\\
&-qlnsemrnsrm[gqlnse]mrns-gmrsnr[m^2]rns- gmrsn[r^2]ns- gmrs[n^2]s\\
&- gmr[s^2]-qlnsemrnsr[m^2]regql-qlnsemrns[r^2]egql-qlnse[mrnse]gql\\
&-qlns[e^2]snrmgql-qln[s^2]nrmgql-ql[n^2]rmgql-q[lrmg]ql-[q^2]gmr\\
&-gmrl[q^2]l-gmr[l^2].
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2(qlrm)^jqlnse]&{}=g[g(qlrm)^j qlnse]-[g^2] (qlrm)^j qlnse+[g(mrlq)^jmre]gqlsnrm\\
&+(qlrm)^j qlnsemrnsrmgq[lrmg]qlsnrm +(qlrm)^j qlnsemrns[q^2]lsnrm\\
&+(qlrm)^j qlnsemrns[l^2]snrm+(qlrm)^j qlnsemrnsrmg[q^2]gmrsnrm\\
&+(qlrm)^j qlnsemrnsrm[g^2]mrsnrm+(qlrm)^j qlnsemrnsr[m^2]rsnrm\\
&+(qlrm)^j qlnsemrns[r^2]snrm+(qlrm)^j qlnsemrn[s^2]nrm\\
&+(qlrm)^j qlnsemr[n^2]rm+(qlrm)^j qlnsem[r^2]m+(qlrm)^j qlnse[m^2].
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2(mrlq)^jmre]&{}=g[g(mrlq)^j mre]-[g^2] (mrlq)^j mre+[g(qlrm)^jqlnse]mrnsrmgqlrm\\
&+(mrlq)^j mregq[lrmg]qlrm+(mrlq)^j mreg[q^2]g+(mrlq)^j mre[g^2]\\
&+ (mrlq)^j mremrl[q^2]lrm+(mrlq)^j mremr[l^2]rm+(mrlq)^j mrem[r^2]m\\
&+(mrlq)^j mre[m^2]+ (mrlq)^j mregqlsnr[m^2]rnsrmgqlrm\\
&+(mrlq)^j mregqlsn[r^2]nsrmgqlrm+(mrlq)^j mregqls[n^2]srmgqlrm\\
&+(mrlq)^j mregql[s^2]rmgqlrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2(qlrm)^jnse]&{}=g[g(qlrm)^jnse]-[g^2] (qlrm)^jnse+[g(mrlq)^je]sng\\
&+(qlrm)^jnsegn[s^2]ng+(qlrm)^jnseg[n^2]g+(qlrm)^jnse[g^2].
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g^2(mrlq)^je]&{}=g[g(mrlq)^je]-[g^2] (mrlq)^je+[ g(qlrm)^jnse]gns+(mrlq)^jesn[g^2]ns\\
&+(mrlq)^jes[n^2]s+(mrlq)^je[s^2].
\end{aligned}
\]

