# An answer to the Whitehead’s asphericity question

Elton Pasku
Universiteti i Tiranës
Fakulteti i Shkencave Natyrore
Departamenti i Matematikës
Tiranë, Albania
<EMAIL_ADDRESS>
###### Abstract

The Whitehead asphericity problem, regarded as a problem of combinatorial
group theory, asks whether any subpresentation of an aspherical group
presentation is also aspherical. We give a positive answer to this question by
proving that if $\mathcal{P}=(\mathbf{x},\mathbf{r})$ is an aspherical
presentation of the trivial group, and $r_{0}\in\mathbf{r}$ a fixed relation,
then $\mathcal{P}_{1}=(\mathbf{x},\mathbf{r}_{1})$ is aspherical where
$\mathbf{r}_{1}=\mathbf{r}\setminus\\{r_{0}\\}$.

##  1 Introduction

A 2-dimensional CW-complex $K$ is called aspherical if $\pi_{2}(K)=0$. The
Whitehead asphericity problem (WAP for short), raised as a question in [44],
asks whether any subcomplex of an aspherical 2-complex is also aspherical. The
question can be formulated in group theoretic terms since every group
presentation $\mathcal{P}$ has a geometric realisation as a 2-dimensional CW-
complex $K(\mathcal{P})$ and so $\mathcal{P}$ is called aspherical if
$K(\mathcal{P})$ is aspherical. A useful review of this question is in [42].

The purpose of the present paper is to prove that if
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ is an aspherical presentation of the
trivial group and $r_{0}\in\mathbf{r}$ is a fixed relation, then the
subpresentation $\mathcal{P}_{1}=(\mathbf{x},\mathbf{r}_{1})$ where
$\mathbf{r}_{1}=\mathbf{r}\setminus\\{r_{0}\\}$ is again aspherical. This in
fact implies that WAP has always a positive answer since in Theorem 1 of [32]
Ivanov proves that if the WAP is false, then there is an aspherical
presentation $\mathcal{P}=(\mathcal{A},\mathcal{R}\cup\\{z\\})$ of the trivial
group where the alphabet $\mathcal{A}$ is countable and $z\in\mathcal{A}$ such
that $\mathcal{P}_{1}=(\mathcal{A},\mathcal{R})$ is not aspherical.

An immediate implication of our result and that of Bestvina-Brady [2] is that
the conjecture of Eilenberg and Ganea [11] is false. This conjecture states
that if a discrete group $G$ has cohomological dimension 2, then it has a
2-dimensional Eilenberg-MacLane space $K(G,1)$.

There is a large corpus of results which are related to ours and is mostly
contained in [4], [5], [8], [10], [14], [18], [19], [20], [22], [23], [24],
[25], [26], [30], [32], [16] and [43].

In the first part of our paper we will make use of the review paper [5] of
Brown and Huebschmann which contains several key results about aspherical
group presentations one of which is proposition 14 that gives sufficient and
necessary conditions under which a group presentation
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ is aspherical. It turns out that the
asphericity of $\mathcal{P}$ is encoded in the structure of the free crossed
module $(H/P,\hat{F},\delta)$ that is associated to $\mathcal{P}$. To be
precise we state below proposition 14.

######  Proposition 1.1.

(Proposition 14 of [5]) Let $K(\mathcal{P})$ be the geometric realisation of a
group presentation $\mathcal{P}=(\mathbf{x},\mathbf{r})$ and let $G$ be the
group given by $\mathcal{P}$. The following are equivalent.

(i)

The 2-complex $K(\mathcal{P})$ is aspherical.

(ii)

The module $\pi$ of identities for $\mathcal{P}$ is zero.

(iii)

The relation module $\mathcal{N(P)}$ of $\mathcal{P}$ is a free left
$\mathbb{Z}G$ module on the images of the relators $r\in\mathbf{r}$.

(iv)

Any identity $Y$-sequence for $\mathcal{P}$ is Peiffer equivalent to the empty
sequence.

The last condition is of a particular interest to us. By definition, a
$Y$-sequence for $\mathcal{P}$ is a finite (possibly empty) sequence of the
form
$((^{u_{1}}r_{1})^{\varepsilon_{1}},...,(^{u_{n}}r_{n})^{\varepsilon_{n}})$
where $r\in\mathbf{r}$, $u$ is a word from the free group $F$ over
$\mathbf{x}$ and $\varepsilon=\pm 1$. A $Y$-sequence
$((^{u_{1}}r_{1})^{\varepsilon_{1}},...,(^{u_{n}}r_{n})^{\varepsilon_{n}})$ is
called an identity $Y$-sequence if it is either empty or if
$\prod_{i=1,n}u_{i}r_{i}^{\varepsilon_{i}}u_{i}^{-1}=1$ in $F$. The definition
of Peiffer equivalence is based on Peiffer operations on $Y$-sequences and
reads as follows.

  * (i)

An elementary Peiffer exchange replaces an adjacent pair
$((^{u}r)^{\varepsilon},((^{v}s)^{\delta})$ in a $Y$-sequence by either
$((^{ur^{\varepsilon}u^{-1}v}s)^{\delta},(^{u}r)^{\varepsilon})$, or by
$((^{v}s)^{\delta},((^{vs^{-\delta}v^{-1}u}r)^{\varepsilon})$.

  * (ii)

A Peiffer deletion deletes an adjacent pair
$((^{u}r)^{\varepsilon},(^{u}r)^{-\varepsilon})$ in a $Y$-sequence.

  * (iii)

A Peiffer insertion is the inverse of the Peiffer deletion.

The equivalence relation on the set of $Y$-sequences generated by the above
operations is called Peiffer equivalence. We recall from [5] what does it mean
for an identity $Y$-sequence
$((^{u_{1}}r_{1})^{\varepsilon_{1}},...,(^{u_{n}}r_{n})^{\varepsilon_{n}})$ to
have the primary identity property. This means that the indices $1,2,...,n$
are grouped into pairs $(i,j)$ such that $r_{i}=r_{j}$,
$\varepsilon_{i}=-\varepsilon_{j}$ and $u_{i}=u_{j}$ modulo $N$ where $N$ is
the normal subgroup of $\hat{F}$ generated by $\mathbf{r}$. Proposition 16 of
[5] shows that every such sequence is Peiffer equivalent to the empty
sequence. Given an identity $Y$-sequence $d$ which is equivalent to the empty
sequence 1, we would be interested to know what kind of insertions
$((^{u}r)^{\varepsilon},(^{u}r)^{-\varepsilon})$ are used along the way of
transforming $d$ to 1. It is obvious that keeping track of that information is
vital to tackle the Whitehead problem.

The aim of Section 3 of the present paper is to offer an alternative way in
dealing with the asphericity of a group presentation
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ by considering a new crossed module
$(\mathcal{G}(\Upsilon),\hat{F},\tilde{\theta})$ over the free group $\hat{F}$
on $\mathbf{x}$ where $\mathcal{G}(\Upsilon)$ is the group generated by the
symbols $(^{u}r)^{\varepsilon}$ subject to relations
$(^{u}r)^{\varepsilon}(^{v}s)^{\delta}=(^{{u{r^{\varepsilon}}u^{-1}}v}s)^{\delta}(^{u}r)^{\varepsilon}$,
the action of $\hat{F}$ on $\mathcal{G}(\Upsilon)$ and the map
$\tilde{\theta}$ are defined in the obvious fashion. The advantage of working
with $\mathcal{G}(\Upsilon)$ is that unlike to $H/P$, in
$\mathcal{G}(\Upsilon)$ the images of insertions
$((^{u}r)^{\varepsilon},(^{u}r)^{-\varepsilon})$ do not cancel out and this
enables us to express the asphericity in terms of such insertions. This is
realized by considering the kernel $\tilde{\Pi}$ of $\tilde{\theta}$ which is
the analogue of the module $\pi$ of identities for $\mathcal{P}$ in the
standard theory and is not trivial when $\mathcal{P}$ is aspherical. We call
$\tilde{\Pi}$ the generalized module of identities for $\mathcal{P}$.

To prove our results we apply techniques from the theory of semigroup actions
and to this end we use concepts like the universal enveloping group
$\mathcal{G}(S)$ of a given semigroup $S$, the dominion of a subsemigroup $U$
of a semigroup $S$ and the tensor product of semigroup actions. These concepts
are explained, with references, in Section 2.

##  2 Monoid actions

For the benefit of the reader not familiar with monoid actions we will list
below some basic notions and results that are used in the paper. For further
results on the subject the reader may consult the monograph [27]. Given $S$ a
monoid with identity element 1 and $X$ a nonempty set, we say that $X$ is a
left S-system if there is an action $(s,x)\mapsto sx$ from $S\times X$ into
$X$ with the properties

$\displaystyle(st)x$ $\displaystyle=s(tx)\text{ for all }s,t\in S\text{ and
}x\in X,$ $\displaystyle 1x$ $\displaystyle=x\text{ for all }x\in X.$

Right $S$-systems are defined analogously in the obvious way. Given $S$ and
$T$ (not necessarily different) monoids, we say that $X$ is an (S,T)-bisystem
if it is a left $S$-system, a right $T$-system, and if

$(sx)t=s(xt)\text{ for all }s\in S,t\in T\text{ and }x\in X.$

If $X$ and $Y$ are both left $S$-systems, then an S-morphism or S-map is a map
$\phi:X\rightarrow Y$ such that

$\phi(sx)=s\phi(x)\text{ for all }s\in S\text{ and }x\in X.$

Morphisms of right $S$-systems and of $(S,T)$-bisystems are defined in an
analogue way. If we are given a left $T$-system $X$ and a right $S$-system
$Y$, then we can give the cartesian product $X\times Y$ the structure of an
$(T,S)$-bisystem by setting

$t(x,y)=(tx,y)\text{ and }(x,y)s=(x,ys).$

Let now $A$ be an $(T,U)$-bisystem, $B$ an $(U,S)$-bisystem and $C$ an
$(T,S)$-bisystem. As explained above, we can give to $A\times B$ the structure
of an $(T,S)$-bisystem. With this in mind we say that a $(T,S)$-map
$\beta:A\times B\rightarrow C$ is a bimap if

$\beta(au,b)=\beta(a,ub)\text{ for all }a\in A,b\in B\text{ and }u\in U.$

A pair $(A\otimes_{U}B,\psi)$ consisting of a $(T,S)$-bisystem $A\otimes_{U}B$
and a bimap $\psi:A\times B\rightarrow A\otimes_{U}B$ will be called a tensor
product of A and B over U if for every $(T,S)$-bisystem $C$ and every bimap
$\beta:A\times B\rightarrow C$, there exists a unique $(T,S)$-map
$\bar{\beta}:A\otimes_{U}B\rightarrow C$ such that the diagram

---
$\textstyle{A\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\psi}$$\textstyle{A\otimes_{U}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\beta}}$$\textstyle{C}$

commutes. It is proved in [27] that $A\otimes_{U}B$ exists and is unique up to
isomorphism. The existence theorem reveals that $A\otimes_{U}B=(A\times
B)/\tau$ where $\tau$ is the equivalence on $A\times B$ generated by the
relation

$T=\\{((au,b),(a,ub)):a\in A,b\in B,u\in U\\}.$

The equivalence class of a pair $(a,b)$ is usually denoted by $a\otimes_{U}b$.
To us is of interest the situation when $A=S=B$ where $S$ is a monoid and $U$
is a submonoid of $S$. Here $A$ is clearly regarded as an $(S,U)$-bisystem
with $U$ acting on the right on $A$ by multiplication, and $B$ as an
$(U,S)$-bisystem where $U$ acts on the left on $B$ by multiplication.

Another concept that is important to our approach is that of the dominion
which is defined in [31] from Isbell. By definition, if $U$ is a submonoid of
a monoid $S$, then the dominion $\text{Dom}_{S}(U)$ consists of all the
elements $d\in S$ having the property that for every monoid $T$ and every pair
of monoid homomorphisms $f,g:S\rightarrow T$ that coincide in $U$, it follows
that $f(d)=g(d)$. Related to dominions there is the well known zigzag theorem
of Isbell. We will present here the Stenstrom version of it (theorem 8.3.3 of
[27]) which reads. Let $U$ be a submonoid of a monoid $S$ and let $d\in S$.
Then, $d\in\text{Dom}_{S}(U)$ if and only if $d\otimes_{U}1=1\otimes_{U}d$ in
the tensor product $A=S\otimes_{U}S$. We mention here that this result holds
true if $S$ turns out to be a group and $U$ a subgroup, both regarded as
monoids. A key result (theorem 8.3.6 of [27]) that is used in the next section
is the fact that any inverse semigroup $U$ is absolutely closed in the sense
that for every semigroup $S$ containing $U$ as a subsemigroup,
$\text{Dom}_{S}(U)=U$. It is obvious that groups are absolutely closed as
special cases of inverse monoids (see [28]).

##  3 Peiffer operations and monoid actions

Before we explain how monoid actions are used to deal with the Peiffer
operations on $Y$-sequences, we will introduce several monoids.

The first one is the monoid $\Upsilon$ defined by the monoid presentation
$\mathcal{M}=\langle Y\cup Y^{-1},P\rangle$ where $Y^{-1}$ is the set of group
inverses of the elements of $Y$ and $P$ consists of all pairs
$(ab,{{}^{\theta(a)}}ba)$ where $a,b\in Y\cup Y^{-1}$.

The second one is the group $\mathcal{G}(\Upsilon)$ given by the group
presentation $(Y\cup Y^{-1},\hat{P})$ where $\hat{P}$ is the set of all words
$ab\iota(a)\iota(^{\theta(a)}b)$ where by $\iota(c)$ we denote the inverse of
$c$ in the free group over $Y\cup Y^{-1}$. Before we introduce the next two
monoids and the respective monoid actions, we stop to explain that $\Upsilon$
and $\mathcal{G}(\Upsilon)$ are special cases of a more general situation. If
a monoid $S$ is given by the monoid presentation $\mathcal{M}=\langle
X,R\rangle$, then its universal enveloping group $\mathcal{G}(S)$ (see [1] and
[9]) is defined to be the group given by the group presentation $(X,\hat{R})$
where $\hat{R}$ consists of all words $u\iota(v)$ whenever $(u,v)\in R$ where
$\iota(v)$ is the inverse of $v$ in the free group over $X$. We let for future
use $\sigma:FM(X)\rightarrow S$ the respective canonical homomorphism where
$FM(X)$ is the free monoid on $X$. It is easy to see that there is a monoid
homomorphism $\mu_{S}:S\rightarrow\mathcal{G}(S)$ which satisfies the
following universal property. For every group $G$ and monoid homomorphism
$f:S\rightarrow G$, there is a unique group homomorphism
$\hat{f}:\mathcal{G}(S)\rightarrow G$ such that $\hat{f}\mu_{S}=f$. This
universal property is an indication of an adjoint situation. Specifically, the
functor $\mathcal{G}:\mathbf{Mon}\rightarrow\mathbf{Grp}$ which maps every
monoid to its universal group, is a left adjoint to the forgetful functor
$U:\mathbf{Grp}\rightarrow\mathbf{Mon}$. This ensures that $\mathcal{G}(S)$ is
an invariant of the presentation of $S$.

The third monoid we consider is the submonoid $\mathfrak{U}$ of $\Upsilon$,
having the same unit as $\Upsilon$, and is generated from all the elements of
the form $\sigma(a)\sigma(a^{-1})$ with $a\in Y\cup Y^{-1}$. This monoid, acts
on the left and on the right on $\Upsilon$ by the multiplication in
$\Upsilon$.

The last monoid considered is the subgroup $\hat{\mathfrak{U}}$ of
$\mathcal{G}(\Upsilon)$ generated by $\mu(\mathfrak{U})$. Similarly to above,
$\hat{\mathfrak{U}}$ acts on $\mathcal{G}(\Upsilon)$ by multiplication.

Given $\alpha=(a_{1},...,a_{n})$ an $Y$-sequence over the group presentation
$\mathcal{P}=(\mathbf{x},\mathbf{r})$, then performing an elementary Peiffer
operation on $\alpha$ can be interpreted in a simple way in terms of the
monoids $\Upsilon$ and $\mathfrak{U}$. In what follows we will denote by
$\sigma(\alpha)$ the element
$\sigma(a_{1})\cdot\cdot\cdot\sigma(a_{n})\in\Upsilon$. If
$\beta=(b_{1},...,b_{n})$ is obtained from $\alpha=(a_{1},...,a_{n})$ by
performing an elementary Peiffer exchange, then from the definition of
$\Upsilon$, $\sigma(\alpha)=\sigma(\beta)$, therefore an elementary Peiffer
exchange or a finite sequence of such has no effect on the element
$\sigma(a_{1})\cdot\cdot\cdot\sigma(a_{n})\in\Upsilon$. Before we see the
effect that a Peiffer insertion in $\alpha$ has on $\sigma(\alpha)$ we need
the first claim of the following.

######  Lemma 3.1.

The elements of $\mathfrak{U}$ are central in $\Upsilon$ and those of
$\hat{\mathfrak{U}}$ are central in $\mathcal{G}(\Upsilon)$.

###### Proof.

We see that for every $a\text{ and }b\in Y\cup Y^{-1}$,
$\sigma(a)\sigma(a^{-1})\sigma(b)=\sigma(b)\sigma(a)\sigma(a^{-1})$. Indeed,

$\displaystyle\sigma(a)\sigma(a^{-1})\sigma(b)$
$\displaystyle=~{}^{\theta(a)\theta(a^{-1})}{\sigma(b)}(\sigma(a)\sigma(a^{-1}))$
$\displaystyle=\sigma(b)\sigma(a)\sigma(a^{-1}).$

Since elements $\sigma(b)$ and $\sigma(a)\sigma(a^{-1})$ are generators of
$\Upsilon$ and $\mathfrak{U}$ respectively, then the first claim holds true.
The second claim follows easily. ∎

If we insert $(a,a^{-1})$ at some point in $\alpha=(a_{1},...,a_{n})$ to
obtain $\alpha^{\prime}=(a_{1},...,a,a^{-1},...,a_{n})$, then from lemma 3.1,

$\sigma(\alpha^{\prime})=\sigma(\alpha)\cdot(\sigma(a)\sigma(a^{-1})),$

which means that inserting $(a,a^{-1})$ inside a $Y$-sequence $\alpha$ has the
same effect as multiplying the corresponding $\sigma(\alpha)$ in $\Upsilon$ by
the element $\sigma(a)\sigma(a^{-1})$ of $\mathfrak{U}$. For the converse, it
is obvious that any word $\beta\in FM(Y\cup Y^{-1})$ representing
$\sigma(\alpha)\cdot(\sigma(a)\sigma(a^{-1}))$ is Peiffer equivalent to
$\alpha$. Of course the deletion has the obvious interpretation in our
semigroup theoretic terms as the inverse of the above process. We retain the
same names for our semigroup operations, that is insertion for multiplication
by $\sigma(a)\sigma(a^{-1})$ and deletion for its inverse. Related to these
operations on the elements of $\Upsilon$ we make the following definition.

######  Definition 3.2.

We denote by $\sim_{\mathfrak{U}}$ the equivalence relation in $\Upsilon$
generated by all pairs
$(\sigma(\alpha),\sigma(\alpha)\cdot\sigma(a)\sigma(a^{-1}))$ where
$\alpha\in\text{FM}(Y\cup Y^{-1})$ and $a\in Y\cup Y^{-1}$. We say that two
elements $\sigma(a_{1})\cdot\cdot\cdot\sigma(a_{n})$ and
$\sigma(b_{1})\cdot\cdot\cdot\sigma(b_{m})$ where $m,n\geq 0$ are Peiffer
equivalent in $\Upsilon$ if they fall in the same $\sim_{\mathfrak{U}}$-class.

