# The splitting of generalisations of the Fadell-Neuwirth short exact sequence

Daciberg Lima Gonçalves, John Guaschi, Carolina de Miranda E Pereiro

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## To cite this version:

Daciberg Lima Gonçalves, John Guaschi, Carolina de Miranda E Pereiro. The splitting of generalisations of the Fadell-Neuwirth short exact sequence. 2025. hal-05233425

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https://hal.science/hal-05233425v1

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Distributed under a Creative Commons Attribution 4.0 International License
# THE SPLITTING OF GENERALISATIONS OF THE FADELL-NEUWIRTH
SHORT EXACT SEQUENCE

DACIBERG LIMA GONÇALVES, JOHN GUASCHI, AND CAROLINA DE MIRANDA E PEREIRO

**ABSTRACT.** We study some generalisations to mixed braid groups of the Fadell-Neuwirth short exact sequence and the possible splitting of this sequence. In certain cases, we determine conditions under which the projection from the mixed braid group $B_{n_1, \dots, n_k}(M)$ to $B_{n_1, \dots, n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_1, \dots, n_k, q \in \mathbb{N}$, and $1 \le q \le k-1$. For $k \ge 2$ and $q = k-1$, we show that this projection admits a section if and only if $n_1$ divides $n_i$ for all $i = 2, \dots, k$. We present some partial conclusions in the case $k \ge 3$ and $q = 1$. To obtain our results, we compute and make use of suitable mixed braid groups of $M$, as well as certain key quotients that play a central rôle in our analysis.

## 1. INTRODUCTION

The braid groups of the disc, also known as Artin braid groups, were introduced by E. Artin [1]. If $n \in \mathbb{N}$, the $n$-string Artin braid group, denoted by $B_n$, is generated by the elements $\sigma_1, \dots, \sigma_{n-1}$, illustrated in Figure 1, and known as the Artin generators of $B_n$, that are subject to the Artin relations:

$$ \left\{ \begin{array}{ll} \sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} & \text{for all } 1 \le i \le n-2 \\ \sigma_i\sigma_j = \sigma_j\sigma_i & \text{if } 1 \le i, j \le n-1 \text{ and } |i-j| \ge 2. \end{array} \right. $$

<figure><img src="image_3.png" /><figcaption>FIGURE 1. The Artin generator $\sigma_i$</figcaption></figure>

These groups were later generalised by Fox and Neuwirth using configuration spaces as follows [10]. Let $M$ be a connected surface, and let $n \in \mathbb{N}$. The $n^{th}$ configuration space of $M$, denoted by $F_n(M)$, is defined by:

$$ F_n(M) = \{(x_1, \dots, x_n) : x_i \in M, \text{ and } x_i \neq x_j \text{ if } i \neq j, i, j = 1, \dots, n\}. $$

The $n$-string pure braid group $P_n(M)$ of $M$ is defined by $P_n(M) = \pi_1(F_n(M))$. The symmetric group $S_n$ on $n$ letters acts freely on $F_n(M)$ by permuting coordinates, and the $n$-string (full) braid group $B_n(M)$ of $M$ is defined by $B_n(M) = \pi_1(D_n(M))$, where $D_n(M) = F_n(M)/S_n$. This gives rise to the following short exact sequence:

$$ 1 \to P_n(M) \to B_n(M) \to S_n \to 1. \qquad (1.1) $$

If $M$ is the 2-disc (or the plane $\mathbb{R}^2$), then $B_n(M)$ (resp. $P_n(M)$) is isomorphic to $B_n$ (resp. to the Artin pure braid group $P_n$). If $M$ is a compact surface without boundary, by the work of Fadell and Neuwirth [8], the projection $p: F_{n+m}(M) \to F_n(M)$ defined by $p(x_1, \dots, x_{n+m}) = (x_1, \dots, x_n)$ for all $(x_1, \dots, x_{n+m}) \in F_{n+m}(M)$ is a locally-trivial fibration whose fibre may be identified with

<footnote>Date: 28th July 2025.</footnote>
$F_m(M \setminus \{x_1, \dots, x_n\})$. Taking the associated long exact sequence in homotopy of this fibration, we obtain the Fadell-Neuwirth short exact sequence:

$$1 \to P_m(M \setminus \{x_1, \dots, x_n\}) \to P_{n+m}(M) \xrightarrow{p_*} P_n(M) \to 1, \quad (1.2)$$

where $p_*$ is the homomorphism induced by $p$, $m \ge 1$, and $n \ge 3$ if $M = \mathbb{S}^2$ [7, 9], $n \ge 2$ if $M$ is the real projective plane $\mathbb{RP}^2$ [22], and $n \ge 1$ otherwise [8]. Geometrically, the homomorphism $p_*$ may be interpreted as the map that forgets the last $m$ strings of a pure braid with $m+n$ strings. If $M$ has boundary, $p$ is not a fibration, but the short exact sequence (1.2) nevertheless exists (see for example the proof of [15, Theorem 2(a)]). The sequence (1.2) is an important tool in the study of surface braid groups. Its use leads to presentations of the corresponding pure braid groups, and it allows us to compute their centre and their possible torsion elements, and to analyse their residual properties. In the case of the Artin pure braid groups, (1.2) splits, and gives rise to a decomposition of $P_n$ as a repeated semi-direct product of free groups, known as the Artin ‘combing’ operation [2]. This is the principal result of Artin’s classical theory of braid groups, from which one may obtain normal forms and a solution to the word problem in $B_n$. One of the principal problems regarding (1.2), known as the splitting problem, is to determine for which surfaces and which values of $n$ and $m$ the sequence splits [5, 8, 9, 11, 14, 15, 22]. If (1.2) splits for all $n, m \in \mathbb{N}$, the group $P_{n+m}(M)$ may be decomposed as an iterated semi-direct product, which aids in the study of its properties. In contrast with the case of compact surfaces without boundary of higher genus, in the cases where $M$ is the 2-torus $\mathbb{T}$ or the Klein bottle $\mathbb{K}$, the fibration $p$ admits a cross-section arising from the existence of a non-vanishing vector field on $M$ for all $n, m \in \mathbb{N}$ [8], which gives rise to an algebraic section for $p_*$. Recall that if the fibre of the fibration is an Eilenberg-MacLane space, which is the case here, then the existence of a section for $p_*$ is equivalent to that of a cross-section for $p$ [3, 12, 23].

With respect to the splitting problem, it is natural to study the corresponding full braid groups. Although the short exact sequence (1.2) does not generalise directly to $B_{n+m}(M)$ directly, the projection $p_*$ extends to an intermediate subgroup of $B_{n+m}(M)$, namely the mixed braid group $B_{n,m}(M)$ that is defined by $B_{n,m}(M) = \pi_1(D_{n,m}(M))$, where $D_{n,m}(M) = F_{n+m}(M)/(S_n \times S_m)$. In this case, if $M$ is a compact surface without boundary, the map $p: F_{n+m}(M)/(S_n \times S_m) \to F_n(M)/S_n$ given by forgetting the last $m$ coordinates is a fibration whose fibre may be identified with $F_m(M \setminus \{x_1, \dots, x_n\})/S_n$. As in the case of the pure braid groups, applying the associated long exact sequence in homotopy to this fibration, we obtain the generalised Fadell-Neuwirth short exact sequence:

$$1 \to B_m(M \setminus \{x_1, \dots, x_n\}) \to B_{n,m}(M) \xrightarrow{p_*} B_n(M) \to 1, \quad (1.3)$$

where $p_*$ is the homomorphism induced by $p$, $m \ge 1$, and $n \ge 3$ if $M = \mathbb{S}^2$, $n \ge 2$ if $M = \mathbb{RP}^2$, and $n \ge 1$ otherwise. Once more, the short exact sequence (1.3) exists even if $M$ has boundary. We are interested in deciding whether this sequence splits. Once more, the existence of a section for $p_*$ is equivalent to that of a cross-section for $p$. In the case of the Artin braid groups, it is easy to see that (1.3) splits for all $n$ and $m$. The case where $M = \mathbb{S}^2$ was originally studied in [13], with further results being obtained in [6], and the case where $M = \mathbb{RP}^2$ has been analysed in [19]. The case of orientable surfaces has been studied recently in [20]. In this paper, we solve the splitting problem with respect to (1.3) for the cases where $M = \mathbb{T}$ or $M = \mathbb{K}$, the precise statement being as follows.

**Theorem 1.1.** Let $M$ be the 2-torus or the Klein bottle. Then the generalised Fadell-Neuwirth short exact sequence (1.3) splits if and only if $n$ divides $m$.

Observe that Theorem 1.1 implies the result of [20, Theorem 1] in the case where $M = \mathbb{T}$.

To prove that the condition of the statement of Theorem 1.1 is sufficient, we make use of the existence of a non-vanishing vector field on $\mathbb{T}$ and $\mathbb{K}$ to construct a geometric section on the level of the associated configuration spaces, which implies the existence of an algebraic section for (1.3). The proof of the converse is algebraic in nature, and for this we determine presentations of the groups appearing in (1.3) as well as some of their quotients.
The mixed braid groups defined above may be generalised to any number of factors. To do so, for $k, n_1, \dots, n_k \in \mathbb{N}$, let:

$$B_{n_1, \dots, n_k}(M) = \pi_1(D_{n_1, \dots, n_k}(M)),$$

where $D_{n_1, \dots, n_k}(M) = F_{n_1+\dots+n_k}(M)/(S_{n_1} \times \dots \times S_{n_k})$. One may obtain short exact sequences similar to that of (1.3) by forgetting one, or several blocks, of strings. More precisely, if $q = 1, \dots, k-1$, then there exists a short exact sequence:

$$1 \to \mathrm{Ker}(p_*) \to B_{n_1, \dots, n_k}(M) \xrightarrow{p_*} B_{n_1, \dots, n_{k-q}}(M) \to 1, \quad (1.4)$$

where $p_*$ is the homomorphism induced by the map $p: D_{n_1, \dots, n_k}(M) \to D_{n_1, \dots, n_{k-q}}(M)$ that forgets the last $n_{k-q+1}+\dots+n_k$ points, and where $\mathrm{Ker}(p_*)$ may be identified with the group $B_{n_{k-q+1}, \dots, n_k}(M \setminus \{x_1, \dots, x_{n_1+\dots+n_{k-q}}\})$. Once more, our aim is to decide when (1.4) splits. As in the situation of Theorem 1.1, in this paper we restrict our attention to the cases where $M = \mathbb{T}$ or $M = \mathbb{K}$, in which case the existence of a splitting for (1.4) is equivalent to that of a geometric section on the level of the corresponding configuration spaces. As a first step in the resolution of this splitting problem, we analyse the extreme values of $q$, namely $q=1$ and $q=k-1$. In the latter case, we solve the problem completely (and the answer is similar to that of the case $k=2$ of Theorem 1.1), while in the former case, we give a partial answer.

**Theorem 1.2.** Let $M$ be the 2-torus or the Klein bottle. If $q = k-1$, with the above notation, the short exact sequence (1.4) splits if and only if $n_1$ divides $n_i$ for all $i = 2, \dots, k$.

The case $q=1$ is more subtle. We currently have the following partial result.

**Theorem 1.3.** Let $M$ be the 2-torus or the Klein bottle, let $k \ge 2$, and let $n_1, \dots, n_k \in \mathbb{N}$.

(a) If there exist $l_1, \dots, l_{k-1} \in \mathbb{N}$ such that $n_k = l_1 n_1 + \dots + l_{k-1} n_{k-1}$, then the homomorphism $p_*: B_{n_1, \dots, n_k}(M) \to B_{n_1, \dots, n_{k-1}}(M)$ admits a section.

(b) If the homomorphism $p_*: B_{n_1, \dots, n_k}(M) \to B_{n_1, \dots, n_{k-1}}(M)$ admits a section, then there exist $l_1, \dots, l_{k-1} \in \mathbb{Z}$ such that $n_k = l_1 n_1 + \dots + l_{k-1} n_{k-1}$.

The obstruction that occurs in part (b) of Theorem 1.3 to proving the converse of part (a) is that our (algebraic) methods do not allow us to decide whether the integers $l_1, \dots, l_{k-1}$ are positive. In theory, some of these integers could be negative, but in that case, the section does not arise as the induced homomorphism of a cross-section for the map $p$. However, the following result shows that in one of the simplest such situations, where $k=3$, $q=1$, $n_1, n_2 \ge 2$, $n_3=1$, and $M = \mathbb{T}$ or $\mathbb{K}$, the converse of Theorem 1.3(a) holds, and if $n_1$ and $n_2$ are coprime then the conclusion of Theorem 1.3(b) is also satisfied.

**Theorem 1.4.** Let $M$ be either the 2-torus or the Klein bottle, and let $t, s \ge 2$. Then the projection $p_*: B_{t,s,1}(M) \to B_{t,s}(M)$ does not admit a section.

This gives some evidence to support the conjecture that the converse to Theorem 1.3(a) holds in general. Note that if either $t=1$ or $s=1$ then $p_*: B_{t,s,1}(M) \to B_{t,s}(M)$ admits a section by Theorem 1.3(a).

**Acknowledgements:** the first author was partially supported by the FAPESP Projeto Temático-FAPESP Topologia Algébrica, Geométrica e Diferencial no. 2022/16455-6 (São Paulo-Brazil). His mission to the Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie during the period 14th–27th of May 2023 was also supported by the French-Brazilian network (Réseau Franco-Brésilien de Mathématiques).

## 2. GROUP PRESENTATIONS

In this section, we give presentations of some of the groups that will be studied in this paper. If $M = \mathbb{T}$ or $\mathbb{K}$, we will make use of the presentations of $P_n(M)$ and $B_n(M)$ that appeared in [21] and [16, Theorems 2.1 and 2.2]. Geometric representatives of the generators of $P_n(M)$ are illustrated
in Figure 2, where the figures represent the projection of the braids onto $M$, so that the constant paths in each figure correspond to vertical strings of the braid.

<figure><img src="image_4.png" /><figcaption>FIGURE 2. The generators of $P_n(\mathbb{T})$ and $P_n(\mathbb{K})$</figcaption></figure>

**Theorem 2.1** ([16]). Let $n \ge 1$, and let $M$ be the torus $\mathbb{T}$ or the Klein bottle $\mathbb{K}$. The following constitutes a presentation of the pure braid group $P_n(M)$ of $M$: generating set: $\{a_i, b_i, i = 1, \dots, n\} \cup \{C_{i,j}, 1 \le i < j \le n\}$. relations:

(1) $a_i a_j = a_j a_i$, where $1 \le i < j \le n$.

(2) $a_i^{-1} b_j a_i = b_j a_j C_{i,j}^{-1} C_{i+1,j} a_j^{-1}$, where $1 \le i < j \le n$.