\[
\begin{aligned}
 \delta_2[g(mrlq)^je^2]&{}=g(mrlq)^j[e^2]-[g(mrlq)^je]e-(qlrm)^jnse[gnse]-(qlrm)^jns[e^2]sng\\
 &-(qlrm)^jn[s^2]ng-(qlrm)^j[n^2]g-\sum_{i=1}^j(qlrm)^{j-i}q[lrmg](mrlq)^{i-1}\\
 &-\sum_{i=1}^j(qlrm)^{j-i}[q^2]g(mrlq)^i.
 \end{aligned}
\]
\[
\begin{aligned}
 \delta_2[g(qlrm)^jnse^2]&{}=g(qlrm)^jns[e^2]-[g(qlrm)^jnse]e-(mrlq)^jesn[ge]-(mrlq)^jes[n^2]segns\\
 &-(mrlq)^je[s^2]egns-(mrlq)^j[e^2]gns-\sum_{i=1}^j(mrlq)^{j-i}m[r^2]mg(qlrm)^{i}ns\\
 &-\sum_{i=1}^j(mrlq)^{j-i}mr[lqg](qlrm)^{i-1}ns-\sum_{i=1}^j(mrlq)^{j-i}[m^2]g(qlrm)^{i}ns.
 \end{aligned}
\]
\[
\begin{aligned}
\delta_2[g(qlrm)^jqlnse^2]&{}=g(qlrm)^jqlns[e^2]-[g(qlrm)^jqlnse]e-(mrlq)^jmregqlsnr[me]\\
&-(mrlq)^jmregqlsn[r^2]nsemrns-(mrlq)^jmregqls[n^2]semrns\\
&-(mrlq)^jmregql[s^2]emrns-(mrlq)^jmregq[le]mrns\\
&-(mrlq)^jgqlr[m^2]rns-(mrlq)^jmreg[qrnse]mrnkmgqksrmgqlns\\
&-(mrlq)^jmre[gnse]gnsrmgqlns-(mrlq)^jmr[e^2]rmgqlns\\
&-(mrlq)^jm[r^2]mgqlns-(mrlq)^jmrsn[g^2]nsrmgqlns\\
&-(mrlq)^jmrs[n^2]srmgqlns-(mrlq)^jmr[s^2]rmgqlns\\
&-(mrlq)^jmrskqgmknr[m^2]rnkmgqksrmgqlns\\
&-(mrlq)^jmrskqgmkn[r^2]nkmgqksrmgqlns\\
&-(mrlq)^jmrskqgmk[n^2]kmgqksrmgqlns-(mrlq)^jgql[r^2]ns\\
&-(mrlq)^jmrskqgm[k^2]mgqksrmgqlns-(mrlq)^j[m^2]gqlns\\
&-(mrlq)^jmrskqg[m^2]gqksrmgqlns-(mrlq)^jmrs[k^2]srmgqlns\\
&-(mrlq)^jmrskq[g^2]qksrmgqlns-(mrlq)^jmrsk[q^2]ksrmgqlns\\
&-\sum_{i=1}^j(mrlq)^{j-i}m[r^2]mg(qlrm)^{i}qlns-\sum_{i=1}^j(mrlq)^{j-i}mr[lqg](qlrm)^{i-1}qlns\\
&-\sum_{i=1}^j(mrlq)^{j-i}[m^2]g(qlrm)^{i}qlns.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[g(mrlq)^jmre^2]&{}=g(mrlq)^jmr[e^2]-[g(mrlq)^jmre]e-(qlrm)^jqlnsemrnsrmgqlr[me]\\
&-(qlrm)^jqlnsemrnsrmgql[r^2]nsemrns-g(mrlq)^jmrsn[r^2]ns\\
&-(qlrm)^jqlnsemrnsrm[gqlnse]mrns-g(mrlq)^jmrsnr[m^2]rns\\
&- g(mrlq)^jmrs[n^2]s-g(mrlq)^jmr[s^2]-(qlrm)^jqlnsemrnsr[m^2]regql\\
&-(qlrm)^jqlnsemrns[r^2]egql-(qlrm)^jqlnse[mrnse]gql\\
&-(qlrm)^jqlns[e^2]snrmgql-(qlrm)^jqln[s^2]nrmgql-(qlrm)^jql[n^2]rmgql\\
&-(qlrm)^jq[lrmg]ql-(qlrm)^j[q^2]gmr-g(mrlq)^jmrl[q^2]l-g(mrlq)^jmr[l^2]\\
&-\sum_{i=1}^j(qlrm)^{j-i}[q^2]g(mrlq)^{i}mr-\sum_{i=1}^j(qlrm)^{j-i}q[lrmg](mrlq)^{i-1}mr.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqg(mrlq)^je]&{}=lq[g(mrlq)^je]-[lqg](mrlq)^je+l[q^2]lrm(qlrm)^{j-1}nsegns\\