From what we said before it is obvious that two $Y$-sequences $\alpha$ and
$\beta$ are Peiffer equivalent in the usual sense if and only if
$\sigma(\alpha)\sim_{\mathfrak{U}}\sigma(\beta)$. For this reason we decided
to make the following convention. If $\alpha=(a_{1},...,a_{n})$ is a
$Y$-sequence (resp. an identity $Y$-sequence), then its image in $\Upsilon$,
$\sigma(\alpha)$ will again be called a $Y$-sequence (resp. an identity
$Y$-sequence). In the future instead of working directly with an $Y$-sequence
$\alpha$, we will work with its image $\sigma(\alpha)$.

We note that it should be mentioned that the study of $\sim_{\mathfrak{U}}$
might be as hard as the study of Peiffer operations on $Y$-sequences, and at
this point it seems we have not made any progress at all. In fact this
definition will become useful later in this section and yet we have to prove a
few more things before we utilize it.

The process of inserting and deleting generators of $\mathfrak{U}$ in an
element of $\Upsilon$ is related to the following new concept. Given $U$ a
submonoid of a monoid $S$ and $d\in S$, then we say that $d$ belongs to the
weak dominion of $U$, shortly written as $d\in\text{WDom}_{S}(U)$, if for
every group $G$ and every monoid homomorphisms $f,g:S\rightarrow G$ such that
$f(u)=g(u)$ for every $u\in U$, then $f(d)=g(d)$. An analogue of the Stenström
version of Isbell’s theorem for weak dominion holds true. The proof of the if
part of its analogue is similar to that of Isbell theorem apart from some
minor differences that reflect the fact that we are working with $WDom$ rather
than $Dom$ and that will become clear along the proof, while the converse
relies on the universal property of $\mu:S\rightarrow\mathcal{G}(S)$.

######  Proposition 3.3.

Let $S$ be a monoid, $U$ a submonoid and let $\hat{U}$ be the subgroup of
$\mathcal{G}(S)$ generated by elements $\mu(u)$ with $u\in U$. Then
$d\in\text{WDom}_{S}(U)$ if and only if $\mu(d)\in\hat{U}$.

###### Proof.

The set $\hat{A}=\mathcal{G}(S)\otimes_{\hat{U}}\mathcal{G}(S)$ has an obvious
$(\mathcal{G}(S),\mathcal{G}(S))$-bisystem structure. The free abelian group
$\mathbb{Z}\hat{A}$ on $\hat{A}$ inherits a
$(\mathcal{G}(S),\mathcal{G}(S))$-bisystem structure if we define

$g\cdot\sum z_{i}(g_{i}\otimes_{\hat{U}}h_{i})=\sum
z_{i}(gg_{i}\otimes_{\hat{U}}h_{i})\text{ and }\left(\sum
z_{i}(g_{i}\otimes_{\hat{U}}h_{i})\right)\cdot g=\sum
z_{i}(g_{i}\otimes_{\hat{U}}h_{i}g).$

The set $\mathcal{G}(S)\times\mathbb{Z}\hat{A}$ becomes a group by defining

$(g,\sum z_{i}g_{i}\otimes_{\hat{U}}h_{i})\cdot(g^{\prime},\sum
z^{\prime}_{i}g^{\prime}_{i}\otimes_{\hat{U}}h^{\prime}_{i})=(gg^{\prime},\sum
z_{i}g_{i}\otimes_{\hat{U}}h_{i}g^{\prime}+\sum
z^{\prime}_{i}gg^{\prime}_{i}\otimes_{\hat{U}}h^{\prime}_{i}).$

The associativity is proved easily. The unit element is $(1,0)$ and for every
$(g,\sum z_{i}g_{i}\otimes_{\hat{U}}h_{i})$ its inverse is the element
$(g^{-1},-\sum z_{i}g^{-1}g_{i}\otimes_{\hat{U}}h_{i}g^{-1})$. Let us now
define

$\beta:S\rightarrow\mathcal{G}(S)\times\mathbb{Z}\hat{A}\text{ by
}s\mapsto(\mu(s),0),$

which is clearly a monoid homomorphism, and

$\gamma:S\rightarrow\mathcal{G}(S)\times\mathbb{Z}\hat{A}\text{ by
}s\mapsto(\mu(s),\mu(s)\otimes_{\hat{U}}1-1\otimes_{\hat{U}}\mu(s)),$

which is again seen to be a monoid homomorphism. These two coincide on $U$
since for every $u\in U$

$\gamma(u)=(\mu(u),\mu(u)\otimes_{\hat{U}}1-1\otimes_{\hat{U}}\mu(u))=(\mu(u),0)=\beta(u).$

The last equality and the assumption that $d\in\text{WDom}_{S}(U)$ imply that
$\beta(d)=\gamma(d)$, therefore

$(\mu(d),0)=(\mu(d),\mu(d)\otimes_{\hat{U}}1-1\otimes_{\hat{U}}\mu(d)),$

which shows that $\mu(d)\otimes_{\hat{U}}1=1\otimes_{\hat{U}}\mu(d)$ in the
tensor product $\mathcal{G}(S)\otimes_{\hat{U}}\mathcal{G}(S)$ and therefore
theorem 8.3.3, [27], applied for monoids $\mathcal{G}(S)$ and $\hat{U}$,
implies that $\mu(d)\in\text{Dom}_{\mathcal{G}(S)}(\hat{U})$. But
$\text{Dom}_{\mathcal{G}(S)}(\hat{U})=\hat{U}$ as from theorem 8.3.6, [27]
every inverse semigroup is absolutely closed, whence $\mu(d)\in\hat{U}$.

Conversely, suppose that $\mu(d)\in\hat{U}$ and we want to show that
$d\in\text{WDom}_{S}(U)$. Let $G$ be a group and $f,g:S\rightarrow G$ two
monoid homomorphisms that coincide in $U$, therefore the group homomorphisms
$\hat{f},\hat{g}:\mathcal{G}(S)\rightarrow G$ of the universal property of
$\mu$ coincide in $\hat{U}$ which, from our assumption, implies that
$\hat{f}(\mu(d))=\hat{g}(\mu(d))$, and then $f(d)=g(d)$ proving that
$d\in\text{WDom}_{S}(U)$. ∎

Given a presentation $\mathcal{P}=(\mathbf{x},\mathbf{r})$ for a group $G$, we
consider the following crossed module. If $\mathcal{G}(\Upsilon)$ is the
universal group associated with $\mathcal{P}$ and $\hat{F}$ is the free group
on $\mathbf{x}$, then we define

$\tilde{\theta}:\mathcal{G}(\Upsilon)\rightarrow\hat{F}\text{ by
}\mu\sigma(^{u}{r})^{\varepsilon}\mapsto ur^{\varepsilon}u^{-1}.$

An action of $\hat{F}$ on $\mathcal{G}(\Upsilon)$ is given by
${}^{v}(\mu\sigma(^{u}r)^{\varepsilon})=\mu\sigma(^{vu}r)^{\varepsilon}$ for
every $v\in\hat{F}$ and every generator $\mu\sigma((^{u}r)^{\varepsilon})$ of
$\mathcal{G}(\Upsilon)$. It is easy to check that the triple
$(\mathcal{G}(\Upsilon),\hat{F},\tilde{\theta})$ is a crossed module over
$\hat{F}$. The elements of $\text{Ker}(\tilde{\theta})$ are central, therefore
$\text{Ker}(\tilde{\theta})$ is an abelian subgroup of $\mathcal{G}(\Upsilon)$
on which $G$ acts on the left by the rule

${}^{g}(\mu\sigma(a_{1},...,a_{n})\iota\mu\sigma(b_{1},...,b_{m}))=\mu\sigma(^{w}{a_{1}},...,^{w}{a_{n}})\iota\mu\sigma(^{w}{b_{1}},...,^{w}{b_{m}}),$

where $w$ is a word in $\hat{F}$ representing $g$. With this action
$\text{Ker}(\tilde{\theta})$ becomes a left $G$-module which we call the
generalized module of identities for $\mathcal{P}$ and is denoted by
$\tilde{\Pi}$. Also we note that $\hat{\mathfrak{U}}$ is a sub $G$-module of
$\tilde{\Pi}$. The module of identities $\pi$ for $\mathcal{P}$ is obtained
from $\tilde{\Pi}$ by factoring out $\hat{\mathfrak{U}}$. In terms of
$\tilde{\Pi}$ and $\hat{\mathfrak{U}}$ we prove the following analogue of
theorem 3.1 of [38].

######  Theorem 3.4.

The following assertions are equivalent.

  * (i)

The presentation $\mathcal{P}=(\mathbf{x},\mathbf{r})$ is aspherical.

  * (ii)

For every identity $Y$-sequence $d$,
$d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$.

  * (iii)

$\tilde{\Pi}=\hat{\mathfrak{U}}$.

###### Proof.

$(i)\Rightarrow(ii)$ Let
$d=\sigma(a_{1})\cdot\cdot\cdot\sigma(a_{n})\in\Upsilon$ be any identity
$Y$-sequence and as such it has to be Peiffer equivalent to 1. We proceed by
showing that $d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$. Let $G$ be any group
and $f,g:\Upsilon\rightarrow G$ two monoid homomorphisms that coincide in
$\mathfrak{U}$ and we want to show that $f(d)=g(d)$. The proof will be done by
induction on the minimal number $h(d)$ of insertions and deletions needed to
transform $d=\sigma(a_{1})\cdot\cdot\cdot\sigma(a_{n})$ to $1$. If $h(d)=1$,
then $d\in\mathfrak{U}$ and $f(d)=g(d)$. Suppose that $h(d)=n>1$ and let
$\tau$ be the first operation performed on $d$ in a series of operations of
minimal length. After $\tau$ is performed on $d$, it is obtained an element
$d^{\prime}$ with $h(d^{\prime})=n-1$. By induction hypothesis,
$f(d^{\prime})=g(d^{\prime})$ and we want to prove that $f(d)=g(d)$. There are
two possible cases for $\tau$. First, $\tau$ is an insertion and let
$u=\sigma(a)\sigma(a^{-1})\in\mathfrak{U}$ be the element inserted. It follows
that $f(d^{\prime})=f(d)f(u)$ and $g(d^{\prime})=g(d)g(u)$, but $f(u)=g(u)$,
therefore from cancellation law in the group $G$ we get $f(d)=g(d)$. Second,
$\tau$ is a deletion and let $u=\sigma(a)\sigma(a^{-1})\in\mathfrak{U}$ be the
element deleted, that is $d=d^{\prime}u$. It follows immediately from the
assumptions that $f(d)=g(d)$ proving that
$d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$.

$(ii)\Rightarrow(iii)$ Let $\tilde{d}\in\tilde{\Pi}$. We may assume without
loss of generality that no $\iota(\mu\sigma(^{u}{r})^{\varepsilon})$ is
represented in $\tilde{d}$ for if there is any such occurrence, we can
multiply $\tilde{d}$ by
$\mu\sigma((^{u}{r})^{\varepsilon}(^{u}{r})^{-\varepsilon})$ to obtain in
return $\tilde{d}^{\prime}$ where $\iota(\mu\sigma(^{u}{r})^{\varepsilon})$ is
now replaced by $\mu\sigma((^{u}{r})^{-\varepsilon})$. It is obvious that if
$\tilde{d}^{\prime}\in\hat{\mathfrak{U}}$, then
$\tilde{d}\in\hat{\mathfrak{U}}$ and conversely. Let now $d$ be any preimage
of $\tilde{d}$ under $\mu$. It is clear that $d$ is an identity $Y$-sequence
and as such $d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$. Then proposition 3.3
implies that $\tilde{d}=\mu(d)\in\hat{\mathfrak{U}}$.

$(iii)\Rightarrow(i)$ Assume that $\tilde{\Pi}=\hat{\mathfrak{U}}$ and we want
to show that any identity $Y$-sequence $d$ is Peiffer equivalent to 1. From
the assumption for $d$ we have that $\mu(d)\in\hat{\mathfrak{U}}$ and then
proposition 3.3 implies that $d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$.
Consider the group $H/P$ as a quotient of $\mathcal{G}(\Upsilon)$ obtained by
identifying $\iota(\mu\sigma({{}^{u}{r}}))$ with $\mu\sigma((^{u}{r})^{-1})$
and let $\nu:\mathcal{G}(\Upsilon)\rightarrow H/P$ be the respective quotient
morphism. Writing $\tau$ for the zero morphism from $\Upsilon$ to $H/P$, we
see that $\tau$ and the composition $\nu\mu$ coincide in $\mathfrak{U}$,
therefore since $d\in\text{WDom}_{\Upsilon}(\mathfrak{U})$, it follows that
$\nu\mu(d)=1$ in $H/P$. The asphericity of $\mathcal{P}$ now follows from
theorem 2.7, p.71 of [17]. ∎

Before we prove our next result we recall the definition of the relation
module $\mathcal{N}(\mathcal{P})$. Given $\mathcal{P}=(\mathbf{x},\mathbf{r})$
a presentation for a group $G$, we let $\alpha:\hat{F}\rightarrow G$ and
$\beta:N\rightarrow N/[N,N]$ be the canonical homomorphisms where $N$ is the
normal closure of $\mathbf{r}$ in $\hat{F}$ and $[N,N]$ its commutator
subgroup. There is a well defined $G$-action on
$\mathcal{N}(\mathcal{P})=N/[N,N]$ given by

$w^{\alpha}\cdot s^{\beta}=(w^{-1}sw)^{\beta}$

for every $w\in\hat{F}$ and $s\in N$. This action extends to an action of
$\mathbb{Z}G$ over $\mathcal{N}(\mathcal{P})$ by setting

$(w_{1}^{\alpha}\pm w_{2}^{\alpha})\cdot
s^{\beta}=(w_{1}^{-1}sw_{1}w_{2}^{-1}s^{\pm 1}w_{2})^{\beta}.$

When $\mathcal{P}$ is aspherical, the basis of $\mathcal{N}(\mathcal{P})$ as a
free $\mathbb{Z}G$ module is the set of elements $r^{\beta}$ with
$r\in\mathbf{r}$.

######  Proposition 3.5.

If $\mathcal{P}$ is aspherical, then $\hat{\mathfrak{U}}$ is a free $G$-module
with bases equipotent to the set $\mathbf{r}$.

###### Proof.

The result follows if we show that
$\hat{\mathfrak{U}}\cong\mathcal{N}(\mathcal{P})$ as $G$-modules. For this we
define

$\Omega:\mathcal{N}(\mathcal{P})\rightarrow\hat{\mathfrak{U}}$

on free generators by $r^{\beta}\mapsto\mu\sigma(rr^{-1})$ which is clearly
well defined and a surjective morphism of $G$-modules. Now we prove that
$\Omega$ is injective. Let

$\xi=\sum_{i=1}^{n}u_{i}^{\alpha}\cdot
r_{i}^{\beta}-\sum_{j=n+1}^{m}v_{j}^{\alpha}\cdot
r_{j}^{\beta}\in\text{Ker}(\Omega),$

which means that

$\prod_{i=1}^{n}\mu\sigma(^{u_{i}}r_{i}(^{u_{i}}r_{i})^{-1})\iota\left(\prod_{j=n+1}^{m}\mu\sigma(^{v_{j}}r_{j}(^{v_{j}}r_{j})^{-1})\right)=1.$
(1)

To prove that $\xi=0$ we will proceed as follows. Define

$\gamma:FM(Y\cup Y^{-1})\rightarrow\mathcal{N}(\mathcal{P})$

on free generators as follows

$(^{u}r)^{\varepsilon}\mapsto u^{\alpha}\cdot r^{\beta}.$

It is easy to see that $\gamma$ is compatible with the defining relations of
$\Upsilon$, hence there is $g:\Upsilon\rightarrow\mathcal{N}(\mathcal{P})$ and
then the universal property of $\mu$ implies the existence of
$\hat{g}:\mathcal{G}(\Upsilon)\rightarrow\mathcal{N}(\mathcal{P})$ such that
$\hat{g}\mu=g$. If we apply now $\hat{g}$ on both sides of (1) obtain

$2\cdot\sum_{i=1}^{n}u_{i}^{\alpha}\cdot
r_{i}^{\beta}-2\cdot\sum_{j=n+1}^{m}v_{j}^{\alpha}\cdot r_{j}^{\beta}=0,$

proving that $\xi=0$. ∎

##  4 Proof of the main theorem

The proof of our main theorem is heavily based on two papers. The first one is
[36] where McGlashan et al extended the Squier complex of a monoid
presentation to a 3-complex and obtained a short exact sequence involving data
from this complex. This sequence will be crucial in the proof of our theorem.
The second one is [40] where Pride realizes the second homotopy group
associated with a group presentation as the first homotopy group of a certain
extension of the Squire complex arising from that presentation. For the sake
of completeness we have added below a number of sections which tend to explain
the material that is used in our proofs. Section 4.1 gives some basic material
about rewriting systems since they are used in the construction of our
complexes and in our proofs. In Section 4.2 we explain in some details how the
Squier complex of a monoid presentation is defined and the cellular chain
complex associated with it. Further in section 4.3 we give the definition of
the extended Squier complex as it appears in [36] and some of the homological
consequences that will be used in our proofs. Section 4.4 shows how the 0 and
the 1-skeleton of the Squier complex is well ordered, and in the case when the
rewriting system is complete, it shows how these well orders induce another
well order in the set of all 2-cells of the extended 3-complex. This new well
order will be used further in section 4.6. Section 4.5 is about the Knuth-
Bendix completion procedure since it is used to give a new and shorter proof
of the key result of [36] regarding the short exact sequence we mentioned
above. This proof is given in section 4.6. Section 4.7 is devoted to
introducing the Pride complex associated with a group presentation and to
explain ideas and results from [40] since we make extensive use of them in our
proofs.

Finally, it is important to mention that theorem 6.6 of [33] is vital in the
proof of key lemma 4.14.

###  4.1 Some basic concepts from rewriting systems

A rewriting system is a pair $\mathcal{P}=(\mathbf{x},\mathbf{r})$ where
$\mathbf{x}$ is a non empty set and $\mathbf{r}$ is a set of rules
$r=(r_{+1},r_{-1})\in F\times F$ where $F$ is the free monoid on $\mathbf{x}$.
Related with $\mathbf{r}$ there is the so called the one single step reduction
of words

$\rightarrow_{\mathbf{r}}=\\{(ur_{+1}v,ur_{-1}v)|r\in\mathbf{r}\text{ and
}u,v\in F\\}.$

The reflexive and transitive closure of $\rightarrow_{\mathbf{r}}$ is denoted
by $\rightarrow_{\mathbf{r}}^{\ast}$, and the reflexive, transitive and
symmetric closure is denoted by $\leftrightarrow_{\mathbf{r}}^{\ast}$ and is
also known as the Thue congruence generated by $\mathbf{r}$. The quotient
$F/\leftrightarrow_{\mathbf{r}}^{\ast}$ forms a monoid $S$ whose elements are
the congruence classes $\bar{u}$ of words $u\in F$, and the multiplication is
given by $\bar{u}\cdot\bar{v}=\overline{uv}$. We say that the monoid $S$ is
given by $\mathcal{P}$, or that $\mathcal{P}$ is a presentation for $S$.