(3) $a_i^{-1} C_{j,k} a_i = \begin{cases} C_{j,k}, & \text{where } 1 \le i < j < k \le n \text{ or } 1 \le j < k < i \le n \\ a_k C_{i+1,k}^{-1} C_{i,k} a_k^{-1} C_{j,k} C_{i,k}^{-1} C_{i+1,k}, & \text{where } 1 \le j \le i < k \le n. \end{cases}$

(4) $C_{i,l}^{-1} C_{j,k} C_{i,l} = \begin{cases} C_{j,k}, & \text{where } 1 \le i < l < j < k \le n \text{ or } 1 \le j \le i < l < k \le n \\ C_{i,k} C_{l+1,k}^{-1} C_{l,k} C_{i,k}^{-1} C_{j,k} C_{l,k}^{-1} C_{l+1,k}, & \text{where } 1 \le i < j \le l < k \le n. \end{cases}$

(5) $\begin{cases} \prod_{j=i+1}^{n} C_{i,j}^{-1} C_{i+1,j} = a_i b_i C_{1,i} a_i^{-1} b_i^{-1}, & \text{where } 1 \le i \le n \text{ and } M = \mathbb{T} \\ \prod_{j=i+1}^{n} C_{i,j} C_{i+1,j}^{-1} = b_i C_{1,i} a_i^{-1} b_i^{-1} a_i^{-1}, & \text{where } 1 \le i \le n \text{ and } M = \mathbb{K}. \end{cases}$

(6) $b_j b_i = \begin{cases} b_i b_j, & \text{where } 1 \le i < j \le n \text{ and } M = \mathbb{T} \\ b_i b_j C_{i,j} C_{i+1,j}^{-1}, & \text{where } 1 \le i < j \le n \text{ and } M = \mathbb{K}. \end{cases}$

(7) $b_i^{-1} a_j b_i = \begin{cases} a_j b_j C_{i,j} C_{i+1,j}^{-1} b_j^{-1}, & \text{where } 1 \le i < j \le n \text{ and } M = \mathbb{T} \\ a_j b_j (C_{i,j} C_{i+1,j}^{-1})^{-1} b_j^{-1}, & \text{where } 1 \le i < j \le n \text{ and } M = \mathbb{K}. \end{cases}$

(8) $b_i^{-1} C_{j,k} b_i = \begin{cases} C_{j,k}, & \text{where } 1 \le i < j < k \le n \text{ or } 1 \le j < k < i \le n \text{ and } M = \mathbb{T} \\ C_{i+1,k} C_{i,k}^{-1} C_{j,k} b_k C_{i,k} C_{i+1,k}^{-1} b_k^{-1}, & \text{where } 1 \le j \le i < k \le n \text{ and } M = \mathbb{K}. \end{cases}$

**Remark 2.2.** For $1 \le i < j \le n$, the elements $C_{i,j}$ may be described in terms of the Artin generators of $B_n(M)$, where $C_{i,i+1} = \sigma_i^2$, for $1 \le i \le n-1$, and $C_{i,j} = \sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+1}\sigma_i^2\sigma_{i+1}\cdots\sigma_{j-2}\sigma_{j-1}$ for $1 \le i$ and $i+1 < j \le n$. If $i=j$, then by convention we take $C_{i,i}$ to be the trivial braid. So if
$3 \le j \le n$ then using the Artin relations, we obtain:

$$
\begin{align*}
C_{1,j}C_{2,j}^{-1} &= (\sigma_{j-1} \cdots \sigma_2 \sigma_1^2 \sigma_2 \cdots \sigma_{j-1})(\sigma_{j-1}^{-1} \cdots \sigma_3^{-1} \sigma_2^{-2} \sigma_3^{-1} \cdots \sigma_{j-1}^{-1}) \\
&= \sigma_{j-1} \cdots \sigma_2 \sigma_1^2 \sigma_2^{-1} \cdots \sigma_{j-1}^{-1} = \sigma_1^{-1} \cdots \sigma_{j-2}^{-1} \sigma_{j-1}^2 \sigma_{j-2} \cdots \sigma_1,
\end{align*}
$$

and

$$
\begin{align*}
C_{1,j}^{-1} C_{2,j} &= (\sigma_{j-1}^{-1} \cdots \sigma_2^{-1} \sigma_1^{-2} \sigma_2^{-1} \cdots \sigma_{j-1}^{-1})(\sigma_{j-1} \cdots \sigma_3 \sigma_2^2 \sigma_3 \cdots \sigma_{j-1}) \\
&= \sigma_{j-1}^{-1} \cdots \sigma_2^{-1} \sigma_1^{-2} \sigma_2 \cdots \sigma_{j-1} = \sigma_1 \cdots \sigma_{j-2} \sigma_{j-1}^{-2} \sigma_{j-2}^{-1} \sigma_1^{-1}.
\end{align*}
$$

Thus for all $2 \le k \le n$:

$$
\begin{align}
\prod_{j=2}^{k} C_{1,j} C_{2,j}^{-1} &= \sigma_1^2 (\sigma_1^{-1} \sigma_2^2 \sigma_1) (\sigma_2^{-1} \sigma_2^{-1} \sigma_3^2 \sigma_2 \sigma_1) \cdots (\sigma_1^{-1} \cdots \sigma_{k-2}^{-1} \sigma_{k-1}^2 \sigma_{k-2} \cdots \sigma_1) \nonumber \\
&= \sigma_1 \cdots \sigma_{k-2} \sigma_{k-1}^2 \sigma_{k-2} \cdots \sigma_1, \tag{2.1}
\end{align}
$$

and

$$
\begin{align*}
\prod_{j=2}^{k} C_{1,j}^{-1} C_{2,j} &= \sigma_1^{-2} (\sigma_1 \sigma_2^{-2} \sigma_1^{-1}) (\sigma_2 \sigma_2 \sigma_3^{-2} \sigma_2^{-1} \sigma_1^{-1}) \cdots (\sigma_1 \cdots \sigma_{k-2} \sigma_{k-1}^{-2} \sigma_{k-2}^{-1} \cdots \sigma_1^{-1}) \\
&= \sigma_1^{-1} \cdots \sigma_{k-2}^{-1} \sigma_{k-1}^{-2} \sigma_{k-2}^{-1} \cdots \sigma_1^{-1} = \left( \prod_{j=2}^{k} C_{1,j} C_{2,j}^{-1} \right)^{-1}.
\end{align*}
$$

**Theorem 2.3** ([16]). Let $n \ge 1$, and let $M$ be the torus $T$ or the Klein bottle $K$. The following constitutes a presentation of the braid group $B_n(M)$ of $M$:
generating set: $\{a, b, \sigma_1, \dots, \sigma_{n-1}\}$.
relations:

$$
(1) \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \quad \text{if } i = 1, \dots, n-2.
$$

$$
(2) \sigma_j \sigma_i = \sigma_i \sigma_j \quad \text{if } 1 \le i, j \le n-1 \text{ and } |i-j| \ge 2.
$$

$$
(3) a\sigma_j = \sigma_j a \quad \text{if } 2 \le j \le n-1.
$$

$$
(4) b\sigma_j = \sigma_j b \quad \text{if } 2 \le j \le n-1.
$$

$$
(5) b^{-1}\sigma_1 a = \sigma_1 a\sigma_1 b^{-1}\sigma_1.
$$

$$
(6) a(\sigma_1 a \sigma_1) = (\sigma_1 a \sigma_1) a.
$$

$$
(7) \left\{
\begin{array}{ll}
b(\sigma_1^{-1} b \sigma_1^{-1}) = (\sigma_1^{-1} b \sigma_1^{-1}) b & \text{if } M = \mathbb{T} \\
b(\sigma_1^{-1} b \sigma_1) = (\sigma_1^{-1} b \sigma_1^{-1}) b & \text{if } M = \mathbb{K}.
\end{array}
\right.
$$

$$
(8) \sigma_1 \sigma_2 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_2 \sigma_1 = \begin{cases} bab^{-1}a^{-1} & \text{if } M = \mathbb{T} \\ ba^{-1}b^{-1}a^{-1} & \text{if } M = \mathbb{K}. \end{cases}
$$

Remark 2.4. In terms of the generators of $P_n(M)$, the generators $a$ and $b$ of Theorem 2.3 are equal to the generators $a_1$ and $b_1$ of Theorem 2.1 respectively.

In order to obtain a presentation of $B_m(M \setminus \{x_1, \dots, x_n\})$, we require first a presentation of $P_m(M \setminus \{x_1, \dots, x_n\})$.

**Proposition 2.5.** Let $M = \mathbb{T}$ or $M = \mathbb{K}$, $n, m \ge 1$. The group $P_m(M \setminus \{x_1, \dots, x_n\})$ admits the following presentation:
generating set: $\{a_i, b_i, n+1 \le i \le n+m\} \cup \{C_{i,j}, 1 \le i < j, n+1 \le j \le n+m\}$.
relations:

$$
(1) a_i a_j = a_j a_i, \text{ where } n+1 \le i < j \le n+m.
$$

$$
(2) a_i^{-1} b_j a_i = b_j a_j C_{i,j}^{-1} C_{i+1,j} a_j^{-1}, \text{ where } n+1 \le i < j \le n+m.
$$
$$
(3) a_i^{-1}C_{j,k}a_i = \begin{cases} C_{j,k} & \text{if } i < j < k \text{ or } j < k < i \\ a_k C_{i+1,k}^{-1}C_{i,k}a_k^{-1}C_{j,k}C_{i,k}^{-1}C_{i+1,k} & \text{if } j \le i < k, \end{cases} \\
\text{where } n+1 \le i \le n+m-1, 1 \le j < k \le n+m \text{ and } n+1 \le k.
$$

$$
(4) C_{i,l}^{-1}C_{j,k}C_{i,l} = \begin{cases} C_{j,k} & \text{if } i < l < j < k \text{ or } j \le i < l < k \\ C_{i,k}C_{l+1,k}^{-1}C_{l,k}C_{i,k}^{-1}C_{j,k}C_{l,k}^{-1}C_{l+1,k} & \text{if } i \le j \le l < k, \end{cases} \\
\text{where } 1 \le i, j, k, l \le n+m \text{ and } n+1 \le k, l.
$$

$$
(5) \quad \left\{
\begin{array}{l}
\displaystyle \prod_{j=i+1}^{n+m} C_{i,j}^{-1} C_{i+1,j} = a_i b_i C_{1,i} a_i^{-1} b_i^{-1} \quad \text{if } n+1 \le i \le n+m \text{ and } M = \mathbb{T} \\
\displaystyle \prod_{j=i+1}^{n+m} C_{i,j} C_{i+1,j}^{-1} = b_i C_{1,i} a_i^{-1} b_i^{-1} a_i^{-1} \quad \text{if } n+1 \le i \le n+m \text{ and } M = \mathbb{K}.
\end{array}
\right.
$$

$$
(6) \quad b_j b_i = \begin{cases} b_i b_j & \text{if } n+1 \le i < j \le n+m \text{ and } M = \mathbb{T} \\ b_i b_j C_{i,j} C_{i+1,j}^{-1} & \text{if } n+1 \le i < j \le n+m \text{ and } M = \mathbb{K}. \end{cases}
$$

$$
(7) b_i^{-1} a_j b_i = \begin{cases} a_j b_j C_{i,j} C_{i+1,j}^{-1} b_j^{-1} & \text{if } n+1 \le i < j \le n+m \text{ and } M = \mathbb{T} \\ a_j b_j (C_{i,j} C_{i+1,j}^{-1})^{-1} b_j^{-1} & \text{if } n+1 \le i < j \le n+m \text{ and } M = \mathbb{K}. \end{cases}
$$

$$
(8) \quad b_i^{-1} C_{j,k} b_i = \left\{
\begin{array}{ll}
\begin{cases} C_{j,k} & \text{if } i < j < k \text{ or } j < k < i \\ C_{i+1,k} C_{i,k}^{-1} C_{j,k} b_k C_{i,k} C_{i+1,k}^{-1} b_k^{-1}, & \text{if } j \le i < k \end{cases} & \text{and } M = \mathbb{T} \\[1em]
\begin{cases} C_{j,k} & \text{if } i < j < k \text{ or } j < k < i \\ C_{i+1,k} C_{i,k}^{-1} C_{j,k} b_k (C_{i,k} C_{i+1,k}^{-1})^{-1} b_k^{-1} & \text{if } j \le i < k \end{cases} & \text{and } M = \mathbb{K},
\end{array}
\right.
\\[1em]
\text{where } n+1 \le i \le n+m, \ 1 \le j < k \le n+m \text{ and } n+1 \le k.
$$

Remark 2.6. With respect to the short exact sequence (1.2) and the presentation of $P_{n+m}(M)$ given by Theorem 2.1, the generating set of $P_m(M \setminus \{x_1, \dots, x_n\})$ given in Proposition 2.5 is obtained by taking those generators of $P_{n+m}(M)$ that belong to $P_m(M \setminus \{x_1, \dots, x_n\})$, and the relations of $P_m(M \setminus \{x_1, \dots, x_n\})$ are those relations of $P_{n+m}(M)$ that contain only elements of the given generating set. Another way of expressing this is that the presentation of $P_{n+m}(M)$ of Theorem 2.1 is obtained by applying the methods of [17, Proposition 1, p. 139] to the short exact sequence (1.2), where $P_m(M \setminus \{x_1, \dots, x_n\})$ is taken to be equipped with the presentation given by Proposition 2.5.

**Proof of Proposition 2.5.** We prove the result by induction on $m \ge 1$. If $m=1$ then a generating set for $P_1(M \setminus \{x_1, \dots, x_n\})$ is $\{a_{n+1}, b_{n+1}, C_{i,n+1}, 1 \le i \le n\}$, subject to the (single) surface relation $a_{n+1}b_{n+1}C_{1,n+1}a_{n+1}^{-1}b_{n+1}^{-1} = 1$ (resp. $b_{n+1}C_{1,n+1}a_{n+1}^{-1}b_{n+1}^{-1}a_{n+1}^{-1} = 1$) if $M = \mathbb{T}$ (resp. $M = \mathbb{K}$) (this is relation (5) of Theorem 2.1 in the case $i=n+1$), which is the presentation given in the statement of the proposition (in the case $m=1$, note that the only relation of (1)–(8) that exists is relation (5)).