&+[l^2]rm(qlrm)^{j-1}nsegns-rmgqlr[m^2]rlq(mrlq)^{j-1}e-rm[g(mrlq)^{j-1}e]\\
&-rmgql[r^2]lq(mrlq)^{j-1}e-rmgq[l^2]q(mrlq)^{j-1}e-rmg[q^2](mrlq)^{j-1}e
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmg(mrlq)^je]&{}=lrm[g(mrlq)^je]-[lrmg](mrlq)^je-[q^2]lrm(qlrm)^{j}nsegns\\
&-q[g(mrlq)^{j+1}e]\\
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqg(qlrm)^jnse]&{}=lq[g(qlrm)^jnse]-[lqg](qlrm)^jnse-r[m^2]rlq(mrlq)^{j}esng\\
&-rm[g(qlrm)^{j+1}nse]-[r^2]lq(mrlq)^{j}esng.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmg(qlrm)^jnse]&{}=lrm[g(qlrm)^jnse]-[lrmg](qlrm)^jnse-qgmrl[q^2]lrm(qlrm)^{j-1}nse\\
&-qgmr[l^2]rm(qlrm)^{j-1}nse-qgm[r^2]m(qlrm)^{j-1}nse\\
&-qg[m^2](qlrm)^{j-1}nse-q[g(qlrm)^{j-1}nse]+lr[m^2]rlq(mrlq)^{j-1}esng\\
&+l[r^2]lq(mrlq)^{j-1}esng+[l^2]q(mrlq)^{j-1}esng.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmg(mrlq)^jmre]&{}=lrm[g(mrlq)^jmre]-[lrmg](mrlq)^jmre-q[g(mrlq)^{j+1}mre]\\
&-[q^2]lrm(qlrm)^{j}qlnsemrnsrmgqlrm.\\
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqg(mrlq)^jmre]&{}=lq[g(mrlq)^jmre]-[lqg](mrlq)^jmre-rm[g(mrlq)^{j-1}mre]\\
&+l[q^2]lrm(qlrm)^{j-1}qlnsemrnsrmgqlrm-rmg[q^2](mrlq)^{j-1}mre\\
&+[l^2]rm(qlrm)^{j-1}qlnsemrnsrmgqlrm-rmgqlr[m^2]rlq(mrlq)^{j-1}mre\\
&-rmgql[r^2]lq(mrlq)^{j-1}mre-rmgq[l^2]q(mrlq)^{j-1}mre.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqg(qlrm)^jqlnse]&{}=lq[g(qlrm)^jqlnse]-[lqg](qlrm)^jqlnse-r[m^2]rlq(mrlq)^{j}mregqlsnrm\\
&-rm[g(qlrm)^{j+1}qlnse]-[r^2]lq(mrlq)^{j}mregqlsnrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmg(qlrm)^jqlnse]&{}=lrm[g(qlrm)^jqlnse]-[lrmg](qlrm)^jqlnse-q[g(qlrm)^{j-1}qlnse]\\
&-qgmrl[q^2]lrm(qlrm)^{j-1}qlnse+l[r^2]lq(mrlq)^{j-1}mregqlsnrm\\
&-qgmr[l^2]rm(qlrm)^{j-1}qlnse-qgm[r^2]m(qlrm)^{j-1}qlnse\\
&-qg[m^2](qlrm)^{j-1}qlnse+lr[m^2]rlq(mrlq)^{j-1}mregqlsnrm\\
&+[l^2]q(mrlq)^{j-1}mregqlsnrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqge]&{}=lq[ge]-[lqg]e+l[qnse]gns-rmgqlr[me]+[lrnse]mrnkmgqks\\
&-rmgql[r^2]nsemrns-rm[gqlnse]mrns-r[m^2]regql-[r^2]egql-egqlsnr[m^2]rns\\
&-egqlsn[r^2]ns-egqls[n^2]s-egql[s^2]+egqlskqgmknr[m^2]rnkmgqks\\
&+egqlskqgmkn[r^2]nkmgqks+egqlskqgmk[n^2]kmgqks+egqlskqgm[k^2]mgqks\\