A rewriting system $\mathcal{P}=(\mathbf{x},\mathbf{r})$ is noetherian if
there is no infinite chain

$w\rightarrow_{\mathbf{r}}w^{\prime}\rightarrow_{\mathbf{r}}\dots$

and is confluent if whenever we have $w\rightarrow_{\mathbf{r}}^{\ast}w_{1}$
and $w\rightarrow_{\mathbf{r}}^{\ast}w_{2}$, then there is $z\in F$ such that
$w_{1}\rightarrow_{\mathbf{r}}^{\ast}z$ and
$w_{2}\rightarrow_{\mathbf{r}}^{\ast}z$. A rewriting system
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ is complete if it is both noetherin and
confluent.

Let $\mathcal{P}=(\mathbf{x},\mathbf{r})$ be a presentation for a monoid $S$.
The natural epimorphism

$F\rightarrow S\text{ such that }w\mapsto\bar{w},$

where $F$ is the free monoid on $\mathbf{x}$, extends linearly to a ring
epimorphism

$\mathbb{Z}F\rightarrow\mathbb{Z}S$

of the corresponding integral monoid rings. The kernel of this epimorphism is
denoted by $J$ which as an abelian group is generated by all

$u(r_{+1}-r_{-1})v\text{ where }u,v\in F\text{ and }r\in\mathbf{r}.$

As a $(\mathbb{Z}F,\mathbb{Z}F)$-bimodule $J$ is generated by all
$r_{+1}-r_{-1}$.

###  4.2 The Squier complex of a monoid presentation

The material included in this section is taken from [36] (see also [35]). At
the end of the section we give shortly the respective terminology used in [33]
which differs slightly from ours. The reason we explain this terminology is
the use of theorem 6.6 of [33] in the proof of our key lemma 4.14.

For every rewriting system $\mathcal{P}=(\mathbf{x},\mathbf{r})$ we can define
its graph of derivations $\Gamma(\mathcal{P})$ whose vertices are the elements
of $F$, and the edges are all quadruples

$e=(w,r,\varepsilon,w^{\prime})\text{ where }w,w^{\prime}\in F,\varepsilon=\pm
1,r\in\mathbf{r},$

with initial, terminal and inverse functions

$\iota e=wr_{\varepsilon}w^{\prime},\tau e=wr_{-\varepsilon}w^{\prime}\text{
and }e^{-1}=(w,r,-\varepsilon,w^{\prime}).$

The edge $e$ is called positive if $\varepsilon=1$. We can think of
$\Gamma(\mathcal{P})$ as a one dimensional cw-complex with 0-cells all the
elements of $F$ and with 1-cells all positive edges. We note here that
$e^{-1}=(w,r,-1,w^{\prime})$ is not a new edge attached to the complex, but is
defined to mean the topological inverse of the attaching map of
$e=(w,r,1,w^{\prime})$. A path $p$ of length $n$ in $\Gamma(\mathcal{P})$ is a
sequence of edges $p=e_{1}\dots e_{i}e_{i+1}\dots e_{n}$ where $\tau
e_{i}=\iota e_{i+1}$ for $1\leq i\leq n-1$. It is called positive if the edges
are positive, and is called closed if $\iota e_{1}=\tau e_{n}$.

There is a natural two-sided action of $F$ on $\Gamma(\mathcal{P})$. The
action on vertices is given by the multiplication of $F$, and the action of
$z,z^{\prime}\in F$ on edges $e=(w,r,\varepsilon,w^{\prime})$ is given by

$z.e.z^{\prime}=(zw,r,\varepsilon,w^{\prime}z^{\prime}),$

and sometimes is called translation. This action extends to paths in the
obvious way.

Note that there is a 1-1 correspondence between the elements of $S$ given by
$\mathcal{P}$ and the connected components of $\Gamma(\mathcal{P})$ since
$u\leftrightarrow_{\mathbf{r}}^{\ast}v$ if and only if there is a path in
$\Gamma(\mathcal{P})$ connecting $u$ with $v$. Also note that the generators
of $J$ as an abelian group are the elements $\iota e-\tau e$ where $e$ is a
positive edge.

We say that two positive edges $e_{1}$ and $e_{2}$ are disjoint if they can be
written in the form

$e_{1}=f_{1}.\iota f_{2},f_{2}=\iota f_{1}f_{2}$

where $f_{1},f_{2}$ are positive edges. We say that an edge $e$ is left
reduced (resp. right reduced) if it cannot be written in the form $u.f$ (resp.
$f.u$) for some non empty word $u\in F$ and an edge $f$. A pair of positive
edges with the same initial forms a critical pair it either

  * (1)

One of the pair is both left and right reduced (a critical pair of inclusion
type), or

  * (2)

One of the pair is left reduced but not right reduced, and the other is right
reduced but not left reduced (a critical pair of overlapping type).

We say that a critical pair $(e_{1},e_{2})$ is resolvable if there are
positive paths (a resolution of the critical pair) from $\tau e_{1}$ and $\tau
e_{2}$ to a common vertex. It is well known [37] that, when the system
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ is noetherian and if all the critical
pairs are resolvable, then the system is confluent.

The Squier complex $\mathcal{D}(\mathcal{P})$ associated with $\mathcal{P}$ is
a combinatorial 2-complex with 1-skeleton $\Gamma(\mathcal{P})$, to which, for
each pair of positive edges $e,f$ a 2-cell $[e,f]$ is attached along the
closed path

$\partial[e,f]=(e.\iota f)(\tau e.f)(e.\tau f)^{-1}(\iota e.f)^{-1}.$

Sometimes we refer the 2-cell $[e,f]$ as a square 2-cell. The two-sided action
of $F$ on $\Gamma(\mathcal{P})$ extends to the 2-cells by

$w.[e,f].w^{\prime}=[w.e,f.w^{\prime}]\text{ where }w,w^{\prime}\in
F,e,f\text{ are positive edges}.$

We have the chain complex

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.8173pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{C}(\mathcal{D}):C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
29.95915pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\partial_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 47.8173pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.8173pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.62163pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\partial_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 88.47978pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
88.47978pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
129.14226pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
129.14226pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $C_{0}$, $C_{1}$, $C_{2}$ and $C_{3}$ are the free abelian groups
generated by all 0-cells, positive edges, 2-cells, and 3-cells respectively.
The boundary maps are given by

$\partial_{1}e=\iota e-\tau e\text{ where }e\text{ is a positive edge},$
$\partial_{2}[e,f]=e.(\iota f-\tau f)-(\iota e-\tau e).f\text{ where
}e,f\text{ are positive edges}.$

In the paper [33] of Otto and Kobayashi, a monoid presentation is denoted by
$(\Sigma,R)$ and the rewriting rules of $R$ are denoted by $r\rightarrow\ell$.
The edges of the graph of derivations in [33] are denoted by $(x,u,v,y)$ where
$x,y\in\Sigma^{\ast}$ and $(u\rightarrow v)\in E=R\cup R^{-1}$. In [33] it is
considered the set of closed paths

$D=\\{e_{1}xu_{2}\circ v_{1}xe_{2}\circ e_{1}^{-1}xv_{2}\circ
u_{1}xe_{2}^{-1}|e_{1}=(u_{1},v_{1})\in R,e_{2}=(u_{2},v_{2})\in
R,x\in\Sigma^{\ast}\\}.$

It is important to observe that each circuit of $D$ is in fact the boundary of
a square 2-cell as the following shows

$e_{1}xu_{2}\circ v_{1}xe_{2}\circ e_{1}^{-1}xv_{2}\circ
u_{1}xe_{2}^{-1}=\partial[(1,r_{1},1,1),(x,r_{2},1,1)],$

where $r_{1}=e_{1}$ and $r_{2}=e_{2}$. The free $\mathbb{Z}\Sigma^{\ast}$ bi-
module $\mathbb{Z}\Sigma^{\ast}\cdot R\cdot\mathbb{Z}\Sigma^{\ast}$ considered
in [33] is the abelian group $C_{1}$ of our complex $\mathbf{C}(\mathcal{D})$,
and the maps $\partial_{1}$ are the same in both papers. On the other hand,
the free $\mathbb{Z}\Sigma^{\ast}$ bi-module $\mathbb{Z}\Sigma^{\ast}\cdot
D\cdot\mathbb{Z}\Sigma^{\ast}$ of [33] is the abelian group $C_{2}$ of
$\mathbf{C}(\mathcal{D})$, and the maps $\partial_{2}$ are the same in both
papers. Finally, the exact sequence of theorem 6.6 of [33] in our notations
will be

$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{J.R.\mathbb{Z}\Sigma^{\ast}+\mathbb{Z}\Sigma^{\ast}.R.J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{J^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The interpretation of the exactness in the middle of the above sequence is
that
$Ker\partial_{1}\cap(J.R.\mathbb{Z}\Sigma^{\ast}+\mathbb{Z}\Sigma^{\ast}.R.J)=Im\partial_{2}$.

###  4.3 The extended Squier complex

Assume now that $\mathbf{p}$ is a set closed paths in
$\mathcal{D}(\mathcal{P})$. In [36] the complex $\mathcal{D}(\mathcal{P})$ has
been extended to a 3-complex $(\mathcal{D},\mathbf{p})$ in the following way.
We add to $\mathcal{D}(\mathcal{P})$ additional 2-cells $[u,p,v]$ attached
along the closed path

$\partial[u,p,v]=u.p.v\text{ where }u,v\in F,\text{ and }p\in\mathbf{p}.$

The construction is then completed by adding 3-cells as follows. For each
positive edge $f$ and each 2-cell $\sigma$ with
$\partial\sigma=e_{1}^{\varepsilon_{1}}\dots e_{n}^{\varepsilon_{n}}$, 3-cells
$[f,\sigma]$ and $[\sigma,f]$ are attached to the 2-skeleton by mapping their
boundaries to respectively:

  * (1)

the 2-cells $\iota f.\sigma$, $\tau f.\sigma$ together with 2-cells
$[f,e_{i}]$ for $1\leq i\leq n$,

  * (2)

the 2-cells $\sigma.\iota f$, $\sigma.\tau f$ together with 2-cells
$[e_{i},f]$ for $1\leq i\leq n$.

The 2-sided action of $F$ on the 2-skeleton extends naturally to the 3-cells.
For $[f,\sigma]$, $[\sigma,f]$ and $u,v\in F$,

$u.[f,\sigma].v=[u.f,\sigma.v]\text{ and }u.[\sigma,f].v=[u.\sigma,f.v].$

The complex $\mathbf{C}(\mathcal{D})$ now extends to

$\textstyle{\mathbf{C}(\mathcal{D},\mathbf{p}):0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}^{\mathbf{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}_{3}}$$\textstyle{C_{2}^{\mathbf{p}}\oplus
C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}_{2}}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $C_{3}^{\mathbf{p}}$ is the free abelian group generated by the set of
all 3-cells, and $C_{2}^{\mathbf{p}}$ is the free abelian group generated by
the set of all newly added 2-cells $\sigma=[u,p,v]$. The boundary map
$\tilde{\partial}_{2}$ restricted to $C_{2}$ is $\partial_{2}$, and for every
$[u,p,v]$ where $p\in\mathbf{p}$ with $\partial p=f_{1}^{\delta_{1}}\dots
f_{n}^{\delta_{n}}$, it is defined

$\tilde{\partial}_{2}[u,p,v]=\sum_{i=1}^{n}\delta_{i}u.f_{i}.v.$

Finally, the definition of $\tilde{\partial}_{3}$ is done in the following
way. For every positive edge $f$ and every 2-cell $\sigma$ with
$\tilde{\partial}_{2}\sigma=\sum_{i=1}^{n}\varepsilon_{i}e_{i}$ we have

$\tilde{\partial}_{3}[f,\sigma]=(\iota f-\tau
f).\sigma+\sum_{i=1}^{n}\varepsilon_{i}[f,e_{i}],$ (2)

and

$\tilde{\partial}_{3}[\sigma,f]=\sigma.(\iota f-\tau
f)-\sum_{i=1}^{n}\varepsilon_{i}[e_{i},f].$ (3)

The definition of the 2-cells $[u,p,v]$ where $u,v\in F,p\in\mathbf{p}$
suggests that $C_{2}^{\mathbf{p}}$ can be regarded as a free
$(\mathbb{Z}F,\mathbb{Z}F)$-bimodule with basis

$\hat{\mathbf{p}}=\\{[1,p,1]|p\in\mathbf{p}\\}.$

This enables us to define a $(\mathbb{Z}F,\mathbb{Z}F)$-homomorphism

$\varphi:C_{2}\oplus
C_{2}^{\mathbf{p}}\rightarrow\mathbb{Z}S\mathbf{p}\mathbb{Z}S$

by mapping $C_{2}$ to 0, and every 2-cell $[u,p,v]$ to $\bar{u}.p.\bar{v}$.
The kernel of $\varphi$ is denoted by $K^{\mathbf{p}}$.

It is shown in [36] that

$K^{\mathbf{p}}=C_{2}+J.\hat{\mathbf{p}}.\mathbb{Z}F+\mathbb{Z}F.\hat{\mathbf{p}}.J.$

Also it is shown that $B_{2}(\mathcal{D},\mathbf{p})\subseteq K^{\mathbf{p}}$
and that the restriction of $\tilde{\partial}_{2}$ on $K^{\mathbf{p}}$ sends
$K^{\mathbf{p}}$ onto $B_{1}(\mathcal{D})$, therefore we have the complex

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl.}$$\textstyle{K^{\mathbf{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}_{2}}$$\textstyle{B_{1}(\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(4)

It is proved in Proposition 14 of [36] that when $\mathbf{p}$ is a homology
trivializer, then the sequence (4) is exact. We will give a new proof in
section 4.6 for the exactness of (4). Since the proof uses the so called
Knuth-Bendix completion procedure, we will explain this procedure in some
details in section 4.5. Before doing that we will introduce in the next
section some useful orders in the skeleta of $\mathcal{D}(\mathcal{P})$.

###  4.4 Ordering the Squier complex

As before $\mathcal{P}=(\mathbf{x},\mathbf{r})$ is a rewriting system and
$\mathcal{D}(\mathcal{P})$ its Squier complex. Assume in addition that for
every $(r_{+1},r_{-1})\in\mathbf{r}$, $r_{+1}\neq r_{-1}$. Let
$\vartriangleright$ be a well ordering on $\mathbf{x}$. The corresponding
length-lexicographical ordering on $F$ is defined as follows. For $u,v\in F$,
we write $u>_{llex}v$ if and only if $|u|>|v|$, or $|u|=|v|$, $u=au^{\prime}$,
$v=bv^{\prime}$ where $a,b\in\mathbf{x}$, $u^{\prime},v^{\prime}\in F$, and
one of the following holds:

  * (i)

$a\vartriangleright b$,

  * (ii)

$a=b$, $u^{\prime}>_{llex}v^{\prime}$.

It turns out that $>_{llex}$ is a well ordering on $F$ (see [3]). We can
always assume that $>_{llex}$ is compatible with $\mathbf{r}$ in the sense
that $r_{+1}>_{llex}r_{-1}$, for if there are rules $(r_{+1},r_{-1})$
satisfying the opposite, we can exchange $r_{+1}$ with $r_{-1}$. Well
orderings in $F$ that are compatible with $\mathbf{r}$ are usually called
reduction well ordering and are the basis to start the Knuth Bendix completion
procedure.

So far we have defined a reduction well order on the 0-skeleton of
$\mathcal{D}(\mathcal{P})$ which will be denoted by $\prec_{0}$. This order
induces a noetherian (well founded) partial order in the 1-skeleton of
$\mathcal{D}(\mathcal{P})$ in the following way. For $e=(u,r,+1,v)$ and
$f=(u^{\prime},r^{\prime},+1,v^{\prime})$ positive edges in by
$\mathcal{D}(\mathcal{P})$, we define $e\prec_{1}f$ if and only if $\iota
e=\iota f$, and one of the following occurs:

  * (i)

$v^{\prime}$ is a proper suffix of $v$, or

  * (ii)

$v=v^{\prime}$ and $|r_{+1}|<|r^{\prime}_{+1}|$, or

  * (iii)

$v=v^{\prime}$, $r_{+1}=r^{\prime}_{+1}$ and $r_{-1}\prec_{0}r^{\prime}_{-1}$.

It turns out that $\prec_{1}$ is a partial order and that it is well founded.

Further, assume that $\mathcal{P}=(\mathbf{x},\mathbf{r})$ is confluent, so
that all the critical pairs of positive edges resolve. In that case, we attach
to $\mathcal{D}(\mathcal{P})$ 2-cells $\mathbf{p}$ by choosing resolutions for
every critical pair of positive edges $(e,f)$ in the following way. If
$p_{e}$, $p_{f}$ are positive paths from $\tau e$ and $\tau f$ respectively to
a common vertex, then the boundary of the 2-cell $\sigma$ corresponding to
$(e,f)$ is

$\partial\sigma=ep_{e}p_{f}^{-1}f^{-1}.$

Also we attach 2-cells $u.\sigma.v$ for every $u,v\in F$ along the loop
$u.\partial\sigma.v$. As it is explained in the section 4.3, this new
2-complex extends to a 3-complex denoted there by $(\mathcal{D},\mathbf{p})$.
It is important to mention that every 2-cell of $(\mathcal{D},\mathbf{p})$,
including the square 2-cells, is uniquely determined by the pair $(e,f)$ of
edges meeting its maximal vertex $w=\iota_{e}=\iota_{f}$ (according to
$\prec_{0}$). For this reason, we write the 2-cell as $[w;(e,f)]$. Now we
extend the orders $\prec_{0}$ and $\prec_{1}$ to the 2-skeleton of the
3-complex $(\mathcal{D},\mathbf{p})$ as follows. For every two 2-cells
$[w;(e,f)]$ and $[w^{\prime};(e^{\prime},f^{\prime})]$ we say that
$[w;(e,f)]\prec_{2}[w^{\prime};(e^{\prime},f^{\prime})]$ if and only if:

  * (i)

$w\prec_{0}w^{\prime}$; or

  * (ii)

$w=w^{\prime}$ and $f\prec_{1}f^{\prime}$; or

  * (iii)

$f=f^{\prime}$ and $e\prec_{1}e^{\prime}$.

This is a well founded total order in the set of all 2-cells of
$(\mathcal{D},\mathbf{p})$.

Under the current assumptions, similarly to 2-cells, every 3-cell is uniquely
determined by three positive edges $e_{1}\prec_{1}e_{2}\prec_{1}e_{3}$ with
initial the maximal vertex $w$ of the 3-cell, where either $e_{1}$ is disjoint
from $e_{2}$ and $e_{3}$, or $e_{3}$ is disjoint from $e_{1}$ and $e_{2}$. For
this reason we write the 3-cell by $[w;(e_{1},e_{2},e_{3})]$. By (2) and (3)
we see that

$\tilde{\partial}_{3}[w;(e_{1},e_{2},e_{3})]=[w;(e_{2},e_{3})]-[w;(e_{1},e_{3})]+[w;(e_{1},e_{2})]+\varsigma$
(5)

where $\varsigma$ is a 2-chain made up of 2-cells all of which have maximal
vertices less than $w$. Also note that the maximal 2-cell represented in
$\tilde{\partial}_{3}[w;(e_{1},e_{2},e_{3})]$ is $[w;(e_{2},e_{3})]$.