So let $m \ge 1$, and suppose that the presentation of the statement is valid for $P_m(M \setminus \{x_1, \dots, x_n\})$. Making use of the short exact sequence (1.2) for both $M$ and $M \setminus \{x_1, \dots, x_n\}$, we obtain the following commutative diagram of short exact sequences:

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    & \multicolumn{2}{c}{\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {1};
        \node at (-0.5,0.2) {1};
        \node at (0.5,1.2) {};
        \node at (-0.5,1.2) {};
    \end{tikzpicture}} & \\
    \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$P_1(M \setminus \{x_1, \dots, x_{n+m}\})$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$P_1(M \setminus \{x_1, \dots, x_{n+m}\})$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$1$};
        \node at (-0.5,0.2) {$1$};
    \end{tikzpicture} & \\
    \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$1 \to P_{m+1}(M \setminus \{x_1, \dots, x_n\})$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$P_{n+m+1}(M) \xrightarrow{p_*} P_n(M) \to 1$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$1 \to P_m(M \setminus \{x_1, \dots, x_n\})$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \\
    \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$1 \to P_m(M \setminus \{x_1, \dots, x_n\})$};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$P_{n+m}(M) \xrightarrow{p_*} P_n(M) \to 1$,};
        \node at (0.5,1.2) {};
    \end{tikzpicture} & \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
        \node (down) at (0,0) {};
        \node (up) at (0,1) {};
        \draw[->] (down) -- (up);
        \node at (0.5,0.2) {$1$};
        \node at (-0.5,0.2) {$1$};
    \end{tikzpicture} &
\end{tikzcd}
\qquad (2.2)
$$

where each of the homomorphisms $p_*$ is that of (1.2) for either $M$ or $M \setminus \{x_1, \dots, x_n\}$, obtained by forgetting the appropriate number of strings. We now apply the methods of [17, Proposition 1,
p. 139] to the leftmost column of (2.2). Taking the union of the set {$a_i, b_i, n + 1 \le i \le n + m$} $\cup \{C_{i,j}, 1 \le i < j, n + 1 \le j \le n + m\}$ of coset representatives in $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ of the given generating set of $P_m(M \setminus \{x_1, \dots, x_n\})$ with the generating set {$a_{n+m+1}, b_{n+m+1}, C_{i,n+m+1}, 1 \le i \le n + m$} of $P_1(M \setminus \{x_1, \dots, x_{n+m}\})$, we obtain the generating set of $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ of the statement. The corresponding relations are obtained as follows:

• all of the relations (1)–(8) of $P_m(M \setminus \{x_1, \dots, x_n\})$ lift directly to relations of $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$, with the exception of relation (5). We analyse the lift of this relation in $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$. Considering the inclusion of this group in $P_{n+m+1}(M)$, and using relation (5) of Theorem 2.1, for $n + 1 \le i \le n + m$, we have:

$$ \left( \prod_{j=i+1}^{n+m} C_{i,j}^{-1} C_{i+1,j} \right)^{-1} a_i b_i C_{1,i} a_i^{-1} b_i^{-1} = C_{i,n+m+1}^{-1} C_{i+1,n+m+1} \quad \text{if } M = \mathbb{T} $$

$$ \left( \prod_{j=i+1}^{n+m} C_{i,j} C_{i+1,j}^{-1} \right)^{-1} b_i C_{1,i} a_i^{-1} b_i^{-1} a_i^{-1} = C_{i,n+m+1} C_{i+1,n+m+1}^{-1} \quad \text{if } M = \mathbb{K}. $$

Since the right-hand side of each of these equalities belongs to $P_1(M \setminus \{x_1, \dots, x_{n+m}\})$, we obtain relation (5) in $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ for all $i = n + 1, \dots, n + m$. In particular, this yields relations (1)–(8) of $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ for the possible values of the indices excluding the cases where some of the indices are equal to $n + m + 1$.

• if $M = \mathbb{T}$ (resp. $M = \mathbb{K}$), the single relation $a_{n+m+1}b_{n+m+1}C_{1,n+m+1}a_{n+m+1}^{-1}b_{n+m+1}^{-1} = 1$ (resp. $b_iC_{1,i}a_i^{-1}b_i^{-1}a_i^{-1} = 1$) of $P_1(M \setminus \{x_1, \dots, x_{n+m}\})$ gives rise to relation (5) of $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ for the case $i = n + m + 1$.

• the conjugates of the elements of the generating set $P_1(M \setminus \{x_1, \dots, x_{n+m}\})$ by the coset representatives of the elements of the generating set of $P_m(M \setminus \{x_1, \dots, x_n\})$. Using the corresponding relations of Theorem 2.1, we obtain relations (1)–(4) and (6)–(8) of the given presentation in the cases where some of the indices are equal to $n + m + 1$.

Combining these relations, we obtain the presentation of $P_{m+1}(M \setminus \{x_1, \dots, x_n\})$ given in the statement of the proposition. $\square$

The next step is to obtain a presentation for the group $B_m(M \setminus \{x_1, \dots, x_n\})$ that appears in the short exact sequence (1.3).

**Proposition 2.7.** Let $M = \mathbb{T}$ or $M = \mathbb{K}$, $n \ge 1$ and $m \ge 2$. The group $B_m(M \setminus \{x_1, \dots, x_n\})$ admits the following presentation: generating set: {$a_i, b_i, n + 1 \le i \le n + m$} $\cup \{C_{i,j}, 1 \le i < j, n + 1 \le j \le n + m\} \cup \{\sigma_{n+1}, \dots, \sigma_{n+m-1}\}$. relations:

(1) relations (1)–(8) of Proposition 2.5.

(2) $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ if $i = n + 1, \dots, n + m - 2$.

(3) $\sigma_i\sigma_j = \sigma_j\sigma_i$ if $n + 1 \le i, j \le n + m - 1$ and $|i - j| \ge 2$.

(4) $\sigma_i^{-1}a_j\sigma_i = \begin{cases} a_j & \text{if } j \neq i, i + 1 \\ \sigma_i^{-2}a_{i+1} & \text{if } j = i \\ a_i\sigma_i^2 & \text{if } j = i + 1 \end{cases}$ where $n + 1 \le i \le n + m - 1$ and $n + 1 \le j \le n + m$.

(5) $\sigma_i^{-1}b_j\sigma_i = \begin{cases} b_j & \text{if } j \neq i, i + 1 \\ b_{i+1}\sigma_i^2 & \text{if } j = i \\ \sigma_i^{-2}b_i & \text{if } j = i + 1 \end{cases}$ where $n + 1 \le i \le n + m - 1$ and $n + 1 \le j \le n + m$.
$$ (6) \quad \sigma_i^{-1} C_{l,j} \sigma_i = \begin{cases} C_{l,j} & \text{if } i+1 < l < j, l \le i < j-1 \text{ or } l < j < i \\ C_{j,j+1}^{-1} C_{l,j+1} & \text{if } i = j \\ C_{l,j-1} C_{j-1,j} & \text{if } i = j-1 \\ C_{l-1,j} C_{l,j}^{-1} C_{l+1,j} & \text{if } l = i+1 \end{cases} $$

where $n+1 \le i \le n+m-1$, $1 \le l < j$ and $n+1 \le j \le n+m$.

$$ (7) \quad C_{i,i+1} = \sigma_i^2, \text{ where } n+1 \le i \le n+m-1. $$

**Proof.** We apply the methods of [17, Proposition 1, p. 139] to the short exact sequence (1.1), where we replace $M$ by $M \setminus \{x_1, \dots, x_n\}$ and $n$ by $m$. A generating set of $B_m(M \setminus \{x_1, \dots, x_n\})$ is given by the union of a generating set of $P_m(M \setminus \{x_1, \dots, x_n\})$ with a set of coset representatives for the projection $B_m(M \setminus \{x_1, \dots, x_n\}) \to S_m$ of a generating set of $S_m$, and by Theorem 2.1, we may take $\{a_i, b_i, n+1 \le i \le n+m\} \cup \{C_{i,j}, 1 \le i < j, n+1 \le j \le n+m\}$ and $\{\sigma_{n+1}, \dots, \sigma_{n+m-1}\}$ respectively for these generating sets, which yields the generating set of the statement.

The first type of relation among the elements of the given generating set is obtained by taking the relations of $P_m(M \setminus \{x_1, \dots, x_n\})$ given by Proposition 2.5, which are relations (1) of the statement. We obtain the second type of relation by rewriting the relations of $S_m$ in terms of the given coset representatives, and expressing the corresponding element as words in terms of the generators of $B_m(M \setminus \{x_1, \dots, x_n\})$. The group $S_m$ is generated by elements $s_1, \dots, s_{m-1}$, where for $i = 1, \dots, m-1$, $\sigma_i$ is a coset representative of $s_i$, and the generators are subject to the relations $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$ if $1 \le i \le m-2$, $s_i s_j = s_j s_i$ if $1 \le i, j \le m-1$ and $|i-j| \ge 2$, and $s_i^2 = 1$ if $1 \le i \le m-1$. This yields relations (2), (3) and (7) respectively. The third type of relation is obtained by writing the conjugates of the generators of the kernel by the coset representatives as words written entirely in terms of the generators of the kernel. This rewriting process may be carried out using the geometric description of the braids given in Figures 1 and 2 (see also [16, equation (5.7)]), which yields relations (4) and (5). We may use the Artin relations and Remark 2.2 to obtain relation (6). $\square$

**Remark 2.8.** The presentation of $B_m(M \setminus \{x_1, \dots, x_n\})$ given in Proposition 2.7 may be simplified by eliminating some of the generators ($a_i$ and $b_i$, where $i = n+1, \dots, n+m-1$, and $C_{i,j}$, where $n+1 < i < j < n+m$, for example), but we shall not do so here.

In what follows, we will make use of certain quotients of the group $B_m(M \setminus \{x_1, \dots, x_n\})$, one of which is described in the following proposition. If $G$ is a group, let $G^{\text{Ab}}$ denote its Abelianisation.

**Proposition 2.9.** Let $M = \mathbb{T}$ or $M = \mathbb{K}$, and let $m \ge 2$ and $n \ge 1$. Then $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}^{n+1}$, where the factors of this decomposition are generated by elements $\sigma, x, y, \rho_2, \dots, \rho_n \in B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ respectively, where for $i = 2, \dots, n$, $j = n+1, \dots, n+m$ and $k = n+1, \dots, n+m-1$, the elements $\sigma_k, a_j, b_j$ and $C_{i,j}$ of $B_m(M \setminus \{x_1, \dots, x_n\})$ are coset representatives of $\sigma, x, y$ and $\rho_i$ respectively, and $\sigma$ is of order 2.

**Proof.** Let $m \ge 2$ and $n \ge 1$. To obtain a presentation of $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$, we Abelianise the presentation of $B_m(M \setminus \{x_1, \dots, x_n\})$ given in Proposition 2.7, making use of the presentation given by Proposition 2.5 whose relations are relations (1) of Proposition 2.7. By relation (2) of that proposition, it follows that $\sigma_j = \sigma_{j+1}$ in $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ for all $j = n+1, \dots, n+m-2$: we denote the coset of $\sigma_j$ by $\sigma$. By relation (2) of Proposition 2.5, we have $C_{i,j}^{-1} C_{i+1,j} = 1$ in $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ for all $n+1 \le i < j \le n+m$. In particular, if $i+1 = j$, since $C_{j,j} = 1$ by Remark 2.2, we see that $C_{i,i+1} = 1$, and by induction we obtain $C_{i,j} = 1$ for all $n+1 \le i < j \le n+m$. It follows from relation (7) that $\sigma^2 = 1$. Applying this to relations (4) and (5), we see that $a_j = a_{j+1}$ and $b_j = b_{j+1}$ in $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ for all $n+1 \le j \le n+m-1$: we denote the coset of these elements by $x$ and $y$ respectively. Taking $i = j$ in relation (6), where $n+1 \le i \le n+m-1$, we see that $C_{l,j} = C_{l,j+1}$ for all $1 \le l \le n$: we denote the coset of the element $C_{l,j}$ by $\rho_l$. By relation (5) of Proposition 2.5 we have $\rho_1 = C_{1,n+m} = 1$ if $M = \mathbb{T}$, and $\rho_1 = C_{1,n+m} = a_{n+m}^2 = x^2$ if $M = \mathbb{K}$. Using the information that we have already obtained, the remaining relations of Proposition 2.7 yield no new relations in $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$. It follows that $B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ is generated by
the elements $\sigma, x, y, \rho_1, \dots, \rho_n$, subject to the relations that these elements commute pairwise, that $\sigma^2 = 1$, and that $\rho_1 = 1$ if $M = \mathbb{T}$, and $\rho_1 = x^2$ if $M = \mathbb{K}$. We may thus remove $\rho_1$ from the generating set, and apart from the fact that the elements commute pairwise, the only relation is $\sigma^2 = 1$. The proposition then follows. $\square$

Remark 2.10. If $m = 1$, then $B_1(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ is a free Abelian group of rank $n + 1$.

Using the same method to obtain a presentation of a group extension, the following result gives a presentation of the mixed braid group $B_{n,m}(M)$.

**Proposition 2.11.** Let $M = \mathbb{T}$ or $M = \mathbb{K}$, and let $m \ge 2$ and $n \ge 1$. Then $B_{n,m}(M)$ admits the following presentation:

generating set: $\{a_i, b_i, n+1 \le i \le n+m\} \cup \{C_{i,j}, 1 \le i < j, n+1 \le j \le n+m\} \cup \{a, b\} \cup \{\sigma_i, 1 \le i \le n+m-1, i \ne n\}$.

relations:

* **Type I:** relations (1)–(7) of Proposition 2.7.

* **Type II:** relations (1)–(7) of Theorem 2.3, together with:

(1) the surface relation:

$$ \prod_{i=1}^{m} C_{1,n+i} C_{2,n+i}^{-1} = \begin{cases} (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} b a b^{-1} a^{-1} & \text{if } M = \mathbb{T} \\ (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} b a^{-1} b^{-1} a^{-1} & \text{if } M = \mathbb{K}. \end{cases} $$

* **Type III:** the conjugates of the generators of $B_m(M \setminus \{x_1, \dots, x_n\})$ by the coset representatives of the generators of $B_n(M)$:

(2) for $n+1 \le j \le n+m$, $a^{-1}a_ja = a_j$.

(3) for $n+1 \le j \le n+m$, $a^{-1}b_ja = b_ja_jC_{1,j}^{-1}C_{2,j}a_j^{-1}$.

(4) for $n+1 \le j \le n+m$, $b^{-1}a_jb = \begin{cases} a_jb_jC_{1,j}C_{2,j}^{-1}b_j^{-1} & \text{if } M = \mathbb{T} \\ a_jb_j(C_{1,j}C_{2,j}^{-1})^{-1}b_j^{-1} & \text{if } M = \mathbb{K}. \end{cases}$

(5) for $n+1 \le j \le n+m$, $b^{-1}b_jb = \begin{cases} b_j & \text{if } M = \mathbb{T} \\ b_jC_{1,j}C_{2,j}^{-1} & \text{if } M = \mathbb{K}. \end{cases}$

(6) for $1 \le i \le n-1$ and $n+1 \le j \le n+m$, $\sigma_i^{-1}a_j\sigma_i = a_j$ and $\sigma_i^{-1}b_j\sigma_i = b_j$.

(7) for $1 \le i < j, n+1 \le j \le n+m$, $a^{-1}C_{i,j}a = \begin{cases} a_jC_{2,j}^{-1}C_{1,j}a_j^{-1}C_{2,j} & \text{if } i=1 \\ C_{i,j} & \text{otherwise.} \end{cases}$

(8) for $1 \le i < j, n+1 \le j \le n+m$:

$$ b^{-1}C_{i,j}b = \begin{cases} C_{2,j}b_jC_{1,j}C_{2,j}^{-1}b_j^{-1} & \text{if } i=1 \text{ and } M = \mathbb{T} \\ C_{2,j}b_jC_{2,j}C_{1,j}^{-1}b_j^{-1} & \text{if } i=1 \text{ and } M = \mathbb{K} \\ C_{i,j} & \text{otherwise.} \end{cases} $$

(9) for $1 \le i \le n-1$ and $1 \le l < j, n+1 \le j \le n+m$:

$$ \sigma_i^{-1} C_{l,j} \sigma_i = \begin{cases} C_{l-1,j} C_{l,j}^{-1} C_{l+1,j} & \text{if } l=i+1. \\ C_{l,j} & \text{otherwise.} \end{cases} $$

(10) for all $1 \le i \le n-1$ and $n+1 \le j \le n+m-1$, $[a, \sigma_j] = [b, \sigma_j] = [\sigma_i, \sigma_j] = 1$.