&+egqlskqg[m^2]gqks+egqlskq[g^2]qks+egqlsk[q^2]ks+egqls[k^2]s+egql[s^2]\\
&+egqls[n^2]s+egqlsn[g^2]ns
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqgnse]&{}=lq[gnse]-[lqg]nse-rm[gqlrmnse]-r[m^2]rlqesng-[r^2]lqesng.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqgqlnse]&{}=lq[gqlnse]-[lqg]qlnse-rm[gqlrmqlnse]-r[m^2]rlqmregqlsnrm\\
&-[r^2]lqmregqlsnrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lqgmre]&{}=lq[gmre]-[lqg]mre+l[q^2]lnsemrnsrmgqlrm+[l^2]nsemrnsrmgqlrm\\
&-rmgqlr[m^2]re-rmgql[r^2]e-rmgq[le]-rmg[qrnse]mrnkmgqksrmgqlrm\\
&-rm[gnse]gnsrmgqlrm-r[me]rmgqlrm-[r^2]nsemrnsrmgqlrm\\
&-nsemrnkqgmkn[r^2]nkmgqksrmgqlrm-nsemrnkqgmk[n^2]kmgqksrmgqlrm\\
&-nsemrnkqgm[k^2]mgqksrmgqlrm-nsemrnkqg[m^2]gqksrmgqlrm\\
&-nsemrnkq[g^2]qksrmgqlrm-nsemrnk[q^2]ksrmgqlrm-nsemrn[k^2]srmgqlrm\\
&-nsemrnkqgmknr[m^2]rnkmgqksrmgqlrm-nsemr[g^2]nsrmgqlrm\\
&-nsemrn[n^2]srmgqlrm-nsemrns[s^2]rmgqlrm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmge]&{}=lrm[ge]-[lrmg]e-q[gmrlqe]-[q^2]lrmnsegns.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmgnse]&{}=lrm[gnse]-[lrmg]nse+lr[me]sng-qgmrl[qnse]\\
&-qgmr[lrnse]mrnkmgqkng-q[gmre]gqlsng-[q^2]lnsemrg\\
&-lnsemrnsrmgq[lrmg]qlsng-lnsemrnsl[q^2]lsng-lnsemrns[l^2]sng\\
&-lnsemrn[s^2]ng-lnsemr[n^2]g-lnsemrns[r^2]sng-lnsemrnsr[m^2]rsng\\
&-lnsemrnsrm[g^2]mrsng-lnsemrnsrmg[q^2]gmrsng\\
&-lnsemrkqgmknr[m^2]rnkmgqkng-lnsemrkqgmkn[r^2]nkmgqkng\\
&-lnsemrkqgmk[n^2]kmgqkng-lnsemrkqgm[k^2]mgqkng\\
&-lnsemrkqg[m^2]gqkng-lnsemrk[q^2]kng-lnsemr[k^2]ng\\
&+lnsemrn[s^2]ng+lnsemr[n^2]g+l[r^2]nsemrg-lnsemrkq[g^2]qkng.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmgqlnse]&{}=lrm[gqlnse]-[lrmg]qlnse+lr[m^2]regqlsnrm+l[r^2]egqlsnrm\\
&+[le]gqlsnrm+rnsemrnkmgqksrmgq[lrmg]qlsnrm\\
&+rnsemrnkmgqksrmg[q^2]gmrsnrm+rnsemrnkmgqksrm[g^2]mrsnrm\\
&+rnsemrnkmgqksr[m^2]rsnrm+rnsemrnkmgqks[r^2]snrm\\
&+rnsemrnkmgqk[s^2]nrm+rnsemrnkmgqksl[q^2]lsnrm\\
&+rnsemrnkmgqks[l^2]snrm-qgmrl[q^2]lnse-qgmr[l^2]nse-qg[mrnse]\\
&-q[ge]snrm-[qnse]grm-rnsemrnkmgqkn[g^2]rm\\
&-rnsemrnkmgqk[s^2]nrm-rnsemrnkmgqkn[n^2]rm.
\end{aligned}
\]
\[
\begin{aligned}
\delta_2[lrmgmre]&{}=lrm[gmre]-[lrmg]mre-q[gmrlqmre]-[q^2]lrmqlnsemrnsrmgqlrm.
\end{aligned}
\]