###  4.5 The Knuth-Bendix completion procedure

The Knuth-Bendix procedure [3], produces a complete system out of any given
system and equivalent to it. Given a rewriting system
$\mathcal{P}=(\mathbf{x},\mathbf{r})$ and a reduction well order $\succ$ on
$F$ that is compatible with $\mathbf{r}$ (there is always such one as
explained in section 4.4), one can produce a complete rewriting system
$\mathcal{P}^{\infty}$ that is equivalent to $\mathcal{P}$ in the following
way. Put $\mathbf{r}_{0}=\mathbf{r}$. For each non-resolvable pair of edges
$(e,f)$ in $\mathcal{D}(\mathcal{P})$ we chose positive path $p_{e}$, $p_{f}$
from $\tau e$ and $\tau f$ respectively to distinct irreducibles. Let
$\mathbf{r}_{1}$ be the set of rules obtained from $\mathbf{r}$ by adding for
each such critical pair $(e,f)$ the rule $(\tau p_{e},\tau p_{f})$ if $\tau
p_{e}\succ\tau p_{f}$, otherwise adding the rule $\tau p_{f}\succ\tau p_{e}$.
It is clear that $\mathcal{P}_{1}=(\mathbf{x},\mathbf{r}_{1})$ is equivalent
to $\mathcal{P}=(\mathbf{x},\mathbf{r})$ and that
$\mathbf{r}\subseteq\mathbf{r}_{1}$ where the inclusion is strict if
$\mathcal{P}$ is not complete. Assume by induction that we have defined a
sequence of equivalent rewriting systems

$\mathcal{P}=(\mathbf{x},\mathbf{r}_{0}),...,\mathcal{P}_{n-1}=(\mathbf{x},\mathbf{r}_{n-1}),\mathcal{P}_{n}=(\mathbf{x},\mathbf{r}_{n}),$

and consequently, an increasing sequence of complexes

$\mathcal{D}(\mathcal{P})\subseteq\dots\subseteq\mathcal{D}(\mathcal{P}_{n-1})\subseteq\mathcal{D}(\mathcal{P}_{n}),$

where $\mathcal{P}_{n}=(\mathbf{x},\mathbf{r}_{n})$ is obtained from
$\mathcal{P}_{n-1}=(\mathbf{x},\mathbf{r}_{n-1})$ by resolving all the non-
resolvable critical pairs of $\mathcal{D}(\mathcal{P}_{n-1})$. Put
$\mathbf{r}_{\infty}=\underset{n\geq 0}{\cup}\mathbf{r}_{n}$ and let
$\mathcal{P}_{\infty}=(\mathbf{x},\mathbf{r}_{\infty})$ be the resulting
rewriting system. The corresponding complex
$\mathcal{D}(\mathcal{P}_{\infty})$ will be latter denoted by
$\mathcal{D}^{\infty}$. The rewriting system
$\mathcal{P}_{\infty}=(\mathbf{x},\mathbf{r}_{\infty})$ is obviously
equivalent to $\mathcal{P}$ and it is complete since it is compatible with the
order $\succ$ on $F$ and for every non-resolvable pair $(e,f)$ of edges found
in some $\mathcal{D}(\mathcal{P}_{n})$, there is an edge $g$ in
$\mathcal{D}(\mathcal{P}_{n+1})$ connecting the endpoints of the positive
paths $p_{e}$ and $p_{f}$ of $\mathcal{D}(\mathcal{P}_{n})$.

###  4.6 A shorter proof for the exactness of (4)

The proof that is provided below is valid in the special case when each 2-cell
from $\mathbf{p}$ arises from the resolution of a critical pair. The proof
goes through the following stages. The first stage is the same as that of [36]
and for this reason is not presented here in full. In this stage it is proved
that (4) is exact in the special case when the monoid presentation
$\mathcal{M}=\langle\mathbf{x},\mathbf{r}\rangle$ from which $\mathcal{D}$ is
defined, is complete, and the set $\mathbf{p}$ of homology trivializers is
obtained by choosing resolutions of critical pairs of $\mathbf{r}$. The proof
is roughly as follows. Using (5), it is shown that every 2-cycle $\xi\in
K^{\mathbf{p}}$ is homologous to a 2-cycle $\bar{\xi}\in K^{\mathbf{p}}$ that
is obtained from $\xi$ by replacing the maximal 2-cell $\sigma$ represented in
$\xi$ by a 2-chain made up of lesser 2-cells than $\sigma$. Then we proceed by
Noetherian induction.

In the second stage, differently from the general case that is considered in
[36], we assume that we have a monoid presentation
$\mathcal{M}=\langle\mathbf{x},\mathbf{r}\rangle$ (not necessarily complete)
and that $H_{1}(\mathcal{D})$ of the corresponding Squier complex
$\mathcal{D}$ is trivialized by adding 2-cells $\mathbf{p}$ arising from the
resolution of certain critical pairs. Also, the same as in [36], we assume
that $\mathbf{r}$ is compatible with a length-lexicographic order in the free
monoid $F$ on $\mathbf{x}$. Using the Knuth-Bendix procedure, we obtain a new
presentation
$\mathcal{M}^{\infty}=\langle\mathbf{x},\mathbf{r}^{\infty}\rangle$ with
$\mathbf{r}\subseteq\mathbf{r}^{\infty}$ and where $\mathbf{r}^{\infty}$ is
compatible with the order on $F$. The Squier complex $\mathcal{D}^{\infty}$
has trivializer $\mathbf{p}^{\infty}$ obtained by choosing resolution of all
critical pairs of $\mathbf{r}^{\infty}$ and as a consequence
$\mathbf{p}\subseteq\mathbf{p}^{\infty}$. From the special case of the first
stage, we have the exactness of

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl.}$$\textstyle{K^{\mathbf{p}^{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}^{\infty}_{2}}$$\textstyle{B_{1}(\mathcal{D}^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where
$K^{\mathbf{p}^{\infty}}=C_{2}(\mathcal{D}^{\infty})+J.\mathbf{p}^{\infty}.\mathbb{Z}F+\mathbb{Z}F.\mathbf{p}^{\infty}.J$.
We will use this and the fact that $\mathbf{p}\subseteq\mathbf{p}^{\infty}$ to
prove in a shorter way the exactness of (4).

We begin by pointing out that $(\mathcal{D},\mathbf{p})$ is a subcomplex of
$(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$, therefore for $i=1,2,3$, we have
that $C_{i}(\mathcal{D},\mathbf{p})\leq
C_{i}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$. We will define for
$i=1,2,3$, retractions
$\hat{\rho_{i}}:C_{i}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\rightarrow
C_{i}(\mathcal{D},\mathbf{p})$.

First, for every positive edge $e$ from
$(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ not belonging to
$(\mathcal{D},\mathbf{p})$, we chose a path
$\rho(e)=e_{1}^{\varepsilon_{1}}\dots e_{n}^{\varepsilon_{n}}$ in
$(\mathcal{D},\mathbf{p})$ connecting $\iota e$ with $\tau e$ where every
$\varepsilon_{i}=\pm 1$. Relative to this choice we define

$\hat{\rho_{1}}:C_{1}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\rightarrow
C_{1}(\mathcal{D},\mathbf{p})$

by

$e\mapsto\sum_{i}\varepsilon_{i}e_{i},$

whenever $e$ is from $(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ not
belonging to $(\mathcal{D},\mathbf{p})$, and for positive edges $e$ from
$(\mathcal{D},\mathbf{p})$ we define

$\hat{\rho_{1}}(e)=e.$

Thus $\hat{\rho_{1}}$ is a retraction. Before we define a second retraction
$\hat{\rho_{2}}:C_{2}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\rightarrow
C_{2}(\mathcal{D},\mathbf{p})$, we prove the following.

######  Lemma 4.1.

For every path $\rho=f_{1}^{\beta_{1}}\dots f_{n}^{\beta_{n}}$ in
$(\mathcal{D},\mathbf{p})$ where every $\beta_{j}=\pm 1$, we have that

$\partial_{1}(\beta_{1}f_{1}+\dots+\beta_{n}f_{n})=\iota(\rho)-\tau(\rho).$

###### Proof.

The proof will be done by induction on $n$. For $n=1$,

$\partial_{1}(\beta_{1}f_{1})=\beta_{1}(\iota f_{1}-\tau f_{1}),$

therefore, depending on the sign of $\beta_{1}$, we have that
$\partial_{1}(\beta_{1}f_{1})=\iota(f_{1}^{\beta_{1}})-\tau(f_{1}^{\beta_{1}})$.
For the inductive step, we write

$\rho=f_{1}^{\beta_{1}}\dots f_{n}^{\beta_{n}}\cdot
f_{n+1}^{\beta_{n+1}}=\rho_{1}\cdot f_{n+1}^{\beta_{n+1}}.$

From the assumption for $\rho_{1}$ we have

$\partial_{1}(\beta_{1}f_{1}+\dots+\beta_{n}f_{n})=\iota(\rho_{1})-\tau(\rho_{1})=\iota(\rho)-\iota(f_{n+1}^{\beta_{n+1}}),$

and then

$\displaystyle\partial_{1}(\beta_{1}f_{1}+\dots+\beta_{n}f_{n}+\beta_{n+1}f_{n+1})$
$\displaystyle=\partial_{1}(\beta_{1}f_{1}+\dots+\beta_{n}f_{n})+\partial_{1}(\beta_{n+1}f_{n+1})$
$\displaystyle=\iota(\rho)-\iota(f_{n+1}^{\beta_{n+1}})+(\iota(f_{n+1}^{\beta_{n+1}})-\tau(f_{n+1}^{\beta_{n+1}}))$
$\displaystyle=\iota(\rho)-\tau(f_{n+1}^{\beta_{n+1}})$
$\displaystyle=\iota(\rho)-\tau(\rho).$

∎

Now we define $\hat{\rho_{2}}$ in the following way. If
$z=\sum_{j}\delta_{j}f_{j}\in Z_{1}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$
is a 1-cycle where at least one of $f_{j}$ is from
$(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ not belonging to
$(\mathcal{D},\mathbf{p})$, we have the 1-chain

$\hat{\rho_{1}}\left(\sum_{j}\delta_{j}f_{j}\right)$

in $C_{1}(\mathcal{D},\mathbf{p})$. Let us show that
$\hat{\rho_{1}}\left(\sum_{j}\delta_{j}f_{j}\right)$ is in fact a 1-cycle in
$Z_{1}(\mathcal{D},\mathbf{p})$. Indeed,

$\displaystyle\partial_{1}\hat{\rho_{1}}\left(\sum_{j}\delta_{j}f_{j}\right)$
$\displaystyle=\sum_{j}\delta_{j}\partial_{1}\hat{\rho_{1}}(f_{j})$
$\displaystyle=\sum_{j}\delta_{j}(\iota f_{j}-\tau f_{j})$ (by lemma 4.1)
$\displaystyle=\partial^{\infty}_{1}\left(\sum_{j}\delta_{j}f_{j}\right)$
$\displaystyle=\partial^{\infty}_{1}(z)$ $\displaystyle=0.$

Since $\mathbf{p}$ is a homology trivializer, then for the 1-cycle
$\hat{\rho_{1}}\left(\sum_{j}\delta_{j}f_{j}\right)$ there is a 2-chain
$\varsigma_{z}\in C_{2}(\mathcal{D},\mathbf{p})$ such that

$\tilde{\partial}_{2}(\varsigma_{z})=\hat{\rho_{1}}\left(\sum_{j}\delta_{j}f_{j}\right).$
(6)

We can apply the above for every 2-cell
$\sigma\in(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ not in
$(\mathcal{D},\mathbf{p})$ by taking $z=\tilde{\partial}^{\infty}_{2}(\sigma)$
and writing $\varsigma_{\sigma}$ instead of $\varsigma_{z}$. With these
notations (6) takes the form

$\tilde{\partial}_{2}(\varsigma_{\sigma})=\hat{\rho_{1}}(\tilde{\partial}^{\infty}_{2}(\sigma)).$
(7)

We define

$\hat{\rho_{2}}:C_{2}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\rightarrow
C_{2}(\mathcal{D},\mathbf{p})$

by

$\hat{\rho_{2}}(\sigma)=\sigma$

for every 2-cell $\sigma$ in $(\mathcal{D},\mathbf{p})$, and for every other
2-cell $\sigma$ we define

$\hat{\rho_{2}}(\sigma)=\varsigma_{\sigma}.$

We will explain how this works for 2-cells $[e,f]$ with
$\hat{\rho_{1}}(e)=\sum_{i}\alpha_{i}e_{i}$ and
$\hat{\rho_{1}}(f)=\sum_{j}\beta_{j}f_{j}$ where at least one of the sums has
more than one term (the corresponding edge is not in
$(\mathcal{D},\mathbf{p})$). In this case we have

$\displaystyle\hat{\rho}_{1}\tilde{\partial}^{\infty}_{2}([e,f])$
$\displaystyle=\hat{\rho}_{1}(e.(\iota f-\tau f)-(\iota e-\tau e).f)$
$\displaystyle=\sum_{i}\alpha_{i}e_{i}\cdot\sum_{j}\beta_{j}(\iota f_{j}-\tau
f_{j})-\sum_{i}\alpha_{i}(\iota e_{i}-\tau e_{i})\cdot\sum_{j}\beta_{j}f_{j}$
(lemma 4.1)
$\displaystyle=\sum_{i,j}\alpha_{i}\beta_{j}\tilde{\partial}_{2}[e_{i},f_{j}]$
$\displaystyle=\tilde{\partial}_{2}\left(\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},f_{j}]\right),$

therefore the 2-chain $\varsigma_{[e,f]}$ in this case can be chosen to be
$\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},f_{j}]$. Again $\hat{\rho_{2}}$ as
defined above is a retraction.

Finally, we define

$\hat{\rho_{3}}:C_{3}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\rightarrow
C_{3}(\mathcal{D},\mathbf{p})$

as follows. If $e$ is any edge with
$\hat{\rho_{1}}(e)=\sum_{i}\varepsilon_{i}e_{i}$ and $\sigma$ a 2-cell in
$(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ such that
$\hat{\rho_{2}}(\sigma)=\sum_{j}\mu_{j}\sigma_{j}$, then we define

$\hat{\rho_{3}}([e,\sigma])=\sum_{i,j}\varepsilon_{i}\mu_{j}[e_{i},\sigma_{j}]\text{
and
}\hat{\rho_{3}}([\sigma,e])=\sum_{i,j}\varepsilon_{i}\mu_{j}[\sigma_{j},e_{i}].$

It is obvious that $\hat{\rho_{3}}$ is a retraction.

######  Lemma 4.2.

The following hold true:

  * (i)

$\hat{\rho_{1}}\partial^{\infty}_{2}=\tilde{\partial}_{2}\hat{\rho_{2}}$.

  * (ii)

$\hat{\rho_{2}}\partial^{\infty}_{3}=\tilde{\partial}_{3}\hat{\rho_{3}}$.

###### Proof.

(i) If $\sigma\in C_{2}(\mathcal{D},\mathbf{p})$, then

$\displaystyle\hat{\rho_{1}}\tilde{\partial}^{\infty}_{2}(\sigma)$
$\displaystyle=\hat{\rho_{1}}\tilde{\partial}_{2}(\sigma)=\tilde{\partial}_{2}(\sigma)=\tilde{\partial}_{2}\hat{\rho_{2}}(\sigma).$

Assume now that $\sigma$ is a 2-cell not in $C_{2}(\mathcal{D},\mathbf{p})$,
then from the definition of $\hat{\rho_{2}}$ and from (7) we have

$\displaystyle\tilde{\partial}_{2}\hat{\rho_{2}}(\sigma)=\tilde{\partial}_{2}(\varsigma_{\sigma})=\hat{\rho_{1}}(\tilde{\partial}^{\infty}_{2}(\sigma)).$

(ii) Let $[e,\sigma]$ be a 3-cell where $\sigma$ is any 2-cell in
$(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ with
$\hat{\rho_{2}}(\sigma)=\sum_{j}\mu_{j}\sigma_{j}$, and $e$ is an edge with
$\hat{\rho_{1}}(e)=\sum_{i}\varepsilon_{i}e_{i}$. Assume also that
$\tilde{\partial}^{\infty}_{2}(\sigma)=\sum_{k}\delta_{k}f_{k}$ where for each
$k$, $\hat{\rho_{1}}(f_{k})=\sum_{s}\beta_{ks}g_{ks}$. It follows that from
(i) that

$\tilde{\partial}_{2}\left(\sum_{j}\mu_{j}\sigma_{j}\right)=\tilde{\partial}_{2}(\hat{\rho_{2}}(\sigma))=\hat{\rho_{1}}\tilde{\partial}^{\infty}_{2}(\sigma)=\sum_{k}\delta_{k}\sum_{s}\beta_{ks}g_{ks}.$
(8)

If for each $j$ we let $\mathcal{T}_{\sigma_{j}}$ be the set of terms of
$\tilde{\partial}_{2}(\sigma_{j})$, we see from (8) that for each $k$ and each
$s$, there is some $j$ and some $\alpha_{j}x_{j}\in\mathcal{T}_{\sigma_{j}}$,
such that $\mu_{j}\alpha_{j}x_{j}=\delta_{k}\beta_{ks}g_{ks}$. This implies
that for each edge $e_{i}$, we have that
$\delta_{k}\beta_{ks}[e_{i},g_{ks}]=\mu_{j}\alpha_{j}[e_{i},x_{j}]$. Further
we see that

$\displaystyle\hat{\rho_{2}}\tilde{\partial}^{\infty}_{3}([e,\sigma])$
$\displaystyle=\hat{\rho_{2}}\left((\iota e-\tau
e).\sigma+\sum_{k}\delta_{k}[e,f_{k}]\right)$
$\displaystyle=\sum_{j}\mu_{j}(\iota e-\tau
e).\sigma_{j}+\sum_{k}\delta_{k}\left(\sum_{i}\varepsilon_{i}\sum_{s}\beta_{ks}[e_{i},g_{ks}]\right)$
$\displaystyle=\sum_{j}\mu_{j}\left(\sum_{i}\varepsilon_{i}(\iota e_{i}-\tau
e_{i})\right).\sigma_{j}+\sum_{k}\delta_{k}\left(\sum_{i}\varepsilon_{i}\sum_{s}\beta_{ks}[e_{i},g_{ks}]\right)$
(lemma 4.1) $\displaystyle=\sum_{i}\varepsilon_{i}\left(\sum_{j}\mu_{j}(\iota
e_{i}-\tau
e_{i})\right).\sigma_{j}+\sum_{i}\varepsilon_{i}\left(\sum_{k}\delta_{k}\sum_{s}\beta_{ks}[e_{i},g_{ks}]\right)$
$\displaystyle=\sum_{i}\varepsilon_{i}\left(\sum_{j}\mu_{j}(\iota e_{i}-\tau
e_{i}).\sigma_{j}+\sum_{k}\delta_{k}\sum_{s}\beta_{ks}[e_{i},g_{ks}]\right)$
$\displaystyle=\sum_{i}\varepsilon_{i}\left(\sum_{j}\mu_{j}(\iota e_{i}-\tau
e_{i}).\sigma_{j}+\sum_{j}\mu_{j}\sum_{\alpha_{j}x_{j}\in\mathcal{T}_{\sigma_{j}}}\alpha_{j}[e_{i},x_{j}]\right)$
$\displaystyle=\sum_{i,j}\varepsilon_{i}\mu_{j}\left((\iota e_{i}-\tau
e_{i}).\sigma_{j}+\sum_{\alpha_{j}x_{j}\in\mathcal{T}_{\sigma_{j}}}\alpha_{j}[e_{i},x_{j}]\right)=\tilde{\partial}_{3}\left(\sum_{i,j}\varepsilon_{i}\mu_{j}[e_{i},\sigma_{j}]\right)$
(by (2)) $\displaystyle=\tilde{\partial}_{3}\hat{\rho_{3}}([e,\sigma]).$

The proof for the 3-cell $[\sigma,e]$ is similar to the above and is omitted
here. ∎

######  Proposition 4.3.

(Proposition 14, [36]) If $\mathbf{p}$ is a homology trivializer obtained by
choosing resolutions of certain critical pairs, then the sequence (4) is
exact.