*Proof.* Applying the methods of [17, Proposition 1, p. 139] to the short exact sequence (1.3), a set of generators of $B_{n,m}(M)$ is the union of the set of generators of $B_m(M \setminus \{x_1, \dots, x_n\})$ given by Proposition 2.7 with the set $\{a, b, \sigma_1, \dots, \sigma_{n-1}\}$ of coset representatives for $p_*$ of the generating set of $B_n(M)$ given by Theorem 2.3, and this is the generating set given in the statement. There are three types of relations in $B_{n,m}(M)$. The relations of Type I are those of $B_m(M \setminus \{x_1, \dots, x_n\})$ given
by Proposition 2.7. The relations of Type II are obtained by lifting the relations (1)–(8) of $B_n(M)$ given by Theorem 2.3, and rewriting the result in terms of the generators of $B_m(M \setminus \{x_1, \dots, x_n\})$. With the exception of the surface relation (8) of Theorem 2.3, all of these lifted relations are also relations in $B_{n,m}(M)$. To lift this surface relation, notice that $bab^{-1}a^{-1}$ (resp. $ba^{-1}b^{-1}a^{-1}$) is equal to $\sigma_1 \cdots \sigma_{n+m-2}\sigma_{n+m-1}^2\sigma_{n+m-2}\cdots\sigma_1$ in $B_{n+m}(\mathbb{T})$ (resp. in $B_{n+m}(\mathbb{K})$) by relation (8) of Theorem 2.3. Using once more this relation, and making use of (2.1), we obtain:

$$ \begin{aligned} \prod_{i=1}^{m} C_{1,n+i} C_{2,n+i}^{-1} &= \left( \prod_{i=2}^{n} C_{1,i} C_{2,i}^{-1} \right)^{-1} \sigma_1 \cdots \sigma_{n+m-2} \sigma_{n+m-1}^2 \sigma_{n+m-2} \cdots \sigma_1 \\ &= \begin{cases} (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} bab^{-1} a^{-1} = \rho_2^{-m} & \text{if } M = \mathbb{T} \\ (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} ba^{-1} b^{-1} a^{-1} = \rho_2^{-m} x^{2m} & \text{if } M = \mathbb{K}, \end{cases} \end{aligned} $$

which yields the surface relation (1) of the statement. Finally, the relations of Type III are obtained by conjugating the generators of $B_m(M \setminus \{x_1, \dots, x_n\})$ by the coset representatives of the generators of $B_n(M)$. Using once more Remark 2.4, relations (2)–(4), (7) and (8) of the statement follow from relations (1), (2), (6), (7), (3) and (8) of Theorem 2.1 respectively, and relations (6), (9) and (10) may be obtained geometrically using Figures 1 and 2. $\square$

In order to prove Theorem 1.1, we will make use of the following presentation of the quotient of $B_{n,m}(M)$ by its normal subgroup $\Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\}))$.

**Proposition 2.12.** Let $M$ be the torus or the Klein bottle, and let $m, n \ge 2$. Then the group $B_{n,m}(M)/\Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\}))$ admits the following presentation:

generators: $a, b, x, y, \sigma, \rho_2, \dots, \rho_n, \sigma_1, \dots, \sigma_{n-1}$.

relations:

$$ (1) \text{ the surface relation} \quad \left\{ \begin{array}{ll} (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} bab^{-1} a^{-1} = \rho_2^{-m} & \text{if } M = \mathbb{T} \\ (\sigma_1 \cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1)^{-1} ba^{-1} b^{-1} a^{-1} = \rho_2^{-m} x^{2m} & \text{if } M = \mathbb{K}. \end{array} \right. $$

$$ (2) \text{ the relations (1)–(7) of } B_n(M) \text{ given by Theorem 2.3.} $$

$$ (3) \sigma^2 = 1. $$

$$ (4) \quad \begin{aligned} [x, y] &= [a, x] = [x, \rho_i] = [y, \rho_i] = [a, \rho_i] = [b, \rho_i] = [x, \sigma_j] = [y, \sigma_j] = [\rho_i, \rho_k] \\ &= [\sigma, \sigma_j] = [\sigma, x] = [\sigma, y] = [\sigma, \rho_i] = [\sigma, a] = [\sigma, b] = 1, \text{ for all } i, k = 2, \dots, n \text{ and } j = 1, \dots, n-1. \end{aligned} $$

$$ (5) \quad a^{-1}ya = \begin{cases} y\rho_2 & \text{if } M = \mathbb{T} \\ yx^{-2}\rho_2 & \text{if } M = \mathbb{K}. \end{cases} $$

$$ (6) \quad b^{-1}xb = \begin{cases} x\rho_2^{-1} & \text{if } M = \mathbb{T} \\ x^{-1}\rho_2 & \text{if } M = \mathbb{K}. \end{cases} $$

$$ (7) \quad b^{-1}yb = \begin{cases} y & \text{if } M = \mathbb{T} \\ yx^2\rho_2^{-1} & \text{if } M = \mathbb{K}. \end{cases} $$

$$ (8) \quad \begin{aligned} & \text{for all } i = 1, \dots, n-1 \text{ and } j = 2, \dots, n, \sigma_i^{-1}\rho_j\sigma_i = \begin{cases} \rho_{j-1}\rho_j^{-1}\rho_{j+1} & i+1=j \\ \rho_j & \text{otherwise,} \end{cases} \text{ where } \rho_1 = 1 \\ & \text{(resp. } \rho_1 = x^2) \text{ if } M = \mathbb{T} \text{ (resp. } M = \mathbb{K}\text{), and } \rho_{n+1} \text{ is taken to be equal to } 1. \end{aligned} $$

**Proof.** The result follows by applying the methods of [17, Proposition 1, p. 139] to the following short exact sequence:

$$ 1 \to B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}} \to B_{n,m}(M)/\Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\})) \to B_n(M) \to 1 \quad (2.3) $$

obtained from (1.3), and using Propositions 2.9 and 2.11. Relations (1)–(7) of $B_n(M)$ given by Theorem 2.3 lift directly to $B_{n,m}(M)/\Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\}))$, and the surface relation (7) of $B_n(M)$ given by Theorem 2.3 is a consequence of the surface relation (1) of Proposition 2.11, the
$B_{t,s}(M)$ and their cosets in the respective quotients $B_{t,s,m}(M)/H$ and $B_{t,s}(M)/H'$. We suppose that $H$ and $H'$ are such that the following relations hold:

(I) in $B_m(M \setminus \{x_1, \dots, x_n\})/L$, $a_i = a_{n+1}$ and $b_i = b_{n+1}$ for all $i \in X$, $C_{i,j} = C_{i,n+1}$ for all $i = 1, \dots, n$ and $j \in X$, $a_l$ and $b_l$ commute with $C_{i,l}$ for all $i = 1, \dots, n$ and $l \in X' \cup \{n+1\}$, $b_{n+1}^{-1}a_{n+1}b_{n+1}$ and $b_{n+1}a_{n+1}b_{n+1}^{-1}$ are words in $a_{n+1}$ and $C_{1,n+1}$, $\sigma_k = \sigma_{n+1}$ for all $k \in X \setminus \{n+m\}$, and $\sigma_{n+1}^2 = 1$. Let $\sigma = \sigma_{n+1}$ in $B_m(M \setminus \{x_1, \dots, x_n\})/L$.

(II) in $B_{t,s,m}(M)/H$ and in $B_{t,s}(M)/H'$, the Artin relations hold among the $\sigma_k$ for $k \in \{1, \dots, n-1\} \setminus \{t\}$, $a_1$ and $b_1$ commute with $\sigma_l$ for $l \in X' \setminus \{n\}$, $a_i$ and $a_j$ commute for $i, j \in \{1\} \cup X'$ (or $i, j \in \{1\} \cup X' \cup X$ in $B_{t,s,m}(M)/H$), and for $k \in \{1, \dots, n-1\} \setminus \{t\}$, and $1 \le l < n+m$:

$$ \sigma_k^{-1} C_{l,n+m} \sigma_k = \begin{cases} C_{l,n+m} & \text{if } l \neq k+1 \\ C_{l-1,n+m} C_{l,n+m}^{-1} C_{l+1,n+m} & \text{if } l = k+1. \end{cases} \qquad (2.6) $$

We also have the surface relation:

$$ \prod_{i=1}^{d} C_{1,t+i} C_{2,t+i}^{-1} = \begin{cases} (\sigma_1 \cdots \sigma_{t-1}^2 \cdots \sigma_1)^{-1} b_1 a_1 b_1^{-1} a_1^{-1} & \text{if } M = \mathbb{T} \\ (\sigma_1 \cdots \sigma_{t-1}^2 \cdots \sigma_1)^{-1} b_1 a_1^{-1} b_1^{-1} a_1^{-1} & \text{if } M = \mathbb{K}, \end{cases} \qquad (2.7) $$

where $d=s$ (resp. $d=s+m$) in $B_{t,s}(M)/H'$ (resp. in $B_{t,s,m}(M)/H$).

(III) in $B_{t,s,m}(M)/H$, $\sigma$ is central.

(IV) in $B_{t,s,m}(M)/H$, for all $j = 1, \dots, n$, $k \in X'$ and $1 \le i < k$, $C_{j,n+1}$ commutes with $C_{i,k}$.

Remark 2.14. Let $\rho = C_{1,n+1}^{-1}C_{2,n+1}$. Since $a_j$ and $b_j$ commute with $C_{i,j}$ in $B_{t,s,m}(M)/L$ for all $i = 1, \dots, n$ and $j \in X' \cup \{n+1\}$, it follows from relations (2), (6), (7) and (8) of $P_{n+m}(M)$ (considered as a subgroup of $B_{t,s,m}(M)$) of Theorem 2.1 that the following relations hold in $B_{t,s,m}(M)/L$:

(i) $a_1b_{n+1}a_1^{-1} = b_{n+1}\rho^{-1}$ and $a_1^{-1}b_{n+1}a_1 = b_{n+1}\rho$.

(ii) $b_1a_{n+1}b_1^{-1} = a_{n+1}\rho$ and $b_1^{-1}a_{n+1}b_1 = \begin{cases} a_{n+1}\rho^{-1} & \text{if } M = \mathbb{T} \\ a_{n+1}\rho & \text{if } M = \mathbb{K}. \end{cases}$

(iii) $b_1^{-1}b_{n+1}b_1 = \begin{cases} b_{n+1} & \text{if } M = \mathbb{T} \\ b_1b_{n+1}b_1^{-1} = b_{n+1}\rho^{-1} & \text{if } M = \mathbb{K}. \end{cases}$

(iv) for $j \in X' \cup \{n+1\}$, $C_{1,j}C_{2,j}^{-1} = \begin{cases} a_j^{-1}b_1^{-1}a_jb_1 & \text{if } M = \mathbb{T} \\ (a_j^{-1}b_1^{-1}a_jb_1)^{-1} & \text{if } M = \mathbb{K}. \end{cases}$

For the homomorphism $\hat{\phi}: B_{t,s}(M)/H' \to B_{t,s,m}(M)/H$ of (2.4), we set:

$$ \left\{ \begin{array}{ll} \hat{\phi}(\sigma_i) = \sigma_i \cdot a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} \sigma^{s_{i,3}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} & \text{for } i=1, \dots, t-1, t+1, \dots, n-1 \\ \hat{\phi}(a_i) = a_i \cdot a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} \sigma^{\alpha_{i,3}} C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} & \text{for } i=1, t+1, t+2, \dots, n \\ \hat{\phi}(b_i) = b_i \cdot a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} \sigma^{\beta_{i,3}} C_{1,n+1}^{y_{i,1}} \cdots C_{n,n+1}^{y_{i,n}} & \text{for } i=1, t+1, t+2, \dots, n, \end{array} \right. \qquad (2.8) $$

where $s_{i,j}, r_{i,j}, \alpha_{i,j}, \beta_{i,j}, x_{i,j}, y_{i,j} \in \mathbb{Z}$ for the relevant values of $i$ and $j$. If $w \in B_{t,s}(M)/H'$ is written in terms of (the coset representatives of) the generators of (2.5) then the element of $B_{t,s,m}(M)/H$ written in terms of the corresponding generators, that we also denote by $w$, satisfies $\hat{\phi}(w) = wz$, where $z \in \text{Ker}(\hat{p}_*)$. If $z$ is written in terms of the generators of $B_m(M \setminus \{x_1, \dots, x_n\})/L$, the decomposition $\hat{\phi}(w) = wz$ shall be referred to as the *canonical form* of $\hat{\phi}(w)$.

We will now take the image by $\hat{\phi}$ of some of the relations of Theorem 2.3 to obtain relations in $B_{t,s,m}(M)/H$. This will enable us to obtain information about the coefficients that appear in (2.8) above.

We first compute $\hat{\phi}((\sigma_1 \cdots \sigma_{t-2}\sigma_{t-1}^2\sigma_{t-2}\cdots\sigma_1)^{-1})$ and put the resulting expression into canonical form. We start by analysing the expression $\hat{\phi}(\sigma_1 \cdots \sigma_{t-2}\sigma_{t-1})$. Using the fact that for $1 \le i \le n$,
$C_{i,n+1}$ commutes with $a_{n+1}$ and $b_{n+1}$, and that $\sigma_i$ commutes with $C_{j,n+1}$ for all $1 \le i \le t-1$ and $1 \le j \le n$ for which $j \ne i+1$, we obtain:

$$ \hat{\phi}(\sigma_1 \cdots \sigma_{t-2} \sigma_{t-1}) = \prod_{i=1}^{t-1} \sigma_i \cdot a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} \sigma^{s_{i,3}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} = wu, \quad (2.9) $$

where:

$$ w = \sigma^{\sum_{i=1}^{t-1} s_{i,3}} C_{1,n+1}^{\sum_{i=1}^{t-1} r_{i,1}} \prod_{k=3}^{t} C_{k,n+1}^{\sum_{i=1}^{k-2} r_{i,k}} \prod_{k=t+1}^{n} C_{k,n+1}^{\sum_{l=1}^{t-1} r_{l,k}} \prod_{i=1}^{t-1} a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}}, \quad \text{and} \quad (2.10) $$

$$ u = \prod_{k=1}^{t-1} \sigma_k C_{k+1,n+1}^{\alpha_k}, \quad \text{where} \quad \alpha_k = \sum_{i=k}^{t-1} r_{i,k+1}. $$

If $\alpha \in \mathbb{Z}$, let us show by induction on $1 \le i \le t-1$ that:

$$ \sigma_1 \cdots \sigma_i C_{i+1,n+1}^{\alpha} = C_{1,n+1}^{\alpha} C_{2,n+1}^{-\alpha} C_{i+2,n+1}^{\alpha} \sigma_1 \cdots \sigma_i. \quad (2.11) $$

If $i=1$ then the result follows from the relation:

$$ \sigma_i C_{i+1,n+1}^{\alpha} \sigma_i^{-1} = C_{i,n+1}^{\alpha} C_{i+1,n+1}^{-\alpha} C_{i+2,n+1}^{\alpha}, \quad \text{where} \quad 1 \le i \le t-1. \quad (2.12) $$

So suppose that (2.11) holds for some $1 \le i \le t-2$. Then by (2.12) and the induction hypothesis, we have:

$$ \sigma_1 \cdots \sigma_{i+1} C_{i+2,n+1}^{\alpha} = \sigma_1 \cdots \sigma_i C_{i+1,n+1}^{\alpha} C_{i+2,n+1}^{-\alpha} C_{i+3,n+1}^{\alpha} \sigma_{i+1} = C_{1,n+1}^{\alpha} C_{2,n+1}^{-\alpha} C_{i+3,n+1}^{\alpha} \sigma_1 \cdots \sigma_i \sigma_{i+1}, $$

which proves (2.11) for all $1 \le i \le t-1$.