\newpage
\vspace{220bp}
\centerline{\hspace{150mm}\vbox{%
\begin{picture}(60,-200)
 \put(-50,-320)
  {$
  [e|e]\stackrel{g\otimes 1}\longleftarrow [g|e|e] \stackrel{-1\otimes e}\longrightarrow [g|e]
  $}
  \put(10,-325){\vector(0,-1){20}}
  \put(-5,-340){\footnotesize{$-1$}}
  \put(-8,-355){$[nsegns|e]$}
  \put(14,-360){\vector(0,-1){20}}
  \put(18,-380){\vector(0,1){20}}
  \put(18,-370){\footnotesize{$-1$}}
  \put(-2,-370){\footnotesize{$1$}}
  \put(-63,-390)
  {$
  [segns|e]\stackrel{n\otimes 1}\longleftarrow [n|segns|e]
  $}
  \put(30,-400){\vector(0,-1){20}}
  \put(32,-411){\footnotesize{$1$}}
  \put(25,-433)
  {$
  [n|ng]
  $}
   \put(30,-445){\vector(0,-1){20}}
   \put(35,-465){\vector(0,1){20}}
   \put(38,-460){\footnotesize{$-1$}}
   \put(21,-460){\footnotesize{$1$}}
           \put(25,-485)
  {$
   [n|n|g]\stackrel{-1\otimes g}\longrightarrow[n|n] 
   $}
   \put(-49,-495){\vector(0,-1){20}}
   \put(-45,-415){\vector(0,1){20}}
   \put(-60,-407){\footnotesize{$1$}}
   \put(-40,-407){\footnotesize{$-1$}}
    \put(-119,-430)
  {$
         [egns|e]\stackrel{s\otimes 1}\longleftarrow [s|egns|e]
   $}
   \put(-110,-435){\vector(0,-1){20}}
   \put(-105,-455){\vector(0,1){20}}
   \put(-116,-447){\footnotesize{$1$}}
   \put(-129,-466)
  {$
   [e|gns|e]
   $}
    \put(-185,-466)
  {$
   [gns|e]\stackrel{e\otimes 1}\longleftarrow 
   $}
  \put(-108,-475){\vector(0,-1){20}}
 % \put(-105,-195){\vector(0,1){20}}
  \put(-116,-485){\footnotesize{$1$}}
  %\put(-100,-185){\footnotesize{$-1$}}
  \put(-119,-509)
  {$
   [e|esng]
   $}
    \put(-108,-517){\vector(0,-1){20}}
   \put(-123,-527){\footnotesize{$-1$}}
 \put(-104,-539){\vector(0,1){20}}
    \put(-101,-527){\footnotesize{$1$}}
    \put(-119,-551)
  {$
   [e|e|sng]\stackrel{-1\otimes sng}\longrightarrow[e|e] 
   $}
   \put(-35,-447){\footnotesize{$1$}}
   \put(-40,-435){\vector(0,-1){20}}
   \put(-35,-447){\footnotesize{$1$}}
   \put(-57,-466)
  {$
   [s|sng]
   $}
  \put(-44,-475){\vector(0,-1){20}}
  \put(-38,-495){\vector(0,1){20}}
  \put(-33,-485){\footnotesize{$1$}}
  \put(-60,-485){\footnotesize{$-1$}}
  \put(-60,-509)
  {$
   [s|s|ng]
   $}
  \put(-17,-508){\vector(1,0){35}}
  \put(-19,-505){\footnotesize{$-1\otimes ng$}}
  \put(20,-510)
  {$
   [s|s]
   $}
   \put(-157,-600){FIGURE 2. Morse matching graph for $[ge^2]$}
   