###### Proof.

From the special case of proposition we have the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl.}$$\textstyle{K^{\mathbf{p}^{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}^{\infty}_{2}}$$\textstyle{B_{1}(\mathcal{D}^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(9)

Let $\xi\in Ker\tilde{\partial}_{2}$. Since $K^{\mathbf{p}}\subseteq
K^{\mathbf{p}^{\infty}}$ and the restriction of
$\tilde{\partial}^{\infty}_{2}$ on $K^{\mathbf{p}}$ is $\tilde{\partial}_{2}$,
then $\xi\in Ker\tilde{\partial}^{\infty}_{2}$ and the exactness of (9)
implies the existence of a 3-chain $w\in
C_{3}(\mathcal{D}^{\infty},\mathbf{p}^{\infty})$ such that
$\tilde{\partial}^{\infty}_{3}(w)=\xi$. It follows form lemma 4.2 that

$\xi=\hat{\rho_{2}}(\xi)=\hat{\rho_{2}}(\tilde{\partial}^{\infty}_{3}(w))=\tilde{\partial}_{3}\hat{\rho_{3}}(w),$

which shows that $\xi\in B_{2}(\mathcal{D},\mathbf{p})$ and hence the
exactness of (4). ∎

###  4.7 The Pride complex $\mathcal{D}(\mathcal{P})^{\ast}$ associated with a
group presentation $\mathcal{P}$

In this section we will explain several results of Pride in [40] where it is
proved that the homotopical property FDT for groups is equivalent to the
homological property $FP_{3}$. In order to achieve this, associated to any
group presentation $\hat{\mathcal{P}}=(\mathbf{x},\mathbf{r})$, Pride
considers two crossed modules. The first one is the free crossed module
$(\Sigma,\hat{F},\partial)$ associated to
$\hat{\mathcal{P}}=(\mathbf{x},\mathbf{r})$. To define the second, he
constructs first a complex $\mathcal{D}(\mathcal{M})^{\ast}$ arising from the
monoid presentation

$\mathcal{M}=\langle\mathbf{x},\mathbf{x}^{-1}:R=1(R\in\mathbf{r}),x^{\varepsilon}x^{-\varepsilon}=1(x\in\mathbf{x},\varepsilon=\pm
1)\rangle$

of the same group given by $\hat{\mathcal{P}}=(\mathbf{x},\mathbf{r})$. In
other words, the group is now realized as the quotient of the free monoid $F$
on $\mathbf{x}\cup\mathbf{x}^{-1}$ by the smallest congruence generated by the
relations giving $\mathcal{M}$. We will give below some necessary details on
this complex. We first mention that $\mathcal{D}(\mathcal{M})^{\ast}$ is an
extension of the usual Squier complex $\mathcal{D}(\mathcal{M})$ arising from
$\mathcal{M}$. This complex is called in [12] the Pride complex. We emphasize
here that the definition of $\mathcal{D}(\mathcal{M})$ in [40] requires the
attachment of 2-cells $[e^{\varepsilon},f^{\delta}]$ where $e,f$ are positive
edges and $\varepsilon,\delta=\pm 1$. But as it is observed in the latter
paper [36] (see Remark 8 there), since we are interested in the homotopy and
homology of the complex, the attachment of 2-cells
$[e^{\varepsilon},f^{\delta}]$ is unnecessary in the presence of $[e,f]$
because the boundary of $[e^{\varepsilon},f^{\delta}]$ is a cyclic permutation
of $(\partial[e,f])^{\pm 1}$, hence it is null homotopic. So we assume in what
follows that $\mathcal{D}(\mathcal{M})$ is that one described in section 4.2.

To complete the construction of $\mathcal{D}(\mathcal{M})^{\ast}$ we need to
add to $\mathcal{D}(\mathcal{M})$ certain extra 2-cells along the closed paths

$\mathbf{t}=(1,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon})\circ(x^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,1)^{-1},$

where $x\in\mathbf{x}$ and $\varepsilon=\pm 1$. The attaching of these two
cells is done for every overlap of two trivial edges as depicted below

---
$\textstyle{x^{\varepsilon}x^{-\varepsilon}x^{\varepsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,1)}$$\scriptstyle{(1,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon})}$$\textstyle{x^{\varepsilon}}$

The attached 2-cell has boundary made of the following edges

$\mathbb{A}=(1,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon})\text{ and
}\mathbb{B}=(x^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,1).$

Together with such 2-cells, there are added to the complex all their
”translates”

---
$\textstyle{ux^{\varepsilon}x^{-\varepsilon}x^{\varepsilon}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(ux^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,v)}$$\scriptstyle{(u,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon}v)}$$\textstyle{ux^{\varepsilon}v}$

In our paper the Pride complex $\mathcal{D}(\mathcal{M})^{\ast}$ is denoted
throughout by $(\mathcal{D},\mathbf{t})$.

If $P$ is a path in $(\mathcal{D},\mathbf{t})$ with $\iota(P)=W$ and
$\tau(P)=Z$, there are defined in [40], $T_{W}$ and $T_{Z}$ to be arbitrary
trivial paths from $W^{\ast}$ to $W$ and from $Z^{\ast}$ to $Z$ where
$W^{\ast}$ and $Z^{\ast}$ are the unique reduced words freely equivalent to
$W$ and $Z$ respectively. Then the path $T_{W}PT_{Z}^{-1}$ is denoted by
$P^{\ast}$. The notation is not ambiguous since any two parallel trivial paths
in $(\mathcal{D},\mathbf{t})$ are homotopic.

Pride has defined in [40] an $\hat{F}$-crossed module $\Sigma^{\ast}$ out of
$(\mathcal{D},\mathbf{t})$ in the following way. The elements of
$\Sigma^{\ast}$ are the homotopy classes $\langle P\rangle$ where $P$ is a
path in the 1-skeleton of $(\mathcal{D},\mathbf{t})$ such that $\tau(P)$ is
the empty word and $\iota(P)$ is a freely reduced word from $\hat{F}$. He then
defines a (non commutative) operation $+$ on $\Sigma^{\ast}$ by

$\langle P_{1}\rangle+\langle P_{2}\rangle=\langle(P_{1}+P_{2})^{\ast}\rangle$

and an action of the free group $\hat{F}$ on $\mathbf{x}\cup\mathbf{x}^{-1}$
on $\Sigma^{\ast}$ by

${}^{[W]}\langle P\rangle=\langle(WPW^{-1})^{\ast}\rangle.$

Also he defines

$\partial^{\ast}:\Sigma^{\ast}\rightarrow\hat{F}$

by

$\partial^{\ast}(\langle P\rangle)=[\iota(P)].$

It is proved in [40] that the triple $(\Sigma^{\ast},\hat{F},\partial^{\ast})$
is a crossed module. Further, using the fact that $\Sigma$ is the free crossed
module over $\mathbf{r}$, it is proved that

$\eta:\Sigma\rightarrow\Sigma^{\ast}$

defined by $r\mapsto\langle(1,r,1,1)\rangle$ is an isomorphism of crossed
modules. The inverse $\psi:\Sigma^{\ast}\rightarrow\Sigma$ of $\eta$ is the
map defined in the following way. It is first defined a map $\psi_{0}$ from
the set of edges of $(\mathcal{D},\mathbf{t})$ to $\Sigma$ as follows. Every
trivial edge is mapped to $0$, and every edge $(u,r,\varepsilon,v)$ is mapped
to $(^{[u]}r)^{\varepsilon}$ where $[u]$ is the element of $\hat{F}$
represented by $u$. It is proved that this map extends to paths of
$(\mathcal{D},\mathbf{t})$ and it sends the boundaries of the defining 2-cells
of $(\mathcal{D},\mathbf{t})$ to 0. We thus have a morphism
$\psi:\Sigma^{\ast}\rightarrow\Sigma$ given by

$\langle P\rangle\mapsto\psi_{0}(P)$

which is proved to be the inverse of $\eta$. By restriction it is obtained an
isomorphism between $\text{Ker}\partial$ and $\text{Ker}\partial^{\ast}$. But
$\text{Ker}\partial$ is itself isomorphic to $\pi_{2}(\hat{\mathcal{P}})$, the
second homotopy module of the standard complex associated with
$\hat{\mathcal{P}}$, and $\text{Ker}\partial^{\ast}$ on the other hand, is
isomorphic to $\pi_{1}(\mathcal{D},\mathbf{t},1)$, the first homotopy group of
the connected component of $(\mathcal{D},\mathbf{t})$ at 1. Recollecting, we
have the following isomorphisms

$\textstyle{\pi_{2}(\hat{\mathcal{P}})=\text{Ker}\partial\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\text{Ker}\partial^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{=\pi_{1}(\mathcal{D},\mathbf{t},1)}.}$$\scriptstyle{\psi}$
(10)

The fundamental group $\pi_{1}(\mathcal{D},\mathbf{t},1)$ is abelian being
isomorphic to $\pi_{2}(\hat{\mathcal{P}})$ and therefore isomorphic to its
abelianization $H_{1}(\mathcal{D},\mathbf{t},1)$. The role of the isomorphism
between the two groups will be played by the well known Hurewicz homomorphism
$h:\pi_{1}(\mathcal{D},\mathbf{t},1)\rightarrow
H_{1}(\mathcal{D},\mathbf{t},1)$ which sends the homotopy class of a loop to
the homology class of the corresponding 1-cycle. In our proofs in the
following sections, we will identify the homotopy class of any loop $\xi$ with
$h(\xi)$ without further comment.

###  4.8 A characterization of the asphericity in terms of the Pride complex

Assume now we are given a presentation $\mathcal{P}=(\mathbf{x},\mathbf{r})$
of a group $G$. The new presentation
$\hat{\mathcal{P}}=(\mathbf{x},\mathbf{r}\cup\mathbf{r}^{-1})$ where
$\mathbf{r}^{-1}=\\{r^{-1}|r\in\mathbf{r}\\}$, is still giving $G$. The free
crossed module $(\Sigma,\hat{F},\partial)$ of [40] arising from
$\hat{\mathcal{P}}$ is in fact isomorphic to our crossed module
$(\mathcal{G}(\Upsilon),\hat{F},\tilde{\theta})$. Indeed, there is a morphism
of crossed modules $\alpha:\Sigma\rightarrow\mathcal{G}(\Upsilon)$ induced by
the map $r^{\varepsilon}\mapsto\mu\sigma(r^{\varepsilon})$, whose inverse is
$\beta:\mathcal{G}(\Upsilon)\rightarrow\Sigma$ defined by
$\mu\sigma((^{u}r)^{\varepsilon})\mapsto{{}^{u}}(r^{\varepsilon})$. So there
is no loss of generality if we identify ${{}^{u}}(r^{\varepsilon})\in\Sigma$
with $\mu\sigma((^{u}r)^{\varepsilon})\in\mathcal{G}(\Upsilon)$. The
isomorphism $\Sigma\cong\mathcal{G}(\Upsilon)$ means in particular that
$\text{Ker}\partial\cong\text{Ker}\tilde{\theta}=\tilde{\Pi}$.

We have on the other hand the monoid presentation of $G$

$\mathcal{M}=\langle\mathbf{x},\mathbf{x}^{-1}:\mathbf{s}\rangle$

where

$\mathbf{s}=\\{(r^{\varepsilon},1):r\in\mathbf{r},\varepsilon=\pm
1\\}\cup\left\\{(x^{\varepsilon}x^{-\varepsilon},1):x\in\mathbf{x},\varepsilon=\pm
1\right\\}.$

Related to $\mathcal{M}$ we have the Pride complex $(\mathcal{D},\mathbf{t})$.
Being aspherical for $\mathcal{P}$ means in virtue of theorem 3.4 and of
isomorphisms in (10) that $H_{1}(\mathcal{D},\mathbf{t},1)$ is trivialized as
an abelian group by the homology classes of all 1-cycles corresponding to
$\eta(\hat{\mathfrak{U}})$. This section is devoted to proving that the
asphericity of $\mathcal{P}$ is equivalent to
$H_{1}(\mathcal{D},\mathbf{p})=0$ where $\mathbf{p}=\mathbf{q}\cup\mathbf{t}$
and $\mathbf{q}$ is the set of 1-cycles corresponding to
$\eta(\mu\sigma(rr^{-1}))$ with $r\in\mathbf{r}$.

For every two paths of positive length
$A=e^{\varepsilon_{1}}_{1}\circ\dots\circ e^{\varepsilon_{n}}_{n}$ and
$B=f^{\delta_{1}}_{1}\circ\dots\circ f^{\delta_{m}}_{m}$ in
$(\mathcal{D},\mathbf{t})$ we have two parallel paths:

$A.\iota B\circ\tau A.B\text{ and }\iota A.B\circ A.\tau B.$

In what follows we use the notation $C\sim D$ to mean that two parallel paths
$C$ and $D$ are homotopic to each other.

######  Lemma 4.4.

For every two paths $A=e^{\varepsilon_{1}}_{1}\circ\dots\circ
e^{\varepsilon_{n}}_{n}$ and $B=f^{\delta_{1}}_{1}\circ\dots\circ
f^{\delta_{m}}_{m}$ as above, $(A.\iota B\circ\tau A.B)\sim(\iota A.B\circ
A.\tau B)$.

###### Proof.

The proof is done by induction on the maximum of $n$ and $m$. If $m=n=1$, then
it follows that

$A.\iota B\circ\tau A.B=e^{\varepsilon_{1}}_{1}.\iota
f^{\delta_{1}}_{1}\circ\tau
e^{\varepsilon_{1}}_{1}.f^{\delta_{1}}_{1}\sim\iota
e^{\varepsilon_{1}}_{1}.f^{\delta_{1}}_{1}\circ e^{\varepsilon_{1}}_{1}.\tau
f^{\delta_{1}}_{1}=\iota A.B\circ A.\tau B.$

Indeed, if $\varepsilon_{1}=\delta_{1}=1$, then this is an immediate
consequence of the 2-cell $[e_{1},f_{1}]$. If $\varepsilon_{1}=\delta_{1}=-1$,
then, since

$e_{1}.\iota f_{1}\circ\tau e_{1}.f_{1}\sim\iota e_{1}.f_{1}\circ e_{1}.\tau
f_{1},$ (11)

it follows by taking inverses that

$\displaystyle\iota e_{1}^{\varepsilon_{1}}.f_{1}^{\delta_{1}}\circ
e_{1}^{\varepsilon_{1}}.\tau f_{1}^{\delta_{1}}$ $\displaystyle=\tau
e_{1}.f_{1}^{-1}\circ e_{1}^{-1}.\iota f_{1}$ $\displaystyle\sim
e_{1}^{-1}.\tau f_{1}\circ\iota e_{1}.f_{1}^{-1}$
$\displaystyle=e^{\varepsilon_{1}}_{1}.\iota f^{\delta_{1}}_{1}\circ\tau
e^{\varepsilon_{1}}_{1}.f^{\delta_{1}}_{1}.$

In the case when $\varepsilon_{1}=1$ and $\delta_{1}=-1$, after composing on
the left of (11) by $\iota e_{1}.f_{1}^{-1}$ we obtain

$\iota e_{1}.f_{1}^{-1}\circ e_{1}.\iota f_{1}\circ\tau e_{1}.f_{1}\sim\iota
e_{1}.f_{1}^{-1}\circ\iota e_{1}.f_{1}\circ e_{1}.\tau f_{1}=e_{1}.\tau
f_{1},$

and then after composing the above on the right by $\tau e_{1}.f_{1}^{-1}$ we
get

$\iota e_{1}.f_{1}^{-1}\circ e_{1}.\iota f_{1}\sim e_{1}.\tau f_{1}\circ\tau
e_{1}.f_{1}^{-1},$

which is the same as

$\iota e_{1}^{\varepsilon_{1}}.f_{1}^{\delta_{1}}\circ
e_{1}^{\varepsilon_{1}}.\tau f_{1}^{\delta_{1}}\sim
e_{1}^{\varepsilon_{1}}.\iota f_{1}^{\delta_{1}}\circ\tau
e_{1}^{\varepsilon_{1}}.f_{1}^{\delta_{1}}.$

The proof for the case when $\varepsilon_{1}=-1$ and $\delta_{1}=1$ is
symmetric to the above and is omitted.

For the inductive step, let (for instance) $B$ be the path of maximal length
$m>1$. For the path $B^{\prime}=f_{1}^{\delta_{1}}\circ\dots\circ
f_{m-1}^{\delta_{m-1}}$ we know by induction that

$A.\iota B^{\prime}\circ\tau A.B^{\prime}\sim\iota A.B^{\prime}\circ A.\tau
B^{\prime}.$

Again, by induction for $A$ and $f_{m}^{\delta_{m}}$ we have that

$A.\iota f_{m}^{\delta_{m}}\circ\tau A.f_{m}^{\delta_{m}}\sim\iota
A.f_{m}^{\delta_{m}}\circ A.\tau f_{m}^{\delta_{m}}.$

It follows that

$\displaystyle A.\iota B\circ\tau A.B$ $\displaystyle=A.\iota B\circ\tau
A.B^{\prime}\circ\tau A.f_{m}^{\delta_{m}}$ $\displaystyle=A.\iota
B^{\prime}\circ\tau A.B^{\prime}\circ\tau A.f_{m}^{\delta_{m}}$
$\displaystyle\sim\iota A.B^{\prime}\circ A.\tau B^{\prime}\circ\tau A.f_{m}$
$\displaystyle=\iota A.B^{\prime}\circ A.\iota f_{m}^{\delta_{m}}\circ\tau
A.f_{m}^{\delta_{m}}$ $\displaystyle\sim\iota A.B^{\prime}\circ\iota
A.f_{m}^{\delta_{m}}\circ A.\tau f_{m}^{\delta_{m}}$ $\displaystyle=\iota
A.B\circ A.\tau B.$

There is a similar proof when $A$ is of maximal length. ∎

For every $u\in\hat{F}$, regarded as an element of the free monoid $F$ on
$\mathbf{x}\cup\mathbf{x}^{-1}$, and for every $r\in\mathbf{r}$, we see that

$\displaystyle\eta(\mu\sigma(^{u}r))={{}^{u}}\eta(\mu\sigma(r))$
$\displaystyle={{}^{u}}\langle(1,r,1,1)\rangle$
$\displaystyle=\langle(u,r,1,u^{-1})^{\ast}\rangle$ $\displaystyle=\langle
T_{uru^{-1}}\circ(u,r,1,u^{-1})\circ T^{-1}_{uu^{-1}}\rangle.$

The path $T_{uru^{-1}}\circ(u,r,1,u^{-1})\circ T_{uu^{-1}}$ is a composition
of $T_{uru^{-1}}$ which is a trivial path from the freely reduced word
$(uru^{-1})^{\ast}$ to $uru^{-1}$ followed by the edge $(u,r,1,u^{-1})$ and
then by the inverse of the trivial path $T_{uu^{-1}}$ from $uu^{-1}$ to 1.
Similarly to the above we have that

$\displaystyle\eta(\mu\sigma((^{u}r)^{-1}))={{}^{u}}\eta(\mu\sigma(r^{-1}))$
$\displaystyle={{}^{u}}\langle(1,r^{-1},1,1)\rangle$
$\displaystyle=\langle(u,r^{-1},1,u^{-1})^{\ast}\rangle$
$\displaystyle=\langle T_{ur^{-1}u^{-1}}\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}\rangle.$

Then we have

$\displaystyle\eta(\mu\sigma(^{u}r({{}^{u}}r)^{-1}))$
$\displaystyle=\eta(\mu\sigma(^{u}r))+\eta(\mu\sigma((^{u}r)^{-1}))$
$\displaystyle=\langle((T_{uru^{-1}}\circ(u,r,1,u^{-1})\circ
T^{-1}_{uu^{-1}})+(T_{ur^{-1}u^{-1}}\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}))^{\ast}\rangle$ $\displaystyle=\langle
T_{(ur^{-1}u)^{\ast}(ur^{-1}u^{-1})^{\ast}}\circ
T_{uru^{-1}}\cdot(ur^{-1}u^{-1})^{\ast}\circ(u,r,1,u^{-1})\cdot(ur^{-1}u^{-1})^{\ast}$
$\displaystyle\circ T^{-1}_{uu^{-1}}\cdot(ur^{-1}u^{-1})^{\ast}\circ
T_{ur^{-1}u^{-1}}\circ(u,r^{-1},1,u^{-1})\circ T^{-1}_{uu^{-1}}\rangle.$

Now we define two closed paths in $(\mathcal{D},\mathbf{t})$. First we let

$\displaystyle P(r,u)=T_{(uru^{-1})(ur^{-1}u^{-1})}$
$\displaystyle\circ(u,r,1,u^{-1}ur^{-1}u^{-1})$
$\displaystyle\circ(T^{-1}_{uu^{-1}}\cdot
ur^{-1}u^{-1})\circ(u,r^{-1},1,u^{-1})\circ T^{-1}_{uu^{-1}},$

and second

$Q(r,u)=T_{urr^{-1}u^{-1}}\circ(u,r,1,r^{-1}u^{-1})\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}.$

######  Proposition 4.5.

The presentation $\mathcal{P}$ is aspherical if and only if
$\pi_{1}(\mathcal{D},\mathbf{t},1)$ is generated as an abelian group by the
homotopy classes of loops $Q(r,u)$ with $r\in\mathbf{r}$ and $u\in F$.