Let us prove by induction on $1 \le i \le t-1$ that:

$$ u = C_{1,n+1}^{\sum_{l=1}^{i-1} \alpha_l} C_{2,n+1}^{-\sum_{l=1}^{i-1} \alpha_l} \left( \prod_{j=3}^{i+1} C_{j,n+1}^{\alpha_{j-2}} \right) \sigma_1 \cdots \sigma_i C_{i+1,n+1}^{\alpha_i} \prod_{k=i+1}^{t-1} \sigma_k C_{k+1,n+1}^{\alpha_k}. \quad (2.13) $$

If $i=1$ then (2.13) is just the definition of $u$. So suppose that (2.13) holds for some $1 \le i \le t-2$. By (2.11), we have:

$$ \begin{aligned} u &= C_{1,n+1}^{\sum_{l=1}^{i-1} \alpha_l} C_{2,n+1}^{-\sum_{l=1}^{i-1} \alpha_l} \left( \prod_{j=3}^{i+1} C_{j,n+1}^{\alpha_{j-2}} \right) C_{1,n+1}^{\alpha_i} C_{2,n+1}^{-\alpha_i} C_{i+2,n+1}^{\alpha_i} \sigma_1 \cdots \sigma_i \prod_{k=i+1}^{t-1} \sigma_k C_{k+1,n+1}^{\alpha_k} \\ &= C_{1,n+1}^{\sum_{l=1}^{i} \alpha_l} C_{2,n+1}^{-\sum_{l=1}^{i} \alpha_l} \left( \prod_{j=3}^{i+2} C_{j,n+1}^{\alpha_{j-2}} \right) \sigma_1 \cdots \sigma_{i+1} C_{i+2,n+1}^{\alpha_{i+1}} \prod_{k=i+2}^{t-1} \sigma_k C_{k+1,n+1}^{\alpha_k}, \end{aligned} $$

which is equation (2.13) in the case $i+1$. Taking $i=t-1$ in (2.13) and using (2.11), we obtain:

$$ u = C_{1,n+1}^{\sum_{l=1}^{t-2} \alpha_l} C_{2,n+1}^{-\sum_{l=1}^{t-2} \alpha_l} \left( \prod_{j=3}^{t} C_{j,n+1}^{\alpha_{j-2}} \right) \sigma_1 \cdots \sigma_{t-1} C_{t,n+1}^{\alpha_{t-1}} = v \sigma_1 \cdots \sigma_{t-1}, \quad (2.14) $$

where:

$$ v = C_{1,n+1}^{\sum_{l=1}^{t-1} \alpha_l} C_{2,n+1}^{-\sum_{l=1}^{t-1} \alpha_l} \left( \prod_{j=3}^{t+1} C_{j,n+1}^{\alpha_{j-2}} \right). \quad (2.15) $$

We now analyse the expression $\hat{\phi}(\sigma_{t-1}\sigma_{t-2}\cdots\sigma_1)$. Using the fact that for $1 \le i \le n$, $C_{i,n+1}$ commutes with $a_{n+1}$ and $b_{n+1}$, that $\sigma$ is central, and that $\sigma_i$ commutes with $C_{j,n+1}$ for all $1 \le i \le t-1$, $1 \le j \le n$ for which $j \ne i+1$, we obtain:

$$ \hat{\phi}(\sigma_{t-1} \sigma_{t-2} \cdots \sigma_1) = \prod_{i=1}^{t-1} \sigma_{t-i} \cdot a_{n+1}^{s_{t-i,1}} b_{n+1}^{s_{t-i,2}} \sigma^{s_{t-i,3}} C_{1,n+1}^{r_{t-i,1}} \cdots C_{n,n+1}^{r_{t-i,n}} = w'u', \quad (2.16) $$
where:

$$w' = \sigma^{\sum_{i=1}^{t-1} s_{t-i,3}} \prod_{k=1}^{t-1} C_{k,n+1}^{\sum_{i=k}^{t-1} r_{i,k}} \prod_{k=t+1}^{n} C_{k,n+1}^{\sum_{l=1}^{t-1} r_{l,k}} \prod_{i=1}^{t-1} a_{n+1}^{s_{t-i,1}} b_{n+1}^{s_{t-i,2}} \text{ and } u' = \prod_{k=1}^{t-1} \sigma_{t-k} C_{t-k+1,n+1}^{\beta_{t-k}} \quad (2.17)$$

where $\beta_k = \sum_{i=1}^k r_{i,k+1}$ for $k = 1, \dots, t-1$. If $\alpha \in \mathbb{Z}$, let us show by induction on $1 \le i \le t-1$ that:

$$\sigma_{t-1} \cdots \sigma_{t-i} C_{t-i+1,n+1}^{\alpha} = C_{t+1,n+1}^{\alpha} C_{t,n+1}^{-\alpha} C_{t-i,n+1}^{\alpha} \sigma_{t-1} \cdots \sigma_{t-i}. \quad (2.18)$$

If $i=1$ then the result follows from (2.12). So suppose that (2.18) holds for some $1 \le i \le t-2$. Then by (2.12) and the induction hypothesis, we have:

$$\begin{aligned} \sigma_{t-1} \cdots \sigma_{t-i} \sigma_{t-i-1} C_{t-i,n+1}^{\alpha} &= \sigma_{t-1} \cdots \sigma_{t-i} C_{t-i+1,n+1}^{\alpha} C_{t-i,n+1}^{-\alpha} C_{t-i-1,n+1}^{\alpha} \sigma_{t-i-1} \\ &= C_{t+1,n+1}^{\alpha} C_{t,n+1}^{-\alpha} C_{t-i-1,n+1}^{\alpha} \sigma_{t-1} \cdots \sigma_{t-i} \sigma_{t-i-1}, \end{aligned}$$

which proves (2.18) for all $1 \le i \le t-1$. Let us prove by induction on $1 \le i \le t-1$ that:

$$u' = C_{t+1,n+1}^{\sum_{l=t-i+1}^{t-1} \beta_l} C_{t,n+1}^{-\sum_{l=t-i+1}^{t-1} \beta_l} \left( \prod_{j=t-i+1}^{t-1} C_{j,n+1}^{\beta_j} \right) \sigma_{t-1} \cdots \sigma_{t-i} C_{t-i+1,n+1}^{\beta_{t-i}} \prod_{k=i+1}^{t-1} \sigma_{t-k} C_{t-k+1,n+1}^{\beta_{t-k}} \quad (2.19)$$

If $i=1$ then (2.19) is just the definition of $u'$. So suppose that (2.14) holds for some $1 \le i \le t-2$. By (2.18), we have:

$$\begin{aligned} u' &= C_{t+1,n+1}^{\sum_{l=t-i+1}^{t-1} \beta_l} C_{t,n+1}^{-\sum_{l=t-i+1}^{t-1} \beta_l} \left( \prod_{j=t-i+1}^{t-1} C_{j,n+1}^{\beta_j} \right) C_{t+1,n+1}^{\beta_{t-i}} C_{t,n+1}^{-\beta_{t-i}} C_{t-i,n+1}^{\beta_{t-i}} \sigma_{t-1} \cdots \sigma_{t-i} \prod_{k=i+1}^{t-1} \sigma_{t-k} C_{t-k+1,n+1}^{\beta_{t-k}} \\ &= C_{t+1,n+1}^{\sum_{l=t-i}^{t-1} \beta_l} C_{t,n+1}^{-\sum_{l=t-i}^{t-1} \beta_l} \left( \prod_{j=t-i}^{t-1} C_{j,n+1}^{\beta_j} \right) \sigma_{t-1} \cdots \sigma_{t-i} \sigma_{t-i-1} C_{t-i,n+1}^{\beta_{t-i-1}} \prod_{k=i+2}^{t-1} \sigma_{t-k} C_{t-k+1,n+1}^{\beta_{t-k}} \end{aligned}$$

which is equation (2.19) in the case $i+1$. Taking $i=t-1$ in (2.19) and using (2.18), we see that:

$$u' = C_{t+1,n+1}^{\sum_{l=2}^{t-1} \beta_l} C_{t,n+1}^{-\sum_{l=2}^{t-1} \beta_l} \left( \prod_{j=2}^{t-1} C_{j,n+1}^{\beta_j} \right) \sigma_{t-1} \cdots \sigma_1 C_{2,n+1}^{\beta_1} = v' \sigma_{t-1} \cdots \sigma_1, \quad (2.20)$$

where:

$$v' = C_{t+1,n+1}^{\sum_{l=1}^{t-1} \beta_l} C_{t,n+1}^{-\sum_{l=1}^{t-1} \beta_l} \prod_{j=1}^{t-1} C_{j,n+1}^{\beta_j}. \quad (2.21)$$

So by (2.9), (2.14), (2.16) and (2.20), we obtain:

$$\hat{\phi}((\sigma_1 \cdots \sigma_{t-2} \sigma_{t-1}^2 \sigma_{t-2} \cdots \sigma_1)^{-1}) = \sigma_1^{-1} \cdots \sigma_{t-1}^{-1} \omega^{-1} v^{-1} w^{-1}, \quad (2.22)$$

where $\omega = \sigma_1 \cdots \sigma_{t-1} w' v'$. Let $z = \sigma^{\sum_{i=1}^{t-1} s_{t-i,3}} C_{t+1,n+1}^{\sum_{l=1}^{t-1} \beta_l + \sum_{l=1}^{t-1} r_{l,t+1}} \prod_{k=t+2}^{n} C_{k,n+1}^{\sum_{l=1}^{t-1} r_{l,k}} \prod_{i=1}^{t-1} a_{n+1}^{s_{t-i,1}} b_{n+1}^{s_{t-i,2}}$, and for $k=1, \dots, t-1$, let $\gamma_k = \beta_k + \sum_{i=k}^{t-1} r_{i,k}$. Then by (2.11), (2.17) and (2.21), we have:

$$\begin{aligned} \omega &= z \sigma_1 \cdots \sigma_{t-1} C_{t,n+1}^{-\sum_{l=1}^{t-1} \beta_l} \prod_{k=1}^{t-1} C_{k,n+1}^{\gamma_k} \\ &= z (C_{1,n+1} C_{2,n+1}^{-1} C_{t+1,n+1})^{-\sum_{l=1}^{t-1} \beta_l} C_{1,n+1}^{\gamma_1} \prod_{k=2}^{t-1} (C_{1,n+1} C_{2,n+1}^{-1} C_{k+1,n+1})^{\gamma_k} \sigma_1 \cdots \sigma_{t-1}. \end{aligned} \quad (2.23)$$

The coefficient of $C_{2,n+1}$ in $(\sigma_1 \cdots \sigma_{t-1} w' v')^{-1}$ is thus equal to:

$$\sum_{l=2}^{t-1} \gamma_l - \sum_{l=1}^{t-1} \beta_l = \sum_{l=2}^{t-1} \left( \beta_l + \sum_{i=l}^{t-1} r_{i,l} \right) - \sum_{l=1}^{t-1} \beta_l = \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} - \beta_1 = \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} - r_{1,2}.$$
Combining (2.10), (2.15), (2.22) and (2.23), and making use of the relation $\sigma^2 = 1$ and the fact that $b_{n+1}a_{n+1}b_{n+1}^{-1}$ is a word in $a_{n+1}$ and $C_{1,n+1}$, it follows that:

$$ \hat{\phi}((\sigma_1 \cdots \sigma_{t-2}\sigma_{t-1}^2 \sigma_{t-2} \cdots \sigma_1)^{-1}) = (\sigma_1 \cdots \sigma_{t-2}\sigma_{t-1}^2 \sigma_{t-2} \cdots \sigma_1)^{-1} C_{2,n+1}^{\alpha} \xi, \quad (2.24) $$

where $\xi = \xi(C_{1,n+1}, C_{3,n+1}, \dots, C_{n,n+1}, a_{n+1}, b_{n+1})$ is in canonical form, and $\alpha = \sum_{l=1}^{t-1} \sum_{i=l}^{t-1} r_{i,l+1} + \sum_{l=2}^{t-1} \sum_{i=1}^{t-1} r_{i,l} - r_{1,2}$. Note that $\alpha$ can be simplified:

$$ \begin{aligned} \alpha &= \sum_{l=1}^{t-1} \sum_{i=l}^{t-1} r_{i,l+1} + \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} - r_{1,2} = \sum_{l=2}^{t-1} \sum_{i=l-1}^{t-1} r_{i,l} + \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} - r_{1,2} \\ &= 2 \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} + \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{l-1,l} - r_{1,2} = 2 \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{i,l} + \sum_{l=2}^{t-1} \sum_{i=l}^{t-1} r_{l,l+1}. \end{aligned} \quad (2.25) $$

We now determine $\hat{\phi}(R)$, where $R = b_1 a_1 b_1^{-1} a_1^{-1}$ (resp. $R = b_1 a_1^{-1} b_1^{-1} a_1^{-1}$).