   
   \put(0,-20)
  {$
   [rmg] \stackrel{l\otimes 1} \longleftarrow [l|rmg] \stackrel{1\otimes rmg} \longrightarrow [l]
   $}
  \put(-5,-17){\vector(-1,0){20}}
  \put(-25,-21){\vector(1,0){20}}
  \put(-20,-30){\footnotesize{$-1$}}
  \put(-15,-15){\footnotesize{$1$}}
  \put(-57,-20){$[r|mg]$}
  \put(-40,-25){\vector(0,-1){20}}
  \put(-37,-40){\footnotesize{$1\otimes mg$}}
  \put(-45,-55){$[r]$}
  \put(-58,-19){\vector(-1,0){20}}
  \put(-78,-15){\footnotesize{$r\otimes 1$}}
  \put(-103,-20){$[mg]$}
  
  \put(-105,-17){\vector(-1,0){20}}
  \put(-125,-21){\vector(1,0){20}}
  \put(-120,-30){\footnotesize{$-1$}}
  \put(-115,-15){\footnotesize{$1$}}
  \put(-152,-20){$[m|g]$}
  \put(-140,-25){\vector(0,-1){20}}
  \put(-166,-35){\footnotesize{$1\otimes g$}}
  \put(-147,-55){$[m]$}
   \put(-178,-15){\footnotesize{$m\otimes 1$}}
  \put(-155,-19){\vector(-1,0){20}}
  \put(-193,-20){$[g]$}
  
  
  
  
  