###### Proof.

First note that the presentation $\mathcal{P}$ is aspherical if and only if
the set of all $\eta(\mu\sigma(^{u}r({{}^{u}}r)^{-1}))$ generates
$\pi_{1}(\mathcal{D},\mathbf{t},1)$. The claim follows directly if we prove
that for each $r\in\mathbf{r}$ and $u\in F$,
$\eta(\mu\sigma(^{u}r({{}^{u}}r)^{-1}))=\langle P(r,u)\rangle$ and that
$P(r,u)\sim Q(r,u)$.

Let us prove first that $\eta(\mu\sigma(^{u}r({{}^{u}}r)^{-1}))=\langle
P(r,u)\rangle$. Consider the following paths in $(\mathcal{D},\mathbf{t})$.
First, we let

$\displaystyle a$ $\displaystyle=(uru^{-1})^{\ast}\cdot T_{ur^{-1}u^{-1}},$
$\displaystyle b$ $\displaystyle=T_{uru^{-1}}\cdot(ur^{-1}u^{-1}),$
$\displaystyle d$ $\displaystyle=T_{uru^{-1}}\cdot(ur^{-1}u^{-1})^{\ast},$
$\displaystyle c$ $\displaystyle=(uru^{-1})\cdot T_{ur^{-1}u^{-1}},$

and observe from lemma 4.4 that

$a\circ b\sim d\circ c$ (12)

Second, we let

$\displaystyle e$ $\displaystyle=(u,r,1,u^{-1}(ur^{-1}u^{-1})^{\ast}),$
$\displaystyle c$ $\displaystyle=(uru^{-1})\cdot T_{ur^{-1}u^{-1}},$
$\displaystyle g$ $\displaystyle=(u,r,1,u^{-1}(ur^{-1}u^{-1})),$
$\displaystyle f$ $\displaystyle=(uu^{-1})\cdot T_{ur^{-1}u^{-1}},$

where again from lemma 4.4 we have that $c\circ g\sim e\circ f$. This implies
that

$c^{-1}\sim g\circ f^{-1}\circ e^{-1}.$ (13)

And finally, let

$\displaystyle y$ $\displaystyle=T^{-1}_{uu^{-1}}\cdot(ur^{-1}u^{-1})^{\ast}$
$\displaystyle f$ $\displaystyle=(uu^{-1})\cdot T_{ur^{-1}u^{-1}}$
$\displaystyle x$ $\displaystyle=T^{-1}_{uu^{-1}}\cdot(ur^{-1}u^{-1})$
$\displaystyle z$ $\displaystyle=1\cdot T_{ur^{-1}u^{-1}},$

where as before $f\circ x\sim y\circ z$. This on the other hand implies that

$f^{-1}\sim x\circ z^{-1}\circ y^{-1}.$ (14)

Further we write $\ell=T_{(uru^{-1})^{\ast}(ur^{-1}u^{-1})^{\ast}}$,
$\ell_{1}=T_{(uru^{-1})(ur^{-1}u^{-1})}$, $k=(u,r^{-1},1,u^{-1})$ and
$h=T^{-1}_{uu^{-1}}$. With the above abbreviations we have

$\displaystyle\ell$ $\displaystyle\sim\ell_{1}\circ b^{-1}\circ a^{-1}$
($\parallel$ trivial parallel paths are $\sim$)
$\displaystyle\sim\ell_{1}\circ c^{-1}\circ d^{-1}$ (from (12))
$\displaystyle\sim\ell_{1}\circ(g\circ f^{-1}\circ e^{-1})\circ d^{-1}$ (from
(13)) $\displaystyle\sim\ell_{1}\circ g\circ(x\circ z^{-1}\circ y^{-1})\circ
e^{-1}\circ d^{-1}$ (from (14)).

It follows that

$\displaystyle\eta(\mu\sigma(^{u}r({{}^{u}}r)^{-1}))$
$\displaystyle=\langle\ell\circ(d\circ e\circ y\circ z\circ k\circ h)\rangle$
$\displaystyle=\langle(\ell_{1}\circ g\circ(x\circ z^{-1}\circ y^{-1})\circ
e^{-1}\circ d^{-1})\circ(d\circ e\circ y\circ z\circ k\circ h)\rangle$
$\displaystyle=\langle\ell_{1}\circ g\circ x\circ k\circ h\rangle$
$\displaystyle=\langle P(r,u)\rangle.$

Secondly, we prove that $P(r,u)\sim Q(r,u)$. Indeed, if we consider paths

$\displaystyle A$ $\displaystyle=(u,r,1,u^{-1}ur^{-1}u^{-1})$ $\displaystyle
B$ $\displaystyle=(ur)\cdot T^{-1}_{u^{-1}u}\cdot(r^{-1}u^{-1})$
$\displaystyle C$ $\displaystyle=u\cdot T^{-1}_{u^{-1}u}\cdot(r^{-1}u^{-1})$
$\displaystyle D$ $\displaystyle=(u,r,1,r^{-1}u^{-1}),$

which from lemma 4.4 satisfy $A\circ C\sim B\circ D$, then we have

$\displaystyle P(r,u)$
$\displaystyle=T_{(uru^{-1})(ur^{-1}u^{-1})}\circ(u,r,1,u^{-1}ur^{-1}u^{-1})\circ(T^{-1}_{uu^{-1}}\cdot
ur^{-1}u^{-1})\circ(u,r^{-1},1,u^{-1})\circ T^{-1}_{uu^{-1}}$
$\displaystyle\sim T_{(uru^{-1})(ur^{-1}u^{-1})}\circ A\circ(u\cdot
T^{-1}_{u^{-1}u}\cdot r^{-1}u^{-1})\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}$ $\displaystyle=T_{(uru^{-1})(ur^{-1}u^{-1})}\circ A\circ
C\circ(u,r^{-1},1,u^{-1})\circ T^{-1}_{uu^{-1}}$ $\displaystyle\sim
T_{(uru^{-1})(ur^{-1}u^{-1})}\circ B\circ D\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}$ $\displaystyle\sim T_{urr^{-1}u^{-1}}\circ B^{-1}\circ
B\circ D\circ(u,r^{-1},1,u^{-1})\circ T^{-1}_{uu^{-1}}$
$\displaystyle=T_{urr^{-1}u^{-1}}\circ D\circ(u,r^{-1},1,u^{-1})\circ
T^{-1}_{uu^{-1}}$ $\displaystyle=Q(r,u).$

This concludes the proof. ∎

Passing to homology we have the following

######  Proposition 4.6.

The presentation $\mathcal{P}$ is aspherical if and only if
$H_{1}(\mathcal{D},\mathbf{t},1)$ is generated as an abelian group by the
homology classes of 1-cycles corresponding to loops $Q(r,u)$ with
$r\in\mathbf{r}$ and $u\in F$.

######  Definition 4.7.

In $(\mathcal{D},\mathbf{t})$ we let $\mathbf{q}$ be the set of closed paths
$Q(r,1)$ with $r\in\mathbf{r}$.

If we attach to $(\mathcal{D},\mathbf{t})$ 2-cells $\sigma$ along the closed
paths $u.Q(r,1).v$ with $u,v\in F$ and 3-cells $[e,\sigma]$ and $[\sigma,e]$
for every 2-cell $\sigma$ and each positive edge $e$, then we obtain a new
3-complex $(\mathcal{D},\mathbf{q}\cup\mathbf{t})$. The asphericity of
$\mathcal{P}$ is encoded in the homology of
$(\mathcal{D},\mathbf{q}\cup\mathbf{t})$ as the following shows.

######  Theorem 4.8.

The presentation $\mathcal{P}$ is aspherical if and only if
$H_{1}(\mathcal{D},\mathbf{q}\cup\mathbf{t})=0$.

To prove the theorem we first note the following two lemmas.

######  Lemma 4.9.

For every $\varsigma\in Z_{1}(\mathcal{D},\mathbf{t})$ and every $u,v\in F$
such that $\bar{u}=\bar{v}$, $\varsigma\cdot
u+B_{1}(\mathcal{D},\mathbf{t})=\varsigma\cdot
v+B_{1}(\mathcal{D},\mathbf{t})$.

###### Proof.

It is enough to prove that for every positive edge $f$, we have
$\varsigma\cdot\iota f+B_{1}(\mathcal{D},\mathbf{t})=\varsigma\cdot\tau
f+B_{1}(\mathcal{D},\mathbf{t})$. From Lemma 4.1 of [39] it follows that
$\varsigma\cdot(\iota f-\tau f)\in B_{1}(\mathcal{D})$. But
$B_{1}(\mathcal{D})\subseteq B_{1}(\mathcal{D},\mathbf{t})$ and we are done. ∎

If $u=x_{1}^{\varepsilon_{1}}\dots x_{n}^{\varepsilon_{n}}\in F$ is any word
of length $|u|=n\in\mathbb{N}$, then a trivial path from 1 to $uu^{-1}$ is the
following

$T_{uu^{-1}}=(1,x_{1}^{\varepsilon_{1}}x_{1}^{-\varepsilon_{1}},1,1)^{-1}\circ\dots\circ(x_{1}^{\varepsilon_{1}}\dots
x_{|u|-1}^{\varepsilon_{|u|-1}},x_{|u|}^{\varepsilon_{|u|}}x_{|u|}^{-\varepsilon_{|u|}},1,x_{|u|-1}^{-\varepsilon_{|u|-1}}\dots
x_{1}^{-\varepsilon_{1}})^{-1}.$

We write for short

$\displaystyle t_{u}^{(1)}$
$\displaystyle=(1,x_{1}^{\varepsilon_{1}}x_{1}^{-\varepsilon_{1}},1,1)$
$\displaystyle...$ $\displaystyle t_{u}^{(|u|)}$
$\displaystyle=(x_{1}^{\varepsilon_{1}}\dots
x_{|u|-1}^{\varepsilon_{|u|-1}},x_{|u|}^{\varepsilon_{|u|}}x_{|u|}^{-\varepsilon_{|u|}},1,x_{|u|-1}^{-\varepsilon_{|u|-1}}\dots
x_{1}^{-\varepsilon_{1}})^{-1},$

and let

$\tau_{uu^{-1}}=t_{u}^{(1)}+\dots+t_{u}^{(|u|)}.$

######  Definition 4.10.

For every $r\in\mathbf{r}$ and $u\in F^{\ast}$, we let

$q(r,u)=(u,r,1,r^{-1}u^{-1})+(u,r^{-1},1,u^{-1})+\tau_{uu^{-1}}-\tau_{urr^{-1}u^{-1}},$

be the 1-cycle that corresponds to the closed path $Q(r,u)$. When $u=1$, we
let

$q(r,1)=(1,r,1,r^{-1})+(1,r^{-1},1,1)-\tau_{rr^{-1}}$

be the 1-cycle corresponding to $Q(r,1)$.

######  Lemma 4.11.

For every $r\in\mathbf{r}$ and $u\in F$,
$u.q(r,1).u^{-1}+B_{1}(\mathcal{D},\mathbf{t})=q(r,u)+B_{1}(\mathcal{D},\mathbf{t})$.

###### Proof.

First note that $T^{-1}_{uu^{-1}}\circ T_{urr^{-1}u^{-1}}\sim
u.T_{rr^{-1}}.u^{-1}$ since any two trivial paths with the same end points are
homotopic with each other. For the corresponding 1-chains we have that

$\tau_{uu^{-1}}-\tau_{urr^{-1}u^{-1}}+B_{1}(\mathcal{D},\mathbf{t})=-u.\tau_{rr^{-1}}.u^{-1}+B_{1}(\mathcal{D},\mathbf{t}).$

It follows now that

$\displaystyle u.q(r,1).u^{-1}+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=(u,r,1,r^{-1}u^{-1})+(u,r^{-1},1,u^{-1})-u.\tau_{rr^{-1}}.u^{-1}+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=(u,r,1,r^{-1}u^{-1})+(u,r^{-1},1,u^{-1})+\tau_{uu^{-1}}-\tau_{urr^{-1}u^{-1}}+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=q(r,u)+B_{1}(\mathcal{D},\mathbf{t}),$

proving the claim. ∎

###### Proof.

(of theorem 4.8) If $H_{1}(\mathcal{D},\mathbf{q}\cup\mathbf{t})=0$, then
$H_{1}(\mathcal{D},\mathbf{q}\cup\mathbf{t},1)=0$ which means that the
homology classes of the loops $u.Q(r,1).v$ trivialize
$H_{1}(\mathcal{D},\mathbf{t},1)$. We claim that every 1-cycle corresponding
to a loop $u.Q(r,1).v$ which sits inside $(\mathcal{D},\mathbf{t},1)$ is in
fact homologous to the 1-cycle corresponding to the loop $Q(r,u)$. Indeed,
since $u.Q(r,1).v$ is a loop in $(\mathcal{D},\mathbf{t},1)$, then
$\bar{v}=\bar{u}^{-1}$. It follows from lemma 4.9 and lemma 4.11 that

$\displaystyle u.q(r,1).v+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=u.q(r,1).u^{-1}+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=q(r,u)+B_{1}(\mathcal{D},\mathbf{t}),$

which proves our claim. As a consequence of this we have that the homology
classes of 1-cycles $q(r,u)$ trivialize $H_{1}(\mathcal{D},\mathbf{t},1)$, and
then from proposition 4.6 we get the asphericity of $\mathcal{P}$.

Conversely, if $\mathcal{P}$ is aspherical, then from proposition 4.6 and
lemma 4.11 $H_{1}(\mathcal{D},\mathbf{t},1)$ is generated as an abelian group
by the homology classes of 1-cycles $u.q(r,1).u^{-1}$. But from [40] the
homology group $H_{1}(\mathcal{D},\mathbf{t},w)$ of the connected component
$(\mathcal{D},\mathbf{t},w)$ is isomorphic to
$H_{1}(\mathcal{D},\mathbf{t},1)$, where the isomorphism
$\phi_{w}:H_{1}(\mathcal{D},\mathbf{t},1)\rightarrow
H_{1}(\mathcal{D},\mathbf{t},w)$ maps each homology class of some 1-cycle
$\varsigma$ to the homology class of $\varsigma\cdot w$. This shows that the
set of the homology classes of 1-cycles $u.q(r,1).u^{-1}w$ trivialize
$H_{1}(\mathcal{D},\mathbf{t})$. We prove that this set equals to the set of
homology classes of 1-cycles $u.q(r,1).v$ where $u,v\in F$. Indeed, for every
$u,v\in F$ and every $q(r,1)$, if we take $w=uv$, we get that
$u.q(r,1).u^{-1}uv+B_{1}(\mathcal{D},\mathbf{t})$ is a generator of
$H_{1}(\mathcal{D},\mathbf{t})$. But from lemma 4.9,
$u.q(r,1).u^{-1}uv+B_{1}(\mathcal{D},\mathbf{t})=u.q(r,1).v+B_{1}(\mathcal{D},\mathbf{t})$,
hence $u.q(r,1).v+B_{1}(\mathcal{D},\mathbf{t})$ is a generator of
$H_{1}(\mathcal{D},\mathbf{t})$. For the converse, it is obvious that any
generator $u.q(r,1).u^{-1}w+B_{1}(\mathcal{D},\mathbf{t})$ is of the form
$u.q(r,1).v+B_{1}(\mathcal{D},\mathbf{t})$ with $v=u^{-1}w$. ∎

######  Remark 4.12.

The Squier complex $\mathcal{D}$ of the monoid presentation
$\mathcal{M}=\langle\mathbf{x},\mathbf{x}^{-1}:\mathbf{s}\rangle$ of $G$ has
an important property. As the theorem 4.8 shows, the homology trivializers of
$H_{1}(\mathcal{D})$ are classes of 1-cycles corresponding to loops from
$\mathbf{q}\cup\mathbf{t}$ and each one of them arises from the resolution of
a critical pair. Indeed, if $r\in\mathbf{r}$ has the reduced word form
$r=x_{1}^{\varepsilon_{1}}\dots x_{n}^{\varepsilon_{n}}$ in $\hat{F}$, then
considering $x_{1}^{\varepsilon_{1}}\dots x_{n}^{\varepsilon_{n}}$ as a word
from $F$, we see that the loop $Q(r,1)$ is obtained by resolving the following
overlapping pair of edges

$((1,r,1,r^{-1}),(x_{1}^{\varepsilon_{1}}\dots
x_{n-1}^{\varepsilon_{n-1}},x_{n}^{\varepsilon_{n}}x_{n}^{-\varepsilon_{n}},1,x_{n-1}^{-\varepsilon_{n-1}}\dots
x_{1}^{-\varepsilon_{1}})).$

On the other hand, if
$t=(1,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon})\circ(x^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,1)^{-1}$
is a loop of $\mathbf{t}$, then it arises from the resolution of the
overlapping pair

$((1,x^{\varepsilon}x^{-\varepsilon},1,x^{\varepsilon}),(x^{\varepsilon},x^{-\varepsilon}x^{\varepsilon},1,1)).$

The importance of this remark stands at the fact that when the given
presentation $\mathcal{P}$ is aspherical, then the sequence (4) that is
associated with the complex $(\mathcal{D},\mathbf{q}\cup\mathbf{t})$ is exact.