If $M = \mathbb{T}$, we have:

$$ \begin{aligned} \hat{\phi}(R) &= b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} a_1^{\alpha_{1,1}} b_{n+1}^{\alpha_{1,2}} \sigma^{\alpha_{1,3}} C_{1,n+1}^{x_{1,1}} \cdots C_{n,n+1}^{x_{1,n}} \\ & \quad \sigma^{-\beta_{1,3}} C_{1,n+1}^{-y_{1,1}} \cdots C_{n,n+1}^{-y_{1,n}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_1^{-1} \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \\ &= b_1 a_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} a_{n+1}^{\alpha_{1,1}} b_{n+1}^{\alpha_{1,2}-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} \sigma^{\alpha_{1,3}} C_{1,n+1}^{\beta_{1,2}+x_{1,1}} C_{2,n+1}^{\beta_{1,2}+x_{1,2}} C_{3,n+1}^{x_{1,3}} \cdots C_{n,n+1}^{x_{1,n}} b_1^{-1} \\ & \quad \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \\ &= b_1 a_1 b_1^{-1} a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} a_{n+1}^{\alpha_{1,1}} b_{n+1}^{\alpha_{1,2}-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} C_{1,n+1}^{\beta_{1,2}-\alpha_{1,1}} C_{2,n+1}^{\beta_{1,2}+\alpha_{1,1}} a_1^{-1} \\ &= b_1 a_1 b_1^{-1} a_1^{-1} a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} a_{n+1}^{\alpha_{1,1}} b_{n+1}^{\alpha_{1,2}-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} C_{1,n+1}^{\beta_{1,2}-\alpha_{1,1}} C_{2,n+1}^{\beta_{1,2}+\alpha_{1,1}}. \end{aligned} $$

If $M = \mathbb{K}$, we have:

$$ \begin{aligned} \hat{\phi}(R) &= b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \\ & \quad \sigma^{-\beta_{1,3}} C_{1,n+1}^{-y_{1,1}} \cdots C_{n,n+1}^{-y_{1,n}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_1^{-1} \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \\ &= b_1 a_1^{-1} a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}-\alpha_{1,2}+\beta_{1,2}} C_{2,n+1}^{-x_{1,2}+\alpha_{1,2}-\beta_{1,2}} C_{3,n+1}^{-x_{1,3}} \cdots C_{n,n+1}^{-x_{1,n}} b_1^{-1} \\ & \quad \sigma^{-\alpha_{1,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \\ &= b_1 a_1^{-1} b_1^{-1} a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} \sigma^{-2\alpha_{1,3}} C_{1,n+1}^{\alpha_{1,1}-\beta_{1,2}} C_{2,n+1}^{-2(x_{1,1}+x_{1,2})-\alpha_{1,1}+\beta_{1,2}} \\ & \quad C_{3,n+1}^{-2x_{1,3}} \cdots C_{n,n+1}^{-2x_{1,n}} a_1^{-1} \\ &= b_1 a_1^{-1} b_1^{-1} a_1^{-1} a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} \sigma^{-2\alpha_{1,3}} C_{1,n+1}^{\alpha_{1,1}-\beta_{1,2}-2\alpha_{1,2}} \\ & \quad C_{2,n+1}^{2(\alpha_{1,2}-x_{1,1}-x_{1,2})-\alpha_{1,1}+\beta_{1,2}} C_{3,n+1}^{-2x_{1,3}} \cdots C_{n,n+1}^{-2x_{1,n}}. \end{aligned} $$

So if $M = \mathbb{T}$ or $\mathbb{K}$ then:

$$ \hat{\phi}(R) = R C_{2,n+1}^{\delta} w, \quad (2.26) $$

where:

$$ \delta = \begin{cases} \beta_{1,2} + \alpha_{1,1} & \text{if } M = \mathbb{T} \\ 2(\alpha_{1,2} - x_{1,1} - x_{1,2}) - \alpha_{1,1} + \beta_{1,2} & \text{if } M = \mathbb{K}, \end{cases} \quad (2.27) $$

and $w = w(C_{1,n+1}, C_{3,n+1}, \dots, C_{n,n+1}, a_{n+1}, b_{n+1})$ is in canonical form.

We now determine $\hat{\phi}(\prod_{j=1}^s C_{1,t+j} C_{2,t+j}^{-1})$. If $M = \mathbb{T}$ (resp. $M = \mathbb{K}$), by Remark 2.14(iv) and $i = t+1, \dots, n$, $\hat{\phi}(C_{1,i}C_{2,i}^{-1}) = \hat{\phi}(a_i^{-1}b_1^{-1}a_i b_1)$ (resp. $\hat{\phi}(C_{1,i}C_{2,i}^{-1}) = \hat{\phi}((a_i^{-1}b_1^{-1}a_i b_1)^{-1})$). Let $i = t+1, \dots, n$. Then $a_i^{-1}b_{n+1}a_i = b_{n+1}C_{i,n+1}^{-1}C_{i+1,n+1}$ and $a_i b_{n+1}a_i^{-1} = b_{n+1}C_{i,n+1}C_{i+1,n+1}^{-1}$. If $M = \mathbb{T}$ then:

$$ \begin{aligned} \hat{\phi}(C_{1,i}C_{2,i}^{-1}) &= \sigma^{-\alpha_{i,3}} C_{1,n+1}^{-x_{1,1}} \cdots C_{n,n+1}^{-x_{1,n}} b_{n+1}^{-\alpha_{1,2}} a_{n+1}^{-\alpha_{1,1}} a_1^{-1} \sigma^{-\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{-y_{1,n}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_1^{-1} \\ & \quad a_i a_{n+1}^{\alpha_{1,1}} b_{n+1}^{\alpha_{1,2}} \sigma^{\alpha_{1,3}} C_{1,n+1}^{x_{1,1}} \cdots C_{n,n+1}^{x_{1,n}} b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} \end{aligned} $$
$$
\begin{align*}
&=a_i^{-1}\sigma^{-\alpha_{i,3}-\beta_{1,3}}b_{n+1}^{-\alpha_{i,2}}a_{n+1}^{-\alpha_{i,1}}b_{n+1}^{-\beta_{1,2}}a_{n+1}^{-\beta_{1,1}}C_{1,n+1}^{-x_{i,1}-y_{1,1}}\cdots C_{i-1,n+1}^{-x_{i,i-1}-y_{1,i-1}}C_{i,n+1}^{-x_{i,i}-y_{1,i}-\alpha_{i,2}} \\
&\qquad C_{i+1,n+1}^{-x_{i,i+1}-y_{1,i+1}+\alpha_{i,2}}C_{i+2,n+1}^{-x_{i,i+2}-y_{1,i+2}}\cdots C_{n,n+1}^{-x_{i,n}-y_{1,n}}b_1^{-1} \\
&=a_i a_{n+1}^{\alpha_{i,1}}b_{n+1}^{\alpha_{i,2}}\sigma^{\alpha_{i,3}}C_{1,n+1}^{x_{i,1}}\cdots C_{n,n+1}^{x_{i,n}}b_1 a_{n+1}^{\beta_{1,1}}b_{n+1}^{\beta_{1,2}}\sigma^{\beta_{1,3}}C_{1,n+1}^{y_{1,1}}\cdots C_{n,n+1}^{y_{1,n}} \\
&=a_i^{-1}b_1^{-1}\sigma^{-\alpha_{i,3}-\beta_{1,3}}b_{n+1}^{-\alpha_{i,2}}a_{n+1}^{-\alpha_{i,1}}b_{n+1}^{-\beta_{1,2}}a_{n+1}^{-\beta_{1,1}}C_{1,n+1}^{-x_{i,1}-y_{1,i+1}+\alpha_{i,1}+\beta_{1,1}}C_{2,n+1}^{-x_{i,2}-y_{1,2}-\alpha_{i,1}-\beta_{1,1}} \\
&\qquad C_{3,n+1}^{-x_{i,3}-y_{1,3}}\cdots C_{i-1,n+1}^{-x_{i,i-1}-y_{1,i-1}}C_{i,n+1}^{-x_{i,i}-y_{1,i}-\alpha_{i,2}}C_{i+1,n+1}^{-x_{i,i+1}-y_{1,i+1}+\alpha_{i,2}} \\
&\qquad C_{i+2,n+1}^{-x_{i,i+2}-y_{1,i+2}}\cdots C_{n,n+1}^{-x_{i,n}-y_{1,n}}a_i a_{n+1}^{\alpha_{i,1}}b_{n+1}^{\alpha_{i,2}}\sigma^{\alpha_{i,3}}C_{1,n+1}^{x_{i,1}}\cdots C_{n,n+1}^{x_{i,n}}b_1 a_{n+1}^{\beta_{1,1}}b_{n+1}^{\beta_{1,2}}\sigma^{\beta_{1,3}} \\
&=a_i^{-1}b_1^{-1}a_i\sigma^{-\beta_{1,3}}b_{n+1}^{-\alpha_{i,2}}a_{n+1}^{-\alpha_{i,1}}b_{n+1}^{-\beta_{1,2}}a_{n+1}^{-\beta_{1,1}}b_{n+1}^{\alpha_{i,2}}\sigma^{\alpha_{i,3}}C_{1,n+1}^{y_{1,1}}\cdots C_{n,n+1}^{y_{1,n}} \\
&=a_i^{-1}b_1^{-1}a_i\sigma^{-\beta_{1,3}}b_{n+1}^{-\alpha_{i,2}}a_{n+1}^{-\alpha_{i,1}}b_{n+1}^{-\beta_{1,2}}a_{n+1}^{-\beta_{1,1}}b_{n+1}^{\alpha_{i,2}}\sigma^{\alpha_{i,3}}C_{1,n+1}^{-y_{1,1}+\alpha_{i,1}+\beta_{1,1}}C_{2,n+1}^{-y_{1,2}-\alpha_{i,1}-\beta_{1,1}} \\
&\qquad C_{3,n+1}^{-y_{1,3}}\cdots C_{i-1,n+1}^{-y_{1,i-1}}C_{i,n+1}^{-y_{1,i}+\beta_{1,2}}C_{i+1,n+1}^{-y_{1,i+1}-\beta_{1,2}}C_{i+2,n+1}^{-y_{1,i+2}}\cdots C_{n,n+1}^{-y_{1,n}}b_1 a_{n+1}^{\beta_{1,1}}b_{n+1}^{\beta_{1,2}}\sigma^{\beta_{1,3}} \\
&\qquad C_{1,n+1}^{y_{1,1}}\cdots C_{n,n+1}^{y_{1,n}} \\
&=a_i^{-1}b_1^{-1}a_i b_1 b_{n+1}^{-\alpha_{i,2}}a_{n+1}^{-\alpha_{i,1}}b_{n+1}^{-\beta_{1,2}}a_{n+1}^{-\beta_{1,1}}b_{n+1}^{\alpha_{i,2}}a_{n+1}^{\alpha_{i,1}}b_{n+1}^{\beta_{1,1}}b_{n+1}^{\beta_{1,2}}C_{1,n+1}^{\alpha_{i,1}}C_{2,n+1}^{-\alpha_{i,1}}C_{i,n+1}^{\beta_{1,2}}C_{i+1,n+1}^{-\beta_{1,2}}
\end{align*}
$$

If $M = \mathbb{K}$ then:

$$
\begin{align*}
\hat{\phi}(C_{1,i}C_{2,i}^{-1}) &= \sigma^{-\beta_{1,3}} C_{1,n+1}^{-y_{1,1}} \cdots C_{n,n+1}^{-y_{1,n}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_1^{-1} \sigma^{-\alpha_{i,3}} C_{1,n+1}^{-x_{i,1}} \cdots C_{n,n+1}^{-x_{i,n}} b_{n+1}^{-\alpha_{i,2}} a_{n+1}^{-\alpha_{i,1}} a_i^{-1} \\
&= b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} a_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} \sigma^{\alpha_{i,3}} C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} \\
&= b_1^{-1} \sigma^{-\beta_{1,3}-\alpha_{i,3}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{i,2}} a_{n+1}^{-\alpha_{i,1}} C_{1,n+1}^{y_{1,1}-x_{i,1}+\beta_{1,1}-\beta_{1,2}} C_{2,n+1}^{-y_{1,2}-x_{i,2}-2y_{1,1}-\beta_{1,1}+\beta_{1,2}} \\
&\quad C_{3,n+1}^{-y_{1,3}-x_{i,3}} \cdots C_{n,n+1}^{-y_{1,n}-x_{i,n}} a_i^{-1} b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} \\
&\quad a_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} \sigma^{\alpha_{i,3}} C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} \\
&= b_1^{-1} a_i^{-1} \sigma^{-\beta_{1,3}-\alpha_{i,3}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{i,2}} a_{n+1}^{-\alpha_{i,1}} C_{1,n+1}^{y_{1,1}-x_{i,1}+\beta_{1,1}-\beta_{1,2}} C_{2,n+1}^{-y_{1,2}-x_{i,2}-2y_{1,1}-\beta_{1,1}+\beta_{1,2}} \\
&\quad C_{3,n+1}^{-y_{1,3}-x_{i,3}} \cdots C_{i-1,n+1}^{-y_{1,i-1}-x_{i,i-1}} C_{i,n+1}^{-y_{1,i}-x_{i,i}-\beta_{1,2}-\alpha_{i,2}} C_{i+1,n+1}^{-y_{1,i+1}-x_{i,i+1}+\beta_{1,2}+\alpha_{i,2}} \\
&\quad C_{i+2,n+1}^{-y_{1,i+2}-x_{i,i+2}} \cdots C_{n,n+1}^{-y_{1,n}-x_{i,n}} b_1 a_{n+1}^{\beta_{1,1}} b_{n+1}^{\beta_{1,2}} \sigma^{\beta_{1,3}} C_{1,n+1}^{y_{1,1}} \cdots C_{n,n+1}^{y_{1,n}} a_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} \sigma^{\alpha_{i,3}} \\
&\quad C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} \\
&= b_1^{-1} a_i^{-1} b_1 \sigma^{-\alpha_{i,3}} b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{i,2}} a_{n+1}^{-\alpha_{i,1}} C_{1,n+1}^{y_{1,1}-x_{i,1}+\beta_{1,1}-\beta_{1,2}} C_{2,n+1}^{-x_{i,2}-2x_{i,1}-\alpha_{i,1}+\alpha_{i,2}} \\
&\quad C_{3,n+1}^{-x_{i,3}} \cdots C_{i-1,n+1}^{-x_{i,i-1}} C_{i,n+1}^{-x_{i,i}-\beta_{1,2}-\alpha_{i,2}} C_{i+1,n+1}^{-x_{i,i+1}+\beta_{1,2}+\alpha_{i,2}} C_{i+2,n+1}^{-x_{i,i+2}} \cdots C_{n,n+1}^{-x_{i,n}} \\
&\quad a_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} \sigma^{\alpha_{i,3}} C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} \\
&= b_1^{-1} a_i^{-1} b_1 a_i b_{n+1}^{-\beta_{1,2}} a_{n+1}^{-\beta_{1,1}} b_{n+1}^{-\alpha_{i,2}} a_{n+1}^{-\alpha_{i,1}} C_{1,n+1}^{y_{1,1}+\alpha_{i,1}-\alpha_{i,2}} C_{2,n+1}^{-2x_{i,1}-\alpha_{i,1}+\alpha_{i,2}} \\
&\quad C_{i,n+1}^{-\beta_{1,2}} C_{i+1,n+1}^{\beta_{1,2}}
\end{align*}
$$

Using Remark 2.14(iv) and relation (IV), it follows from these computations that if $M = \mathbb{T}$ or $\mathbb{K}$ and $i = t+1, \dots, n$, $\hat{\phi}(C_{1,i}C_{2,i}^{-1}) = C_{1,i}C_{2,i}^{-1}C_{2,n+1}^{\gamma_i}z_i$, where $z_i = z_i(C_{1,n+1}, C_{3,n+1}, \dots, C_{n,n+1}, a_{n+1}, b_{n+1})$ is in canonical form, and:

$$
\gamma_i = \begin{cases} -\alpha_{i,1} & \text{if } M = \mathbb{T} \\ -2x_{i,1} - \alpha_{i,1} + \alpha_{i,2} & \text{if } M = \mathbb{K}. \end{cases} \tag{2.28}
$$

Applying also relations (3) and (8) of Theorem 2.1, one may check that the word $a_i^{-1}b_1^{-1}a_i b_1$ (resp. $(a_i^{-1}b_1^{-1}a_i b_1)^{-1}$) commutes with $a_{n+1}$, $b_{n+1}$ and $C_{j,n+1}$ for $j = 1, \dots, n$, from which it follows that:

$$
\hat{\phi} \left( \prod_{j=1}^{s} C_{1,t+j} C_{2,t+j}^{-1} \right) = \left( \prod_{j=1}^{s} C_{1,t+j} C_{2,t+j}^{-1} \right) C_{2,n+1}^{\gamma} z, \quad (2.29)
$$
where:

$$ \gamma = \sum_{i=1}^{s} \gamma_{t+i}, \qquad (2.30) $$

and $z = \prod_{i=1}^{s} z_{t+i}$ is in canonical form.