  
  \put(68,-25){\vector(0,-1){20}}
  \put(38,-35){\footnotesize{$-1$}}
  \put(41,-55){$[qgmrlq]$}
  \put(52,-60){\vector(0,-1){20}}
  \put(57,-79){\vector(0,1){20}}
  \put(58,-72){\footnotesize{$-1$}}
  \put(42,-72){\footnotesize{$1$}}
  \put(-21,-88)
  {$
   [gmrlq] \stackrel{q\otimes 1} \longleftarrow [q|gmrlq] \stackrel{1\otimes gmrlq} \longrightarrow [q]
   $}
  \put(5,-94){\vector(0,-1){20}}
  \put(10,-113){\vector(0,1){20}}
  \put(15,-105){\footnotesize{$-1$}}
  \put(-5,-105){\footnotesize{$1$}}
  \put(-60,-125)
  {$
   [mrlq] \stackrel{g\otimes 1} \longleftarrow [g|mrlq] \stackrel{1\otimes mrlq} \longrightarrow [g]
   $}
     \put(-50,-128){\vector(0,-1){20}}
  \put(10,-113){\vector(0,1){20}}
  \put(15,-105){\footnotesize{$-1$}}
  \put(-5,-105){\footnotesize{$1$}}
\put(-45,-148){\vector(0,1){20}}
\put(-45,-140){\footnotesize{$-1$}}
  \put(-63,-140){\footnotesize{$1$}}
\put(-103,-158)
  {$
   [rlq] \stackrel{m\otimes 1} \longleftarrow [m|rlq] \stackrel{1\otimes rlq} \longrightarrow [m]
   $}
\put(-93,-182){\vector(0,1){20}}
\put(-97,-162){\vector(0,-1){20}}
\put(-91,-172){\footnotesize{$-1$}}
  \put(-104,-172){\footnotesize{$1$}}
  \put(-103,-190)
  {$
   [r|lq] \stackrel{1\otimes lq} \longrightarrow [r]
   $}
\put(-104,-189){\vector(-1,0){20}}
\put(-123,-185){\footnotesize{$r\otimes 1$}}
  \put(-138,-190){$[lq]$}
  
  
  \put(-141,-187){\vector(-1,0){20}}
\put(-153,-185){\footnotesize{$1$}}
  \put(-180,-190){$[l|q]$}
   \put(-161,-190){\vector(1,0){20}}
\put(-158,-199){\footnotesize{$-1$}}

\put(-180,-189){\vector(-1,0){20}}
\put(-199,-185){\footnotesize{$l\otimes 1$}}
  \put(-212,-190){$[q]$}
  
  \put(-173,-180){\vector(0,1){20}}
  \put(-178,-157){$[l]$}
  \put(-169,-172){\footnotesize{$1\otimes q$}}
 \put(-157,-240){FIGURE 1. Morse matching graph for $[lrmg]$}
\end{picture}}}
\newpage
\section{Application of Anick resolution on $\Gamma^4_5$}
Let $M=\Bbbk $ be a one-dimensional bimodule over $\Lambda =\Bbbk \Gamma^4_5$. Let us assume the bimodule is symmetric:
$x1=1x=\epsilon(x)$,  where $x\in A_2$
are the generators of $\Gamma_5^4$, $1\in \Bbbk $
is the generator of $M$, and 
$\epsilon: \{e,g,k,l,m,n,q,r,s\} \to \{1,-1\}$
is an arbitrary function.
By Lemma~\ref{lem:Lemma1} an arbitrary function $\epsilon $ like that
completely determines a 1-dimensional bimodule over 
 $\Bbbk \Gamma_5^4$
 denoted by $M_\epsilon $.

Let us fix a function $\epsilon$ and denote the bimodule $M$ obtained as above by $M_\epsilon $.
For $i\ge 0$, consider the $\Lambda $-bilinear map 
$\tau_i : B_{(i)}\to \Bbbk V^{(i)}$ defined by 
\[
\tau_i : \lambda_1\otimes v\otimes \lambda_2 \mapsto 
\epsilon(\lambda_1)\epsilon(\lambda_2)v,
\quad v\in V^{(i)}, \ \lambda_1,\lambda_2\in \Lambda.
\]
Then the map
 $\delta_{m}$
 induces a linear map 
 $\hat \delta_{m}$
from $\Bbbk V^{(m)}$ to $\Bbbk V^{(m-1)}$
such that 
\[
\tau_{m-1}\delta_m = \hat\delta_m \tau_m.
\]
The corresponding conjugate map $\hat \delta_{n}^*$ acts from  
the space $(\Bbbk V^{(m-1)})^*$ to $(\Bbbk V^{(m)})^*$.
Since 
\[
\mathrm{Hom}_{\Lambda{-}\Lambda} (B_{(m)}, M)
\simeq \mathrm{Hom}(\Bbbk V^{(m)}, M)\simeq (\Bbbk V^{(m)})^*,
\]
we have 
\[
Z^m(\Lambda , M)\simeq \Ker \hat\delta_m^*,\quad 
\beta^m (\Lambda , M) \simeq \mathrm{Im}\, \hat\delta_{m-1}^*.
\]
This is straightforward to check that 
$\rank \hat\delta_1 = 9 = |V^{(0)}|$. 
It remains to find $\dim\Ker\hat\delta_2^*$. 
According to the Fredholm principle, 
$\dim\Ker\hat\delta_2^* = \dim (\mathrm{Im}\,\hat\delta_2)^\perp = \mathrm{codim}_{V^{(1)}} \mathrm{Im}\,\hat\delta_2$.
By definition, 
\[
H^{m}:=H^{m}(\Bbbk \Gamma^4_5, M)\simeq \Ker \hat \delta_{m}^*/\mathrm{Im}\,\hat \delta_{m-1}^*.
\]
Hence we obtain  
\begin{equation}\label{eq:HochDim}
\dim H^{2} =  \mathrm{codim}_{\Bbbk V^{(1)}} \mathrm{Im}\,\hat\delta_2 - 9.
\end{equation}
 