###  4.9 A preliminary result

Let $\mathcal{P}=(\mathbf{x},\mathbf{r})$ be an aspherical group presentation
and $\mathcal{P}_{1}=(\mathbf{x},\mathbf{r}_{1})$ a subpresentation of the
first where $\mathbf{r}_{1}=\mathbf{r}\setminus\\{r_{0}\\}$ and
$r_{0}\in\mathbf{r}$ is a fixed relation. We denote by $\Upsilon_{1}$,
$\mathfrak{U}_{1}$ monoids associated with $\mathcal{P}_{1}$ and by
$\mathcal{G}(\Upsilon_{1})$ and $\hat{\mathfrak{U}}_{1}$ their respective
groups and let $\tilde{\theta}_{1}$ be the morphism of the crossed module
$\mathcal{G}(\Upsilon_{1})$ whose kernel is denoted by $\tilde{\Pi}_{1}$. Also
we consider $\hat{\mathfrak{A}}_{1}$ the subgroup of $\hat{\mathfrak{U}}$
generated by all $\mu\sigma(bb^{-1})$ where $b\in Y_{1}\cup Y_{1}^{-1}$.
Finally note that the monomorphism $\varphi:\Upsilon_{1}\rightarrow\Upsilon$
induced by the map $\sigma_{1}(a)\rightarrow\sigma(a)$ induces a homomorphism
$\hat{\varphi}:\mathcal{G}(\Upsilon_{1})\rightarrow\mathcal{G}(\Upsilon)$.
These data fit into a commutative diagram as depicted below.

---
$\textstyle{\mathcal{G}(\Upsilon_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\varphi}}$$\scriptstyle{\tilde{\theta}_{1}}$$\textstyle{\mathcal{G}(\Upsilon)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}}$$\textstyle{F}$

The following will be useful in the proof of our main theorem.

######  Proposition 4.13.

If $\mathcal{P}$ is aspherical, then
$\hat{\varphi}(\tilde{\Pi}_{1})=\hat{\mathfrak{A}}_{1}$.

###### Proof.

Let $\tilde{d}=\mu_{1}\sigma_{1}(a_{1}\cdot\cdot\cdot
a_{n})\in\tilde{\Pi}_{1}$ where as before no $a_{i}$ is equal to any
$\iota(\mu_{1}\sigma_{1}(^{u}{r})^{\varepsilon})$ and assume that

$\displaystyle\hat{\varphi}(\tilde{d})=$
$\displaystyle(\mu\sigma(b_{1}b_{1}^{-1})\cdot\cdot\cdot\mu\sigma(b_{s}b_{s}^{-1}))\cdot(\iota(\mu\sigma(b_{s+1}b_{s+1}^{-1}))\cdot\cdot\cdot\iota(\mu\sigma(b_{r}b_{r}^{-1})))$
$\displaystyle(\mu\sigma(c_{1}c_{1}^{-1})\cdot\cdot\cdot\mu\sigma(c_{t}c_{t}^{-1}))\cdot(\iota(\mu\sigma(d_{1}d_{1}^{-1}))\cdot\cdot\cdot\iota(\mu\sigma(d_{k}d_{k}^{-1}))),$

where the first half involves elements from $Y_{1}\cup Y_{1}^{-1}$ and the
second one is

$\mu\sigma(C)\iota(\mu\sigma(D))$

with

$C=c_{1}c_{1}^{-1}\cdot\cdot\cdot c_{t}c_{t}^{-1}\text{ and
}D=d_{1}d_{1}^{-1}\cdot\cdot\cdot d_{k}d_{k}^{-1},$

where $C$ and $D$ involve only elements of the form
$(^{u}{r_{0}})^{\varepsilon}$ with $\varepsilon=\pm 1$. Recalling from above
that in $\mathcal{G}(\Upsilon)$ we have

$\displaystyle\mu\sigma((a_{1}\cdot\cdot\cdot
a_{n})\cdot((b_{s+1}b_{s+1}^{-1})\cdot\cdot\cdot(b_{r}b_{r}^{-1}))\cdot((d_{1}d_{1}^{-1})\cdot\cdot\cdot(d_{k}d_{k}^{-1})))$
$\displaystyle=\mu\sigma(((b_{1}b_{1}^{-1})\cdot\cdot\cdot(b_{s}b_{s}^{-1}))\cdot((c_{1}c_{1}^{-1})\cdot\cdot\cdot(c_{t}c_{t}^{-1}))),$

we can apply $\hat{g}$ defined in proposition 3.5 on both sides and get

$\displaystyle g\sigma((a_{1}\cdot\cdot\cdot
a_{n})\cdot((b_{s+1}b_{s+1}^{-1})\cdot\cdot\cdot(b_{r}b_{r}^{-1}))\cdot((d_{1}d_{1}^{-1})\cdot\cdot\cdot(d_{k}d_{k}^{-1})))$
$\displaystyle=g\sigma(((b_{1}b_{1}^{-1})\cdot\cdot\cdot(b_{s}b_{s}^{-1}))\cdot((c_{1}c_{1}^{-1})\cdot\cdot\cdot(c_{t}c_{t}^{-1}))).$

If we now write each $c_{i}=(^{u_{i}}r_{0})^{\varepsilon_{i}}$ and each
$d_{j}=(^{v_{j}}r_{0})^{\delta_{j}}$ where $\varepsilon_{i}$ and
$\delta_{j}=\pm 1$, while we write each
$a_{\ell}=(^{w_{\ell}}r_{\ell})^{\gamma_{\ell}}$ and each
$b_{p}=(^{\eta_{p}}\rho_{p})^{\epsilon_{p}}$ where all $r_{\ell}$ and
$\rho_{p}$ belong to $\mathbf{r}_{1}$ and $\gamma_{\ell},\epsilon_{p}=\pm 1$,
then the definition of $g$ yields

$\displaystyle(w_{1}^{\alpha}\cdot
r_{1}^{\beta}+\cdot\cdot\cdot+w_{n}^{\alpha}\cdot
r_{n}^{\beta})+(2\eta_{s+1}^{\alpha}\cdot\rho_{s+1}^{\beta}+\cdot\cdot\cdot+2\eta_{r}^{\alpha}\cdot\rho_{r}^{\beta})+(2v_{1}^{\alpha}+\cdot\cdot\cdot+2v_{k}^{\alpha})\cdot
r_{0}^{\beta}$
$\displaystyle=(2\eta_{1}^{\alpha}\cdot\rho_{1}^{\beta}+\cdot\cdot\cdot+2\eta_{s}^{\alpha}\cdot\rho_{s}^{\beta})+(2u_{1}^{\alpha}+\cdot\cdot\cdot+2u_{t}^{\alpha})\cdot
r_{0}^{\beta}$

The freeness of $\mathcal{N}(\mathcal{P})$ on the set of elements $r^{\beta}$
implies in particular that

$(2v_{1}^{\alpha}+\cdot\cdot\cdot+2v_{k}^{\alpha})\cdot
r_{0}^{\beta}=(2u_{1}^{\alpha}+\cdot\cdot\cdot+2u_{t}^{\alpha})\cdot
r_{0}^{\beta}$

from which we see that $k=t$, and after a rearrangement of terms
$u^{\alpha}_{i}=v^{\alpha}_{i}$ for $i=1,...,k$. The easily verified fact that
in $\mathcal{G}(\Upsilon)$, $\mu\sigma(aa^{-1})=\mu\sigma(a^{-1}a)$ and the
fact that if $u^{\alpha}=v^{\alpha}$, then for every $s\in\mathbf{r}$,
$\mu\sigma((^{v}s)^{\delta}(^{v}s)^{-\delta})=\mu\sigma((^{u}s)^{\delta}(^{u}s)^{-\delta})$,
imply easily that

$\mu\sigma((^{v}r_{0})^{\delta}(^{v}r_{0})^{-\delta})=\mu\sigma((^{u}r_{0})^{\varepsilon}(^{u}r_{0})^{-\varepsilon}).$

If we apply the latter to pairs $(c_{i},d_{i})$ for which
$u^{\alpha}_{i}=v^{\alpha}_{i}$, we get that
$\mu\sigma(C)\iota(\mu\sigma(D))=1$ which shows that
$\hat{\varphi}(\tilde{d})\in\hat{\mathfrak{A}}_{1}$. ∎

###  4.10 The proof

Throughout this section we assume that $\mathcal{P}=(\mathbf{x},\mathbf{r})$
is an aspherical presentation of the trivial group. Consider now a sub
presentation $\mathcal{P}_{1}=(\mathbf{x},\mathbf{r}_{1})$ of $\mathcal{P}$
where $\mathbf{r}_{1}=\mathbf{r}\setminus\\{r_{0}\\}$. For each of the above
group presentations, we have a monoid presentation of the same group, namely

$\mathcal{M}=\langle\mathbf{x},\mathbf{x}^{-1}:\mathbf{s}\rangle$

is a monoid presentation of the trivial group, where

$\mathbf{s}=\\{(r^{\varepsilon},1):r\in\mathbf{r},\varepsilon=\pm
1\\}\cup\left\\{(x^{\varepsilon}x^{-\varepsilon},1):x\in\mathbf{x},\varepsilon=\pm
1\right\\},$

and

$\mathcal{M}_{1}=\langle\mathbf{x},\mathbf{x}^{-1}:\mathbf{s}_{1}\rangle$

is a monoid presentation of the group given by $\mathcal{P}_{1}$, where

$\mathbf{s}_{1}=\\{(r_{1}^{\varepsilon},1):r_{1}\in\mathbf{r}_{1},\varepsilon=\pm
1\\}\cup\left\\{(x^{\varepsilon}x^{-\varepsilon},1):x\in\mathbf{x},\varepsilon=\pm
1\right\\}.$

Related to $\mathcal{M}$ we have defined two 2-complexes. The first one is the
usual Squier complex $\mathcal{D}$, and the second one is its extension
$(\mathcal{D},\mathbf{t})$, and similarly we have two 2-complexes arising from
$\mathcal{M}_{1}$, $\mathcal{D}_{1}$ and its extension
$(\mathcal{D}_{1},\mathbf{t})$. Further, $(\mathcal{D},\mathbf{t})$ has been
extended to a 3-complex $(\mathcal{D},\mathbf{q}\cup\mathbf{t})$ by adding
first 2-cells arising from $Q(r,1)$ and their translates, and than adding all
the 3-cells $[e,\sigma]$ or $[\sigma,e]$ for every 2-cell $\sigma$ and every
positive edge $e$. We write for short $(\mathcal{D},\mathbf{q}\cup\mathbf{t})$
by $(\mathcal{D},\mathbf{p})$ where $\mathbf{p}=\mathbf{q}\cup\mathbf{t}$.
Likewise, $(\mathcal{D}_{1},\mathbf{t})$ extends to a 3-complex
$(\mathcal{D}_{1},\mathbf{p}_{1})$ where
$\mathbf{p}_{1}=\mathbf{q}_{1}\cup\mathbf{t}$ and $\mathbf{q}_{1}$ is the set
of 2-cells arising from $Q(r_{1},1)$ with $r_{1}\in\mathbf{r}_{1}$. But
$(\mathcal{D}_{1},\mathbf{p}_{1})$ is a subcomplex of
$(\mathcal{D},\mathbf{p})$, therefore we have the following exact sequence of
abelian groups

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
36.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{2}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
108.42502pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
108.42502pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{H_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
221.67233pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
221.67233pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{1}(\mathcal{D}_{1},\mathbf{p}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
298.94736pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
298.94736pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{1}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
370.62238pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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370.62238pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

We know from theorem 4.8 that $H_{1}(\mathcal{D},\mathbf{p})=0$, so if we
prove that
$H_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))=0$, then the
exactness of the above sequence will imply that
$H_{1}(\mathcal{D}_{1},\mathbf{p}_{1})=0$ and we are done. Before we proceed
with the proof, we explain how the boundary maps for the corresponding
quotient complex are defined. For this we consider the commutative diagram

$\textstyle{C_{3}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{3}}$$\scriptstyle{\tilde{\partial}_{3}}$$\textstyle{C_{2}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\tilde{\partial}_{2}}$$\textstyle{C_{1}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{C_{3}(\mathcal{D},\mathbf{p})/C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\partial}_{3}}$$\textstyle{C_{2}(\mathcal{D},\mathbf{p})/C_{2}(\mathcal{D}_{1},\mathbf{p}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\partial}_{2}}$$\textstyle{C_{1}(\mathcal{D},\mathbf{p})/C_{1}(\mathcal{D}_{1},\mathbf{p}_{1})}$
(15)

where $\mu_{i}$ for $i=1,2,3$ are the canonical epimorphisms. Then, for
$i=2,3$ and for every $\sigma\in C_{i}(\mathcal{D},\mathbf{p})$ we have

$\hat{\partial}_{i}(\mu_{i}\sigma)=\mu_{i-1}\tilde{\partial}_{i}(\sigma).$

We write
$Im(\hat{\partial}_{3})=B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))$,
and similarly
$Im(\hat{\partial}_{2})=B_{1}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))$.
Also we let
$Z_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))=Ker(\hat{\partial}_{2})$
and then

$H_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))=Z_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))/B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1})).$

We note that

$C_{2}(\mathcal{D},\mathbf{p})/C_{2}(\mathcal{D}_{1},\mathbf{p}_{1})\cong\mu_{2}(C_{2}(\mathcal{D}))\oplus
C_{2}^{\mathbf{q}_{0}},$

where $\mu_{2}(C_{2}(\mathcal{D}))$ is the free abelian group generated by all
2-cells $[e,f]$ where at least one of the edges $e$ or $f$ arises from
$r_{0}$, and $C_{2}^{\mathbf{q}_{0}}$ is the free abelian group on 2-cells
$u.\mathbf{q}_{0}.v$ where $\mathbf{q}_{0}$ is the 2-cell attached along
$Q(r_{0},1)$. We can thus regard
$Z_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))$ as a
subgroup of $\mu_{2}(C_{2}(\mathcal{D}))\oplus C_{2}^{\mathbf{q}_{0}}$. Now we
let

$\varphi_{rel}:\mu_{2}(C_{2}(\mathcal{D}))\oplus
C_{2}^{\mathbf{q}_{0}}\rightarrow\mathbb{Z}G.\mathbf{q}_{0}.\mathbb{Z}G$

be the $(\mathbb{Z}F,\mathbb{Z}F)$-homomorphism defined by mapping
$\mu_{2}(C_{2}(\mathcal{D}))$ to 0, and every 2-cell $u.\mathbf{q}_{0}.v$ to
$\bar{u}.\mathbf{q}_{0}.\bar{v}$. Denote the kernel of $\varphi_{rel}$ by
$K_{rel}^{\mathbf{q}_{0}}$. By the same argument as that of [36] we see that

$K_{rel}^{\mathbf{q}_{0}}=\mu_{2}(C_{2}(\mathcal{D}))+J.\mathbf{q}_{0}.\mathbb{Z}F+\mathbb{Z}F.\mathbf{q}_{0}.J.$

Latter we will make use of the fact that $K_{rel}^{\mathbf{q}_{0}}$ can be
regarded as a sub group of $K^{\mathbf{p}}$.

Next we show that
$B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\subseteq
K_{rel}^{\mathbf{q}_{0}}$ and that the restriction of $\hat{\partial}_{2}$ on
$K_{rel}^{\mathbf{q}_{0}}$ sends $K_{rel}^{\mathbf{q}_{0}}$ onto the subgroup
$B_{1}(\mathcal{D},\mathcal{D}_{1})$ of
$C_{1}(\mathcal{D},\mathbf{p})/C_{1}(\mathcal{D}_{1},\mathbf{p}_{1})$ defined
by

$B_{1}(\mathcal{D},\mathcal{D}_{1})=\left\\{\beta+C_{1}(\mathcal{D}_{1})|\beta\in\tilde{\partial}_{2}(C_{2}(\mathcal{D}))\right\\}.$

To see that
$B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\subseteq
K_{rel}^{\mathbf{q}_{0}}$, we must prove that for every 3-cell $[f,\sigma]$ or
$[\sigma,f]$,

$\hat{\partial}_{3}\left([f,\sigma]+C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})\right)\in
K_{rel}^{\mathbf{q}_{0}}$

and similarly,

$\hat{\partial}_{3}\left([\sigma,f]+C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})\right)\in
K_{rel}^{\mathbf{q}_{0}}.$

We prove the second for convenience. Let $[\sigma,f]\notin
C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})$. If $\sigma\in F.\mathbf{q}_{0}.F$ or
$f$ arises from $r_{0}$, then clearly

$\displaystyle\hat{\partial}_{3}\left([\sigma,f]+C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})\right)$
$\displaystyle=\left(\sigma.(\iota f-\tau
f)-\sum_{i}\varepsilon_{i}[e_{i},f]\right)+C_{2}(\mathcal{D}_{1},\mathbf{p}_{1})\in
K_{rel}^{\mathbf{q}_{0}}.$

Otherwise, if $\sigma\notin\mathbf{q}_{0}$ and $f$ arises from
$\mathbf{r}_{1}$, then $\sigma=[g,h]$ where at least $g$ or $h$ arises from
$r_{0}$. Again we see that
$\hat{\partial}_{3}\left([\sigma,f]+C_{3}(\mathcal{D}_{1},\mathbf{p}_{1})\right)\in
K_{rel}^{\mathbf{q}_{0}}$.

Next we prove that the restriction of $\hat{\partial}_{2}$ on
$K_{rel}^{\mathbf{q}_{0}}$ sends $K_{rel}^{\mathbf{q}_{0}}$ onto
$B_{1}(\mathcal{D},\mathcal{D}_{1})$. Indeed, since for every $(\iota f-\tau
f).\sigma_{0}\in J.\mathbf{q}_{0}.\mathbb{Z}F$

$(\iota f-\tau
f).\sigma_{0}=\tilde{\partial}_{3}[f,\sigma_{0}]-\sum_{i}\varepsilon_{i}[f,e_{i}],$

then we can derive that

$\displaystyle\hat{\partial}_{2}\left((\iota f-\tau f).\sigma_{0}\right)$
$\displaystyle=-\hat{\partial}_{2}\left(\sum_{i}\varepsilon_{i}[f,e_{i}]\right)$
$\displaystyle=-\sum_{i}\varepsilon_{i}\tilde{\partial}_{2}[f,e_{i}]+C_{1}(\mathcal{D}_{1})\in
B_{1}(\mathcal{D},\mathcal{D}_{1}).$

In a symmetric way one can show that
$\hat{\partial}_{2}\left(\sigma_{0}.(\iota f-\tau f)\right)\in
B_{1}(\mathcal{D},\mathcal{D}_{1})$. Finally, if
$[e,f]\in\mu_{2}(C_{2}(\mathcal{D}))$, then

$\hat{\partial}_{2}[e,f]=\tilde{\partial_{2}}[e,f]+C_{1}(\mathcal{D}_{1})\in
B_{1}(\mathcal{D},\mathcal{D}_{1}).$

This also shows that $\hat{\partial}_{2}$ is onto.

Therefore we have the complex

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl.}$$\textstyle{K_{rel}^{\mathbf{q}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\partial}_{2}}$$\textstyle{B_{1}(\mathcal{D},\mathcal{D}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(16)

which is exact on the left and on the right.

######  Lemma 4.14.

The complex (16) is exact.