Taking the image by $\hat{\phi}$ of (2.7) with $d = s$ and applying (2.24), (2.26) and (2.29), we obtain the following equality in $B_{t,s,m}(M)/H$:

$$ \left( \prod_{j=1}^{s} C_{1,t+j} C_{2,t+j}^{-1} \right) C_{2,n+1}^{\gamma} z = (\sigma_1 \cdots \sigma_{t-2} \sigma_{t-1}^2 \sigma_{t-2} \cdots \sigma_1)^{-1} C_{2,n+1}^{\alpha} \xi R C_{2,n+1}^{\delta} w. \qquad (2.31) $$

One may check using Proposition 4.8 that $R$ commutes with $a_{n+1}$, $b_{n+1}$ and $C_{j,n+1}$ for all $j = 1, \dots, n$. By relations (I), we have:

$$ \prod_{j=1}^{m} C_{1,n+j} C_{2,n+j}^{-1} = C_{1,n+j}^{m} C_{2,n+j}^{-m}. \qquad (2.32) $$

It follows from (2.7) with $d = s + m$, (2.31) and (2.32) that:

$$ C_{2,n+1}^{\gamma+m} z = C_{2,n+1}^{\alpha} \cdot \xi C_{2,n+1}^{\delta} w, \qquad (2.33) $$

which up to collecting terms in $\mathrm{Ker}(\hat{\phi}_*)$, is in canonical form.

**Lemma 2.15.** With the notation of (2.8):

(a) let $t \ge 4$, and let $i, j \in \{1, \dots, n-1\} \setminus \{t\}$, where $|i-j| \ge 2$. In $\mathrm{Ker}(\hat{p}_*)$ we have:

$$ [b_{n+1}^{-s_{i,2}} a_{n+1}^{-s_{i,1}}, b_{n+1}^{-s_{j,2}} a_{n+1}^{-s_{j,1}}] = C_{i,n+1}^{r_{j,i+1}} C_{i+1,n+1}^{-2r_{j,i+1}} C_{i+2,n+1}^{r_{j,i+1}} C_{j,n+1}^{-r_{i,j+1}} C_{j+1,n+1}^{2r_{i,j+1}} C_{j+2,n+1}^{-r_{i,j+1}} \qquad (2.34) $$

(b) if $t \ge 3$, for all $1 \le i \le t-2$, in $\mathrm{Ker}(\hat{p}_*)$ we have:

$$ \delta = \prod_{\substack{k=1 \\ k \neq i+1, i+2}}^{n} C_{k,n+1}^{r_{i+1,k}-r_{i,k}} \cdot C_{i,n+1}^{-r_{i,i+1}-r_{i,i+2}} C_{i+1,n+1}^{\rho} C_{i+2,n+1}^{-\rho} C_{i+3,n+1}^{r_{i+1,i+1}+r_{i+1,i+2}}, \qquad (2.35) $$

where $\rho = 2r_{i,i+2} + 2r_{i+1,i+1} + r_{i,i+1} + r_{i+1,i+2}$, $\delta = \beta^{-1}\alpha^{-1}\beta^{-1}\alpha\beta\alpha\sigma^{s_{i,3}-s_{i+1,3}}$, with $\alpha = a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}}$ and $\beta = a_{n+1}^{s_{i+1,1}} b_{n+1}^{s_{i+1,2}}$.

**Proof.** Let $t \ge 4$, and let $1 \le i, j \le t-1$, where $|i-j| \ge 2$, and consider the Artin relation $\sigma_i \sigma_j = \sigma_j \sigma_i$ in $B_{t,s}(M)/H'$. By relation (2.6) and relation (6) of Proposition 2.11, the only generators of $B_m(M \setminus \{x_1, \dots, x_n\})/L$ that do not both commute with $\sigma_i$ and $\sigma_j$ in $B_{t,s,m}(M)/H$ are $C_{i+1,n+1}$ and $C_{j+1,n+1}$ respectively. Taking the image of $\sigma_i \sigma_j = \sigma_j \sigma_i$ by $\hat{\phi}$ and making use of (2.8), it follows that the coefficients of $\sigma$ and of the terms $C_{k,n+1}$, $k = 1, \dots, n$, $k \ne i+1, j+1$ cancel pairwise, and applying (2.6), we obtain the following relation:

$$ \sigma_i \sigma_j \tau_i \tau_j C_{j,n+1}^{r_{i,j+1}} C_{j+1,n+1}^{-r_{i,j+1}} C_{j+2,n+1}^{r_{i,j+1}} C_{i+1,n+1}^{r_{i,i+1}+r_{j,i+1}} = \sigma_j \sigma_i \tau_j \tau_i C_{i,n+1}^{r_{j,i+1}} C_{i+1,n+1}^{-r_{j,i+1}} C_{i+2,n+1}^{r_{j,i+1}} C_{j+1,n+1}^{r_{j,j+1}+r_{i,j+1}}, $$

where $\tau_i = a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}}$ and $\tau_j = a_{n+1}^{s_{j,1}} b_{n+1}^{s_{j,2}}$. Equation (2.34) then follows using the lift of relation (2) of Theorem 2.3.

We obtain equation (2.35) in a similar manner by considering the image by $\hat{\phi}$ of the Artin relation $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ for $1 \le i \le t-2$, where $t \ge 3$. $\square$

### 3. PROOF OF THEOREM 1.1

This section is devoted to proving Theorem 1.1. We start by showing that the condition is sufficient.

**Proposition 3.1.** Let $M$ be the torus or the Klein bottle, and let $m, n \ge 1$. If $n$ divides $m$ then the generalised Fadell-Neuwirth short exact sequence (1.3) splits.
**Proof.** Suppose that $m = \ln$ for some $l \in \mathbb{N}$. To conclude that there exists a section, we proceed in a manner similar to that of [8] in the case of the short exact sequence (1.2). If $M$ is the 2-torus or the Klein bottle, let $\nu$ be a non-vanishing vector field in $M$ and let $d$ be a metric on $M$. We shall construct a cross-section on the level of configuration spaces, from which the result will follow by taking the induced homomorphism on the level of fundamental groups. Let $s: D_n(M) \to D_{n,m}(M)$ be the map defined by $s(x) = (x, s_1(x), \dots, s_n(x))$ for all $x = (x_1, \dots, x_n) \in D_n(M)$ (note that notationally, we do not distinguish between ordered and unordered tuples), where for $i = 1, \dots, n$:

$$s_i(x) = \left( x_i + \frac{\nu(x) \cdot \epsilon(x)}{2(l+1)}, x_i + \frac{2\nu(x) \cdot \epsilon(x)}{2(l+1)}, \dots, x_i + \frac{l\nu(x) \cdot \epsilon(x)}{2(l+1)} \right), \quad (3.1)$$

and $\epsilon(x) = \min_{1 \le k < j \le n} \{d(x_k, x_j)\} > 0$. So for all $i = 1, \dots, n$, $s_i(x)$ consists of $l$ distinct unordered points of $M$, and the union of these points yields $m$ distinct unordered points of $M$ that are also distinct from the $n$ points of $x$. Therefore $s$ is a well-defined continuous map, and it is a cross-section for the map $p: D_{n,m}(M) \to D_n(M)$. $\square$

**Proposition 3.1** gives a sufficient condition for the short exact sequence (1.3) to split. We now prove that it is also necessary. Suppose then that the short exact sequence (1.3) splits. If $n = 1$ then there is nothing to prove. So suppose that $n \ge 2$. We will use the computations of Section 2.1 and the commutative diagram (2.4), with $s = 0$, $t = n$, $L = H = \Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\}))$ and $H' = \{1\}$. Note that $X' = \emptyset$ in this case, so relations (IV) do not exist. Also, relation (I) follows from Proposition 2.9, and relations (II) and (III) follow from Proposition 2.12 and Theorems 2.1 and 2.3. Making use of the presentation of $\text{Ker}(\hat{p}_*) = B_m(M \setminus \{x_1, \dots, x_n\})^{\text{Ab}}$ given by Proposition 2.9, it follows that:

$$\left\{ \begin{array}{l} \hat{\phi}(a) = a \cdot x^{k_1} y^{k_2} \sigma^{i_0} \rho_2^{i_2} \cdots \rho_n^{i_n} \\ \hat{\phi}(b) = b \cdot x^{k_3} y^{k_4} \sigma^{j_0} \rho_2^{j_2} \cdots \rho_n^{j_n} \\ \hat{\phi}(\sigma_i) = \sigma_i \cdot x^{l_{i,1}} y^{l_{i,2}} \sigma^{r_{i,0}} \rho_2^{r_{i,2}} \cdots \rho_n^{r_{i,n}} \text{ for } i = 1, \dots, n-1, \end{array} \right. \quad (3.2)$$

where $k_1, \dots, k_4, l_{i,1}, l_{i,2}, i_q, j_q, r_{i,q} \in \mathbb{Z}$ for $q = 0, 2, \dots, n$, and $i_0, j_0, r_{i,0}$ are defined modulo 2. Comparing the notation of (2.8) with that of (3.2), by Proposition 2.9, $a_{n+1} = x$, $b_{n+1} = y$, $\rho_i = C_{i,n+1}$ for $i = 2, \dots, n$, $\rho_1 = C_{1,n+1} = 1$ if $M = \mathbb{T}$ and $\rho_1 = C_{1,n+1} = x^2$ if $M = \mathbb{K}$. Note also that the exponents in (3.2) have been renamed with respect to (2.8). To simplify the notation in what follows, for $q = 0, 2, \dots, n$, let $r_{1,q} = r_q$ and for $p = 1, 2$, let $l_{1,p} = l_p$. We also set $r_{1,1} = 0$. We will now take the image by $\hat{\phi}$ of some of the relations of Theorem 2.3 to obtain relations in $B_{n,m}(M)/\Gamma_2(B_m(M \setminus \{x_1, \dots, x_n\}))$ that we will simplify using Proposition 2.12. This will enable us to obtain information about the coefficients appearing in (3.2) above.

We first apply this procedure to relations (5) and (6) of Theorem 2.3.

**Lemma 3.2.** With the above notation, we have:

(1) $l_1 = 0$ if $M = \mathbb{T}$, and $l_1 = k_4 - r_2$ if $M = \mathbb{K}$.

(2) $l_2 = 0$.

(3) $k_4 = k_1$ if $M = \mathbb{T}$, and $k_2 = 0$ and $k_4 = -k_1 - 2i_2$ if $M = \mathbb{K}$.

(4) If $n \ge 3$, $k_4 = -r_2 - 2r_3$.

**Proof.** We start by studying the image by $\hat{\phi}$ of relation (5) of Theorem 2.3, where we substitute each term of the image by the corresponding term of (3.2). The left- and right-hand sides yield respectively:

$$\hat{\phi}(b^{-1}\sigma_1 a) = (\rho_n^{-j_n} \cdots \rho_2^{-j_2} \sigma^{-j_0} y^{-k_4} x^{-k_3} b^{-1})(\sigma_1 x^{l_1} y^{l_2} \sigma^{r_0} \rho_2^{r_2} \cdots \rho_n^{r_n})(ax^{k_1} y^{k_2} \sigma^{i_0} \rho_2^{i_2} \cdots \rho_n^{i_n}) \quad (3.3)$$

and

$$\hat{\phi}(\sigma_1 a \sigma_1 b^{-1} \sigma_1) = (\sigma_1 x^{l_1} y^{l_2} \sigma^{r_0} \rho_2^{r_2} \cdots \rho_n^{r_n})(ax^{k_1} y^{k_2} \sigma^{i_0} \rho_2^{i_2} \cdots \rho_n^{i_n})(\sigma_1 x^{l_1} y^{l_2} \sigma^{r_0} \rho_2^{r_2} \cdots \rho_n^{r_n}).$$
$$ (\rho_n^{-j_n} \cdots \rho_2^{-j_2} \sigma^{-j_0} y^{-k_4} x^{-k_3} b^{-1}) (\sigma_1 x^{l_1} y^{l_2} \sigma^{r_0} \rho_2^{r_2} \cdots \rho_n^{r_n}). \quad (3.4) $$

Using Proposition 2.12, we see that the conjugate by $a, b$ or $\sigma_1$ of $x, y, \rho_2$ or $\rho_3$ is a word in $x, y, \rho_2$ and $\rho_3$, and that $a, b$ and $\sigma_1$ commute with each of $\rho_4, \dots, \rho_n$ and $\sigma$. In this way, the terms of (3.3) and (3.4) involving $\rho_4, \dots, \rho_n$ and $\sigma$ commute with all of the other terms, so they may be gathered together on the right-hand side of each of the expressions, and the remaining terms in the canonical form do not involve the elements $\rho_4, \dots, \rho_n$ or $\sigma$. In particular, identifying the coefficients of these elements, for $k = 4, \dots, n$, we obtain $i_k - j_k + r_k = i_k - j_k + 3r_k$, in other words:

$$ r_k = 0 \text{ for } k = 4, \dots, n. $$

It follows from (3.3) and (3.4) that:

$$ \begin{aligned} & (\rho_3^{-j_3} \rho_2^{-j_2} y^{-k_4} x^{-k_3} b^{-1}) (\sigma_1 x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) (ax^{k_1} y^{k_2} \rho_2^{i_2} \rho_3^{i_3}) \\ & = (\sigma_1 x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) (ax^{k_1} y^{k_2} \rho_2^{i_2} \rho_3^{i_3}). \end{aligned} \quad (3.5) $$

Let $w_L$ and $w_R$ denote the left- and right-hand side of (3.5) respectively. We now put each of $w_L$ and $w_R$ in canonical form using relations (4)–(8) of Proposition 2.12 and Remarks 2.13(c). First suppose that $M = \mathbb{T}$. Then:

$$ (\sigma_1 x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) (ax^{k_1} y^{k_2} \rho_2^{i_2} \rho_3^{i_3}) = \sigma_1 a (x^{l_1+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}). $$

It follows that:

$$ \begin{aligned} w_L &= b^{-1} (x^{-k_3} y^{-k_4} \rho_2^{-k_3-j_2} \rho_3^{-j_3}) \sigma_1 a (x^{l_1+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \\ &= b^{-1} \sigma_1 (x^{-k_3} y^{-k_4} \rho_2^{k_3+j_2} \rho_3^{-k_3-j_2-j_3}) a (x^{l_1+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \\ &= b^{-1} \sigma_1 a (x^{-k_3+l_1+k_1} y^{-k_4+l_2+k_2} \rho_2^{-k_4+k_3+j_2+l_2+r_2+i_2} \rho_3^{-k_3-j_2-j_3+r_3+i_3}) \\ w_R &= \sigma_1 a (x^{l_1+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \sigma_1 (x^{l_1-k_3} y^{l_2-k_4} \rho_2^{r_2-j_2} \rho_3^{r_3-j_3}) b^{-1} \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 (x^{2l_1+k_1-k_3} y^{2l_2+k_2-k_4} \rho_2^{-l_2-i_2-j_2} \rho_3^{l_2+r_2+i_2+2r_3+i_3-j_3}) b^{-1} \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 b^{-1} (x^{2l_1+k_1-k_3} y^{2l_2+k_2-k_4} \rho_2^{-l_2-i_2-j_2+2l_1+k_1-k_3} \rho_3^{l_2+r_2+i_2+2r_3+i_3-j_3}) \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 b^{-1} \sigma_1 (x^{3l_1+k_1-k_3} y^{3l_2+k_2-k_4} \rho_2^{l_2+i_2+j_2-2l_1-k_1+k_3+r_2} \rho_3^{-j_2+2l_1+k_1-k_3+r_2+3r_3+i_3-j_3}). \end{aligned} $$

Thus $w_L$ and $w_R$ are now in canonical form, and applying relation (2) (the lift of relation (5) of Theorem 2.3) of Proposition 2.12, and comparing the coefficients of $x, y, \rho_2$ and $\rho_3$, we obtain parts (1)–(4) respectively of the statement, and the lemma is proved in the case $M = \mathbb{T}$.