The similar formula for $\dim H^1$ implies the 
following (quite expectable)
 
\begin{corollary}
The first Hochschild cohomology group $H^1(\Bbbk \Gamma_5^4, M_\epsilon )$ is trivial for every~$\epsilon $. 
\end{corollary}


In order to find the dimension of the second Hochschild cohomology group 
$H^2(\Bbbk \Gamma_5^4, M)$ 
for $M=M_\epsilon$ one needs 
to calculate 
$\mathrm{codim}_{\Bbbk V^{(1)}}\mathrm{Im}\,\hat\delta_2$
by means of the formulae from Section~\ref{sec:Complex}.

\begin{corollary}
The second Hochschild cohomology group of $\Bbbk \Gamma_5^4$ with trivial coefficients is 5-dimensional.
\end{corollary}

\begin{proof}
Let 
$x1=1x=\epsilon(x)=1$ for all  $x\in A_2$.
Let us calculate $\hat \delta_2$ and 
use elementary transformations to reduce it (partially) 
to the row echelon form. 
As a result, we obtain the following vectors 
from $\Bbbk V^{(1)}$ to span $\mathrm{Im}\,\hat \delta_2$:
\[
\begin{aligned}
e_1&=2[g^2]+2[k^2]+2[m^2]+2[n^2]+[r^2]+[qrnse]+[qnse]\\
e_2&=[l^2]+2[m^2]+[n^2]+2[q^2]+2[r^2]+[s^2]+[lrmg]+[gqlnse]+[gmre]\\e_3&=2[m^2]+[n^2]+[r^2]+[kse]+[krnse]\\
e_4&=-[n^2]-[r^2]-[s^2]-[me]-[mrnse]\\
e_5&=3[q^2]+2[r^2]+\frac{5}{2}[s^2]+[le]+[lrnse]-\frac{1}{2}[kse]-\frac{1}{2}[krnse]+\frac{3}{2}[me]\\&+\frac{3}{2}[mrnse]-[qrnse]-[qnse]+[lrmg]\\
e_6&=[r^2]-[ge]-[gnse]+[me]+[mrnse]\\
e_7&=-\frac{7}{6}[s^2]-\frac{4}{3}[ge]-\frac{7}{3}[gnse]-\frac{2}{3}[le]+\frac{1}{3}[lrnse]+\frac{5}{6}[kse]+\frac{5}{6}[krsne]+\frac{5}{6}[me]\\
&+\frac{11}{6}[mrnse]-\frac{1}{3}[qrnse]+\frac{2}{3}[qnse]+\frac{1}{3}[lrmg]-[gqlnse]\\
e_8&=\frac{5}{7}[ge]+[gnse]-\frac{1}{7}[le]-\frac{3}{7}[lrnse]-\frac{4}{7}[kse]-\frac{4}{7}[krsne]-\frac{4}{7}[me]-\frac{6}{7}[mrnse]\\&+\frac{3}{7}[qrnse]+\frac{1}{7}[qnse]+[lqg]+\frac{4}{7}[lrmg]+\frac{2}{7}[gqlnse]\\
e_9&=\frac{8}{5}[gnse]+\frac{1}{5}[le]-\frac{2}{5}[lrnse]-\frac{1}{5}[kse]-\frac{1}{5}[krnse]-\frac{1}{5}[me]-\frac{4}{5}[mrnse]\\
&+\frac{2}{5}[qrnse]-\frac{1}{5}[qnse]-\frac{2}{5}[lqg]+\frac{1}{5}[lrmg]+\frac{3}{5}[gqlnse]-[gqlrmnse]\\
e_{10}&=\frac{1}{8}[le]-\frac{1}{4}[lrnse]-\frac{1}{8}[kse]-\frac{1}{8}[krnse]-\frac{1}{8}[me]-\frac{1}{2}[mrnse]+\frac{1}{4}[qrnse]\\
&-\frac{1}{8}[qnse]-\frac{1}{4}[lqg]+\frac{1}{8}[lrmg]+\frac{11}{8}[gqlnse]+\frac{3}{8}[gqlrmnse]-[qdlrmqlnse]\\
e_{11}&=[lrnse]+[kse]+[krnse]+[me]+2[mrnse]-[qrnse]-[lqg]-[lrmg]\\
&-3[gqlnse]-2[gqlrmnse]+2[gqlrmqlnse]-[gmrlqe]\\
e_{12}&=[gqlnse]+[gmre]+[gqlrmqlnse]+[gmrlqmre]\\
e_{13}&=[gmre]-[gmrlqmre]-[g(mrlq)^{j-1}e]+[g(mrlq)^je]\\
e_{14}&=-[gqlrmnse]-[gmrlqe]+[g(mrlq)^je]+[g(qlrm)^j nse]\\
e_{15}&=-[gqlrmqlnse]-[gmrlqmre]+[g(mrlq)^jmre]+[g(qlrm)^jqlnse]\\
e_{16}&=[gmrlqe]-2[g(mrlq)^{j-1}e]+[g(mrlq)^je]\\
e_{17}&=[gmrlqmre]-2[g(mrlq)^{j-1}e]+2[g(mrlq)^je]-[g(mrlq)^jmre]\\
e_{18}&=[g(mrlq)^{j-1}e]-[g(mrlq)^{j-1}mre]-[g(mrlq)^je]+[g(mrlq)^jmre]\\
e_{19}&=-[g(mrlq)^{j-1}mre]-[g(qlrm)^{j-1}qlnse]+[g(mrlq)^jmre]+[g(qlrm)^jqlnse]\\
e_{20}&=[g(qlrm)^{j-1}qlnse]-[g(mrlq)^{j-1}nse]-[g(qlrm)^jqlnse]+[g(mrlq)^jnse]\\
e_{21}&=[g(mrlq)^{j-1}nse]+[g(mrlq)^je]-[g(qlrm)^jnse]-[g(mrlq)^{j+1}e]\\
e_{22}&=-[g(mrlq)^je]+[g(mrlq)^jmre]+[g(mrlq)^{j+1}e]-[g(mrlq)^{j+1}mre]\\
e_{23}&=[g(mrlq)^jmre]+[g(qlrm)^jqlnse]-[g(mrlq)^{j+1}mre]-[g(qlrm)^{j+1}qlnse]\\
e_{24}&=-[g(qlrm)^jqlnse]+[g(qlrm)^jnse]+[g(qlrm)^{j+1}qlnse]-[g(qlrm)^{j+1}nse]
\end{aligned}
\]
Here $j\ge 2$. 
Now we may choose a linear basis of 
$\Bbbk V^{(1)}/\mathrm{Im}\,\hat \delta_2$. 
Note that the classes of $e^2, k^2, kse, krnse, me, mrnse, qrnse, qnse, lqg, lrmg$ must belong to the desired basis. 
Moreover, if we compare $e_{13}$ and $e_{16}$ for $j$ and $j+1$ then 
we conclude that 
\[
g(mrlq)^je \equiv  g(mrlq)^{j-1}e \pmod {\mathrm{Im}\,\hat \delta_2}.
\]
Then proceed in the same way with 
$e_{14}$ to get 
\[
g(qlrm)^j nse \equiv g(qlrm)^{j-1} nse \pmod {\mathrm{Im}\,\hat \delta_2 }.
\]
Similarly, $e_{18}$ and $e_{20}$ provide us with the same 
relations for $g(mrlq)^jmre$ and $g(qlrm)^j qlnse$. 
One may easily see that there are no more relations for 
the elements of $V^{(1)}$, so the total codimension 
of $\mathrm{Im}\,\hat \delta_2 $ is~14.
It follows from \eqref{eq:HochDim} that 
$\dim H^2 = 5$.
\end{proof}

\bigskip
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