###### Proof.

For this we consider the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}(\mathcal{D},\mathbf{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{incl.}$$\textstyle{K^{\mathbf{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\tilde{\partial}_{2}}$$\textstyle{B_{1}(\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl.}$$\textstyle{K_{rel}^{\mathbf{q}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}$$\scriptstyle{\hat{\partial}_{2}}$$\textstyle{B_{1}(\mathcal{D},\mathcal{D}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The top row is exact from proposition 4.3 and from remark 4.12, and
$\mu_{1},\mu_{2}$ are the restrictions of the epimorphisms of (15). Let
$\xi=\sum_{i}z_{i}\mu_{2}(\sigma_{i})\in Ker\hat{\partial}_{2}$. We recall
that $\xi$ can be regarded as an element of $K^{\mathbf{p}}$ with no terms
arising from $\mathbf{t}$ or square 2-cells $[e,f]$ with both $e$ and $f$ in
$\mathcal{D}_{1}$. Further we have that

$\displaystyle 0$
$\displaystyle=\hat{\partial}_{2}\left(\sum_{i}z_{i}\mu_{2}(\sigma_{i})\right)=\sum_{i}z_{i}\hat{\partial}_{2}\mu_{2}(\sigma_{i})$
$\displaystyle=\sum_{i}z_{i}\mu_{1}\tilde{\partial}_{2}(\sigma_{i})=\mu_{1}\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right),$

which implies that
$\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)\in
C_{1}(\mathcal{D}_{1})$, and so
$\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)$ is a 1-cycle in
$Z_{1}(\mathcal{D}_{1})$. It follows that

$\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)\in
Ker\tilde{\partial}_{1}\cap(J.\mathbf{s}_{1}.\mathbb{Z}F+\mathbb{Z}F.\mathbf{s}_{1}.J).$

We note that each term from
$J.\mathbf{s}_{1}.\mathbb{Z}F+\mathbb{Z}F.\mathbf{s}_{1}.J$ that is
represented in $\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)$
arises either from a 2-cell $[e,f]$ where at least one of $e$ or $f$ is a
positive edges that belongs to $\mathcal{D}_{1}$, or arises from an element of
the form $j.\mathbf{q}_{0}.v$ or $u.\mathbf{q}_{0}.j$ with $u,v\in F$ and
$j\in J$. Theorem 6.6 of [33] implies that there is a 2-chain
$\sum_{j}k_{j}\beta_{j}\in C_{2}(\mathcal{D}_{1})$ such that
$\tilde{\partial}_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)=\tilde{\partial}_{2}\left(\sum_{j}k_{j}\beta_{j}\right)$
and then we have the 2-cycle
$\tilde{\xi}=\sum_{i}z_{i}\sigma_{i}-\sum_{j}k_{j}\beta_{j}$ in
$K^{\mathbf{p}}$. It follows that $\tilde{\xi}$ is a 2-boundary since the top
row is exact, and has the property that

$\displaystyle\mu_{2}(\tilde{\xi})$
$\displaystyle=\mu_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)-\mu_{2}\left(\sum_{j}k_{j}\beta_{j}\right)$
$\displaystyle=\mu_{2}\left(\sum_{i}z_{i}\sigma_{i}\right)$ (since each
$\beta_{j}\in C_{2}(\mathcal{D}_{1})$)
$\displaystyle=\sum_{i}z_{i}\mu_{2}(\sigma_{i})$ $\displaystyle=\xi,$

hence for some $w\in C_{3}(\mathcal{D},\mathbf{p})$ we have that

$\xi=\mu_{2}(\tilde{\xi})=\mu_{2}(\tilde{\partial}_{3}(w))=\hat{\partial}_{3}\mu_{3}(w).$

This proves that $\xi$ is a relative 2-boundary and as a consequence the
exactness of the bottom row. ∎

Further we note that $B_{1}(\mathcal{D},\mathcal{D}_{1})$ embeds in
$Im(\hat{\partial}_{2})$. Indeed, any element
$\tilde{\partial}_{2}(\xi)+C_{1}(\mathcal{D}_{1})$ of
$B_{1}(\mathcal{D},\mathcal{D}_{1})$ where $\xi$ is a 2-chain from
$C_{2}(\mathcal{D})$ is in $Im(\hat{\partial}_{2})$ since
$C_{2}(\mathcal{D})\leq C_{2}(\mathcal{D},\mathbf{p})$ and
$C_{1}(\mathcal{D}_{1},\mathbf{p}_{1})=C_{1}(\mathcal{D}_{1})$.

Finally, consider the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{incl.}$$\textstyle{K_{rel}^{\mathbf{q}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\hat{\partial}_{2}}$$\textstyle{B_{1}(\mathcal{D},\mathcal{D}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\mu_{2}(C_{2}(\mathcal{D}))\oplus
C_{2}^{\mathbf{q}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\partial}_{2}}$$\textstyle{Im(\hat{\partial}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the top row is exact from lemma 4.14, and the bottom one is also exact
where $Im(\hat{\partial}_{2})\leq
C_{1}(\mathcal{D},\mathbf{p})/C_{1}(\mathcal{D}_{1},\mathbf{p}_{1})$. From the
Snake Lemma we get the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}((\mathcal{D},\mathbf{p}),(\mathcal{D}_{1},\mathbf{p}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}G.\mathbf{q}_{0}.\mathbb{Z}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{Im(\hat{\partial}_{2})/B_{1}(\mathcal{D},\mathcal{D}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
(17)

where
$\nu(\mathbf{q}_{0})=\hat{\partial}_{2}([1,\mathbf{q}_{0},1])+B_{1}(\mathcal{D},\mathcal{D}_{1})$.
Since $G$ is the trivial group, we have that for every $[u,\mathbf{q}_{0},v]$,

$\hat{\partial}_{2}([u,\mathbf{q}_{0},v])+B_{1}(\mathcal{D},\mathcal{D}_{1})=\hat{\partial}_{2}([1,\mathbf{q}_{0},1])+B_{1}(\mathcal{D},\mathcal{D}_{1}).$
(18)

This follows easily if we prove that for every positive edge $e$, and $v\in F$
we have that

$\hat{\partial}_{2}([\iota
e,\mathbf{q}_{0},v])+B_{1}(\mathcal{D},\mathcal{D}_{1})=\hat{\partial}_{2}([\tau
e,\mathbf{q}_{0},v])+B_{1}(\mathcal{D},\mathcal{D}_{1}),$

and similarly, for every positive edge $f$ and $u\in F$,

$\hat{\partial}_{2}([u,\mathbf{q}_{0},\iota
f])+B_{1}(\mathcal{D},\mathcal{D}_{1})=\hat{\partial}_{2}([u,\mathbf{q}_{0},\tau
f])+B_{1}(\mathcal{D},\mathcal{D}_{1}).$

We prove the first claim for convenience. Since

$(\iota e-\tau
e).\mathbf{q}_{0}.v+C_{2}(\mathcal{D}_{1},\mathbf{p}_{1})=\left(\tilde{\partial}_{3}[e,\mathbf{q}_{0}]-\sum_{i}\varepsilon_{i}[e,e_{i}]\right)+C_{2}(\mathcal{D}_{1},\mathbf{p}_{1}),$

where $\partial\mathbf{q}_{0}=e_{1}^{\varepsilon_{1}}\dots
e_{n}^{\varepsilon_{n}}$, then

$\tilde{\partial}_{2}((\iota e-\tau
e).\mathbf{q}_{0}.v)+C_{1}(\mathcal{D}_{1})=-\tilde{\partial}_{2}\left(\sum_{i}\varepsilon_{i}[e,e_{i}]\right)+C_{1}(\mathcal{D}_{1}).$

But

$\tilde{\partial}_{2}\left(\sum_{i}\varepsilon_{i}[e,e_{i}]\right)+C_{1}(\mathcal{D}_{1})\in
B_{1}(\mathcal{D},\mathcal{D}_{1}),$

consequently

$\displaystyle\left(\hat{\partial}_{2}([\iota
e,\mathbf{q}_{0},v])-\hat{\partial}_{2}([\tau
e,\mathbf{q}_{0},v])\right)+B_{1}(\mathcal{D},\mathcal{D}_{1})$
$\displaystyle=\left(\tilde{\partial}_{2}((\iota e-\tau
e).\mathbf{q}_{0}.v)+C_{1}(\mathcal{D}_{1})\right)+B_{1}(\mathcal{D},\mathcal{D}_{1})$
$\displaystyle=-\left(\tilde{\partial}_{2}\left(\sum_{i}\varepsilon_{i}[e,e_{i}]\right)+C_{1}(\mathcal{D}_{1})\right)+B_{1}(\mathcal{D},\mathcal{D}_{1})$
$\displaystyle=B_{1}(\mathcal{D},\mathcal{D}_{1}),$

which proves the first claim.

An obvious consequence of (18) is that
$Im(\hat{\partial}_{2})/B_{1}(\mathcal{D},\mathcal{D}_{1})$ is a cyclic group
with generator
$\hat{\partial}_{2}([1,\mathbf{q}_{0},1])+B_{1}(\mathcal{D},\mathcal{D}_{1})$.
The key to proving our main theorem is that
$Im(\hat{\partial}_{2})/B_{1}(\mathcal{D},\mathcal{D}_{1})$ is infinite
cyclic. Before that, we need to do some preparatory work.

If we let $G_{1}$ be the group given by $\mathcal{P}_{1}$, then for every
$g\in G_{1}$, we let $(\mathcal{D}_{1},\mathbf{t},g)$ be the connected
component of $(\mathcal{D}_{1},\mathbf{t})$ corresponding to $g$, and let
$H_{1}(\mathcal{D}_{1},\mathbf{t},g)$ be the corresponding homology group. The
homology group $H_{1}(\mathcal{D}_{1},\mathbf{t})$ decomposes as a direct sum

$H_{1}(\mathcal{D}_{1},\mathbf{t})=\oplus_{g\in
G_{1}}H_{1}(\mathcal{D}_{1},\mathbf{t},g).$

Any 1-cycle $\varsigma$ now decomposes uniquely as

$\varsigma=\varsigma_{g_{1}}+\dots+\varsigma_{g_{n}}$

where $\varsigma_{g_{i}}\in Z_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})$, and
$\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t})$ writes uniquely as

$\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t})=(\varsigma_{g_{1}}+B_{1}(\mathcal{D}_{1},\mathbf{t},g_{1}))+\dots+(\varsigma_{g_{n}}+B_{1}(\mathcal{D}_{1},\mathbf{t},g_{n})).$

From [40] we know that each $H_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})$ is
isomorphic to $H_{1}(\mathcal{D}_{1},\mathbf{t},1)$ where the isomorphism

$\theta_{u_{i}}:H_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})\rightarrow
H_{1}(\mathcal{D}_{1},\mathbf{t},1)$

is defined by

$\varsigma_{g_{i}}+B_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})\mapsto\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D}_{1},\mathbf{t},1)$

where $u_{i}$ is any vertex in $(\mathcal{D}_{1},\mathbf{t},g_{i})$.

We let
$\psi_{1}:H_{1}(\mathcal{D}_{1},\mathbf{t},1)\rightarrow\tilde{\Pi}_{1}$ and
$\eta:\tilde{\Pi}\rightarrow H_{1}(\mathcal{D},\mathbf{t})$ be the isomorphism
of [40]. With these notations the following holds true.

######  Lemma 4.15.

For every $\varsigma\in Z_{1}(\mathcal{D}_{1},\mathbf{t},1)$,
$\hat{\varphi}\psi_{1}(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t},1))=\psi(\varsigma+B_{1}(\mathcal{D},\mathbf{t}))$.

###### Proof.

This follows easily from the definitions of $\psi,\psi_{1}$ and
$\hat{\varphi}$. Indeed, assume that

$\varsigma=\sum_{i\in I}z_{i}(u_{i},s_{i},1,v_{i}),$

and let

$J=\\{i\in I:s_{i}=r^{\varepsilon_{i}}_{i}\text{ where
}r^{\varepsilon_{i}}_{i}\in\mathbf{r}_{1}^{\pm 1}\\}.$

Then from the definitions of $\psi_{1}$ and $\psi$ we have that

$\psi_{1,0}(\varsigma)=\prod_{j\in
J}\mu_{1}\sigma_{1}(^{u_{j}}s_{j})^{z_{j}}\text{ and
}\psi_{0}(\varsigma)=\prod_{j\in J}\mu\sigma(^{u_{j}}s_{j})^{z_{j}}$ (19)

where the exponential notations of the right hand sides mean that if
$z_{j}<0$, then
$\mu_{1}\sigma_{1}(^{u_{j}}s_{j})^{z_{j}}=\iota\left(\mu_{1}\sigma_{1}(^{u_{j}}s_{j})\right)^{|z_{j}|}$
and likewise,
$\mu\sigma(^{u_{j}}s_{j})^{z_{j}}=\iota\left(\mu\sigma(^{u_{j}}s_{j})\right)^{|z_{j}|}$.
We used the definitions of $\psi_{1,0}$ and $\psi_{0}$ by regarding
$\varsigma$ as a sum of 1-cycles arising from loops in
$(\mathcal{D}_{1},\mathbf{t},1)$. This is always possible due to lemma 5.1 of
[33]. Further we have that

$\displaystyle\hat{\varphi}\psi_{1}(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t},1))$
$\displaystyle=\hat{\varphi}(\psi_{1,0}(\varsigma))$ (from the definition of
$\psi_{1}$) $\displaystyle=\hat{\varphi}\left(\prod_{j\in
J}\mu_{1}\sigma_{1}(^{u_{j}}s_{j})^{z_{j}}\right)$ (from (19))
$\displaystyle=\prod_{j\in J}\mu\sigma(^{u_{j}}s_{j})^{z_{j}}$ (from the
definition of $\hat{\varphi}$) $\displaystyle=\psi_{0}(\varsigma)$ (from (19))
$\displaystyle=\psi(\varsigma+B_{1}(\mathcal{D},\mathbf{t}))$
$\displaystyle\text{(from the definition of $\psi$)},$

proving the lemma. ∎

With the decomposition $H_{1}(\mathcal{D}_{1},\mathbf{t})=\oplus_{g_{i}\in
G_{1}}H_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})$, consider the following
sequence of homomorphisms

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.8995pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\oplus_{g_{i}\in
G_{1}}H_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.32365pt\raise 6.38873pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.47238pt\hbox{$\scriptstyle{\oplus\theta_{u_{i}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 61.8995pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.8995pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\oplus_{g_{i}\in
G_{1}}H_{1}(\mathcal{D}_{1},\mathbf{t},1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
138.63872pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\oplus\psi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 159.64081pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
159.64081pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\oplus_{g_{i}\in
G_{1}}\tilde{\Pi}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
198.28487pt\raise 6.90279pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.31946pt\hbox{$\scriptstyle{\oplus\hat{\varphi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 219.4849pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
219.4849pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\oplus_{g_{i}\in
G_{1}}\tilde{\Pi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
261.45798pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 276.52896pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
276.52896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{1}(\mathcal{D},\mathbf{t})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where

$\gamma\left(\sum_{g_{i}}d_{g_{i}}\right)=\sum_{g_{i}}\eta(d_{g_{i}}),$

and write for short
$\chi=\gamma(\oplus\hat{\varphi})(\oplus\psi_{1})(\oplus\theta_{u_{i}})$.

######  Lemma 4.16.

For any element $\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t})\in
H_{1}(\mathcal{D}_{1},\mathbf{t})$, we have
$\chi(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t}))=\varsigma+B_{1}(\mathcal{D},\mathbf{t})$.

###### Proof.

If $\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t})$ is expressed as

$\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t})=\sum_{g_{i}}(\varsigma_{g_{i}}+B_{1}(\mathcal{D}_{1},\mathbf{t},g_{i})),$

then we have

$\displaystyle\chi(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t}))$
$\displaystyle=(\gamma(\oplus\hat{\varphi})(\oplus\psi_{1})(\oplus\theta_{u_{i}}))\left(\sum_{g_{i}}(\varsigma_{g_{i}}+B_{1}(\mathcal{D}_{1},\mathbf{t},g_{i}))\right)$
$\displaystyle=(\gamma(\oplus\hat{\varphi})(\oplus\psi_{1}))\left(\sum_{g_{i}}(\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D}_{1},\mathbf{t},1))\right)$
$\displaystyle=(\gamma(\oplus\hat{\varphi}))\left(\sum_{g_{i}}\psi_{1}(\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D}_{1},\mathbf{t},1))\right)$
$\displaystyle=\gamma\left(\sum_{g_{i}}\hat{\varphi}\psi_{1}(\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D}_{1},\mathbf{t},1))\right)$
$\displaystyle=\gamma\left(\sum_{g_{i}}\psi(\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D},\mathbf{t}))\right)$ (by lemma 4.15)
$\displaystyle=\sum_{g_{i}}\eta\psi(\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D},\mathbf{t}))$
$\displaystyle=\sum_{g_{i}}\varsigma_{g_{i}}\cdot
u_{i}^{-1}+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=\sum_{g_{i}}\varsigma_{g_{i}}+B_{1}(\mathcal{D},\mathbf{t})$
(by lemma 4.9) $\displaystyle=\varsigma+B_{1}(\mathcal{D},\mathbf{t}).$

∎

######  Theorem 4.17.

The subpresentation $\mathcal{P}_{1}$ is aspherical.

###### Proof.

We prove first that
$Im(\hat{\partial}_{2})/B_{1}(\mathcal{D},\mathcal{D}_{1})$ is infinite
cyclic. If we assume the contrary, then there is $z\in\mathbb{Z}$ and a
2-chain $\xi\in C_{2}(\mathcal{D})$ such that

$z\tilde{\partial}_{2}([1,\mathbf{q}_{0},1])+C_{1}(\mathcal{D}_{1})=\tilde{\partial}_{2}(\xi)+C_{1}(\mathcal{D}_{1}).$

It follows that
$\varsigma=z\tilde{\partial}_{2}([1,\mathbf{q}_{0},1])-\tilde{\partial}_{2}(\xi)$
is a 1-cycle in $C_{1}(\mathcal{D}_{1})$ and therefore $\varsigma\in
Z_{1}(\mathcal{D}_{1},\mathbf{t})$. Now we see that

$\displaystyle\chi(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t}))$
$\displaystyle=\varsigma+B_{1}(\mathcal{D},\mathbf{t})$ (from lemma 4.16)
$\displaystyle=(z\tilde{\partial}_{2}([1,\mathbf{q}_{0},1])-\tilde{\partial}_{2}(\xi))+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=z\tilde{\partial}_{2}([1,\mathbf{q}_{0},1])+B_{1}(\mathcal{D},\mathbf{t})$
$\displaystyle=zq(r_{0},1)+B_{1}(\mathcal{D},\mathbf{t}).$

But from proposition 4.13 it follows that

$((\oplus\hat{\varphi})(\oplus\psi_{1})(\oplus\theta_{u_{i}}))(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t}))=\sum_{g_{i}}v_{g_{i}},$

where $v_{g_{i}}\in\hat{\mathfrak{A}}_{1}$, say
$v_{g_{i}}=\mu\sigma(^{u_{i}}r_{i}({{}^{u_{i}}}r_{i})^{-1})^{z_{i}}$ where
$r_{i}\in\mathbf{r}_{1}$, $u_{i}\in F$ and $z_{i}\in\mathbb{Z}$. Now we have

$\displaystyle\chi(\varsigma+B_{1}(\mathcal{D}_{1},\mathbf{t}))$
$\displaystyle=\gamma\left(\sum_{g_{i}}v_{g_{i}}\right)$
$\displaystyle=\gamma\left(\sum_{g_{i}}\mu\sigma(^{u_{i}}r_{i}({{}^{u_{i}}}r_{i})^{-1})^{z_{i}}\right)$
$\displaystyle=\sum_{g_{i}}\eta(\mu\sigma(^{u_{i}}r_{i}({{}^{u_{i}}}r_{i})^{-1})^{z_{i}})$
$\displaystyle=\sum_{g_{i}}z_{i}(q(r_{i},u_{i})+B_{1}(\mathcal{D},\mathbf{t}))$
$\displaystyle\text{(Proposition \ref{th})}.$

Recollecting, we have in $H_{1}(\mathcal{D},\mathbf{t})$ the equality