Now suppose that $M = \mathbb{K}$. Then:

$$ \begin{aligned} & (\sigma_1 x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) (ax^{k_1} y^{k_2} \rho_2^{i_2} \rho_3^{i_3}) \\ &= \sigma_1 a (x^{l_1} (y^{l_2} x^{-2l_2} \rho_2^{l_2}) \rho_2^{r_2} \rho_3^{r_3}) (x^{k_1} y^{k_2} \rho_2^{i_2} \rho_3^{i_3}) \\ &= \sigma_1 a (x^{l_1-2l_2+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}), \end{aligned} \quad (3.6) $$

and thus:

$$ \begin{aligned} w_L &= b^{-1} (x^{k_3-2k_4} y^{-k_4} \rho_2^{-k_3+k_4-j_2} \rho_3^{-j_3}) \sigma_1 a (x^{l_1-2l_2+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \\ &= b^{-1} \sigma_1 (x^{-k_3-2j_2} y^{-k_4} \rho_2^{k_3-k_4+j_2} \rho_3^{-k_3+k_4-j_2-j_3}) a (x^{l_1-2l_2+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \\ &= b^{-1} \sigma_1 a (x^{-k_3-2j_2+2k_4+l_1-2l_2+k_1} y^{-k_4+l_2+k_2} \rho_2^{k_3-2k_4+j_2+l_2+r_2+i_2} \rho_3^{-k_3+k_4-j_2-j_3+r_3+i_3}) \end{aligned} $$

$$ \begin{aligned} w_R &= \sigma_1 a (x^{l_1-2l_2+k_1} y^{l_2+k_2} \rho_2^{l_2+r_2+i_2} \rho_3^{r_3+i_3}) \sigma_1 (x^{l_1-k_3} y^{l_2-k_4} \rho_2^{r_2-j_2} \rho_3^{r_3-j_3}) b^{-1} \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 (x^{2l_1+k_1+2r_2+2i_2-k_3} y^{2l_2+k_2-k_4} \rho_2^{-l_2-i_2-j_2} \rho_3^{l_2+r_2+i_2+2r_3+i_3-j_3}) b^{-1} \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 b^{-1} (x^{-2l_1-k_1-2r_2-2i_2+k_3+4l_2+2k_2-2k_4} y^{2l_2+k_2-k_4} \rho_2^{2l_1+k_1+2r_2+i_2-k_3-3l_2-k_2+k_4-j_2} \\ & \qquad \rho_3^{l_2+r_2+i_2+2r_3+i_3-j_3}) \sigma_1 (x^{l_1} y^{l_2} \rho_2^{r_2} \rho_3^{r_3}) \\ &= \sigma_1 a \sigma_1 b^{-1} \sigma_1 (x^{3l_1+k_1+2r_2-k_3-2l_2-2j_2} y^{3l_2+k_2-k_4} \rho_2^{-2l_1-k_1-r_2-i_2+k_3+3l_2+k_2-k_4+j_2} \\ & \qquad \rho_3^{2l_1+k_1+3r_2+2i_2-k_3-2l_2-k_2+k_4-j_2+3r_3+i_3-j_3}). \end{aligned} $$
Thus $w_L$ and $w_R$ are now in canonical form, and applying relation (2) (the lift of relation (5) of Theorem 2.3) of Proposition 2.12, and comparing the coefficients of $x$ and $y$, we obtain parts (1) and (2) respectively of the statement. Comparing the coefficients of $\rho_2$ and using part (2) of the statement, we obtain:

$$ -k_4 = -2(l_1 + r_2) - k_1 - 2i_2 + k_2. \qquad (3.7) $$

It follows from part (1) of the statement and (3.7) that:

$$ k_4 = -k_1 + k_2 - 2i_2. \qquad (3.8) $$

If $n \ge 3$, comparing the coefficients of $\rho_3$ and using parts (1) and (2) of the statement, we see that:

$$ 0 = 2k_4 + k_1 + r_2 + 2i_2 - k_2 + 2r_3. \qquad (3.9) $$

To obtain parts (3) and (4) of the statement, we analyse the image of relation (6) of Theorem 2.3 by $\hat{\phi}$ using the fact that $l_2 = 0$. We need only analyse the coefficients of $y$ and $\rho_2$, since these are the only generators of $B_m(M \setminus \{x_1, \dots, x_n\})^{Ab}$ that do not commute both with $a$ and $\sigma_1$. Let $w'_L = a(y^{k_2}\rho_2^{i_2})\sigma_1(\rho_2^{r_2})a(y^{k_2}\rho_2^{i_2})\sigma_1(\rho_2^{r_2})$, and let $w'_R = \sigma_1(\rho_2^{r_2})a(y^{k_2}\rho_2^{i_2})\sigma_1(\rho_2^{r_2})a(y^{k_2}\rho_2^{i_2})$. By a computation similar to that of (3.6), we have $a(y^{k_2}\rho_2^{i_2})\sigma_1(\rho_2^{r_2}) = a\sigma_1(x^{2i_2}y^{k_2}\rho_2^{r_2-i_2}\rho_3^{i_2})$ and $\sigma_1(\rho_2^{r_2})a(y^{k_2}\rho_2^{i_2}) = \sigma_1 a(y^{k_2}\rho_2^{r_2+i_2})$, so:

$$ \begin{aligned} w'_L &= a\sigma_1(x^{2i_2}y^{k_2}\rho_2^{-i_2+r_2}\rho_3^{i_2})a\sigma_1(x^{2i_2}y^{k_2}\rho_2^{-i_2+r_2}\rho_3^{i_2}) \\ &= a\sigma_1 a(x^{2(i_2-k_2)}y^{k_2}\rho_2^{k_2+r_2-i_2}\rho_3^{i_2})\sigma_1(x^{2i_2}y^{k_2}\rho_2^{r_2-i_2}\rho_3^{i_2}) = a\sigma_1 a\sigma_1(x^{2(i_2+r_2)}y^{2k_2}\rho_2^{-k_2}\rho_3^{i_2+k_2+r_2}) \end{aligned} $$

$$ \begin{aligned} w'_R &= \sigma_1 a(y^{k_2}\rho_2^{r_2+i_2})\sigma_1 a(y^{k_2}\rho_2^{r_2+i_2}) = \sigma_1 a\sigma_1(x^{2(r_2+i_2)}y^{k_2}\rho_2^{-i_2-r_2}\rho_3^{r_2+i_2})a(y^{k_2}\rho_2^{r_2+i_2}) \\ &= \sigma_1 a\sigma_1 a(x^{2(r_2+i_2-k_2)}y^{2k_2}\rho_2^{k_2}\rho_3^{r_2+i_2}). \end{aligned} $$

Part (3) of the statement follows by comparing the coefficients of $x$ and (3.8), and part (4) is a consequence of (3.9) and part (3). $\square$

**Lemma 3.3.**

(a) Let $M = \mathbb{T}$ or $\mathbb{K}$, and let $n \ge 4$. Then $r_{j,k} = 0$ for all $1 \le j \le n-1$ and $k = 2, \dots, j-1, j+3, \dots, n$.

(b) Let $n \ge 3$. If $M = \mathbb{T}$ (resp. $M = \mathbb{K}$), and $2 \le i \le n-2$, then $l_{i,j} = l_{i+1,j} = l_{1,j}$ (resp. $l_{i,j} = l_{i+1,j}$) for $j = 1, 2$, $r_{1,0} \equiv r_{i,0} \equiv r_{i+1,0} \mod 2$, and:

$$ -2r_{i+1,i+1} - r_{i+1,i+2} = -2r_{i,i} - r_{i,i+1}. \qquad (3.10) $$

Further:

$$ -2r_{i,i} - r_{i,i+1} = 2r_3 + r_2 \text{ for all } i = 2, \dots, n-1. \qquad (3.11) $$

(c) If $M = \mathbb{K}$, $l_{k,1} = 0$ for all $2 \le k \le n-1$.

*Proof.* We first prove part (a). Let $M = \mathbb{T}$ or $\mathbb{K}$. Recall from the proof of Proposition 2.9 that $\rho_1 = C_{1,n+1}$ is equal to 1 (resp. to $x^2$) if $M = \mathbb{T}$ (resp. if $M = \mathbb{K}$). First let $n \ge 4$, and let $1 \le i, j \le n-1$ be such that $|i-j| \ge 2$. Applying Proposition 2.9 and (2.34), we have:

$$ \rho_i^{r_{j,i+1}} \rho_{i+1}^{-2r_{j,i+1}} \rho_{i+2}^{r_{j,i+1}} \rho_j^{-r_{i,j+1}} \rho_{j+1}^{2r_{i,j+1}} \rho_{j+2}^{-r_{i,j+1}} = 1, $$

which is in canonical form (possibly up to permutation of some of the factors). Comparing the coefficients of $\rho_{i+1}$ (resp. $\rho_{j+1}$) if $i < j$ (resp. $i > j$) and using once more Proposition 2.9, we see that $r_{j,i+1} = 0$ (resp. $r_{i,j+1} = 0$). So for all $1 \le j \le n-1$, $r_{j,k} = 0$ for all $k = 2, \dots, j-1$ (resp. for all $k = j+3, \dots, n$), which proves part (a).

Now let $n \ge 3$, and let $1 \le i \le n-2$. Using Proposition 2.9, equation (2.35) may be written as:

$$ x^{l_{i+1,1}-l_{i,1}} y^{l_{i+1,2}-l_{i,2}} \sigma^{r_{i+1,0}-r_{i,0}} \prod_{\substack{k=1 \\ k \ne i+1, i+2}}^{n} \rho_k^{r_{i+1,k}-r_{i,k}} \cdot \rho_i^{-r_{i,i+1}-r_{i,i+2}} \rho_{i+1}^{\rho} \rho_{i+2}^{-\rho} \rho_{i+3}^{\rho'} = 1, \qquad (3.12) $$
where $\rho = 2r_{i,i+2} + 2r_{i+1,i+1} + r_{i,i+1} + r_{i+1,i+2}$ and $\rho' = r_{i+1,i+1} + r_{i+1,i+2}$. Now (3.12) is in canonical form, and using the fact that $r_{i+1,i} = r_{i,i+3} = 0$ by part (a), and comparing the coefficients of $\rho_i, \rho_{i+1}, \rho_{i+3}, x, y$ and $\sigma$, we deduce that:

$$r_{i,i} + r_{i,i+1} + r_{i,i+2} = 0 \text{ for } 2 \le i \le n-2 \qquad (3.13)$$

$$-2r_{i+1,i+1} - r_{i+1,i+2} = 2r_{i,i+2} + r_{i,i+1} \text{ for } 1 \le i \le n-2 \qquad (3.14)$$

$$r_{i+1,i+1} + r_{i+1,i+2} + r_{i+1,i+3} = 0 \text{ for } 1 \le i \le n-3.$$

$$l_{i,j} = l_{i+1,j}, \text{ where } j = 1, 2 \text{ and } 1 \le i \le n-2 \text{ (resp. } 2 \le i \le n-2) \text{ if } M = \mathbb{T} \text{ (resp. } M = \mathbb{K}) \qquad (3.15)$$

$$r_{i,0} \equiv r_{i+1,0} \mod 2 \text{ for } 1 \le i \le n-2,$$

from which we obtain the relations involving $l_{i,j}$ and $r_{i,0}$. Equation (3.10) follows from (3.13) and (3.14) for all $2 \le i \le n-2$. Replacing $i$ by $i-1$ in (3.10) and using induction on $i$, we see that $-2r_{i,i} - r_{i,i+1} = -2r_{2,2} - r_{2,3}$ for all $2 \le i \le n-1$. Equation (3.11) then follows from this by applying (3.14) with $i=1$. This proves part (b).

To prove part (c), let $M = \mathbb{K}$. Since $\rho_1 = x^2$, it follows by taking $i=1$ in (3.12) and comparing the coefficients of $\rho_1$ that $l_{2,1} - l_1 + 2(r_{2,1} - r_1 - r_2 - r_3) = 0$. Since $r_1 = 0$ by definition and $r_{2,1} = 0$ by part (b), we see that:

$$2(r_2 + r_3) + l_1 = l_{2,1}, \quad l_{1,2} = l_{2,2}, \text{ and } r_{1,0} \equiv r_{2,0} \mod 2, \qquad (3.16)$$

where we also compare the coefficients of $y$ and $\sigma$ in (3.12). Now by Lemma 3.2(1) and (4), $l_1 = k_4 - r_2 = -2(r_2 + r_3)$, and we deduce from (3.16) that $l_{2,1} = 0$, and then from (3.15) that $l_{k,1} = 0$ for all $2 \le k \le n-1$. This proves part (c) of the statement. $\square$

We now complete the proof of Theorem 1.1.

*Proof of Theorem 1.1.* Recall that $\rho_1 = 1$ (resp. $\rho_1 = x^2$) if $M = \mathbb{T}$ (resp. if $M = \mathbb{K}$). We compare the exponents of $\rho_2$ in (2.33). Using Proposition 2.9, we obtain:

$$\gamma + m = \alpha + \delta. \qquad (3.17)$$

where $\alpha, \gamma$ and $\delta$ are given by (2.25) (where $t=n$), (2.30) and (2.27) respectively. Since $s=0$, $\gamma=0$ by (2.30), and using (2.27), equation (3.2) and Lemma 3.2(3), we see that:

$$\delta = \begin{cases} \beta_{1,2} + \alpha_{1,1} = k_4 + k_1 & \text{if } M = \mathbb{T} \\ 2(\alpha_{1,2} - x_{1,1} - x_{1,2}) - \alpha_{1,1} + \beta_{1,2} = -2i_2 - k_1 + k_4 & \text{if } M = \mathbb{K} \end{cases} \\ = 2k_4, \qquad (3.18)$$

using the fact that in the case $M = \mathbb{K}$ that $x_{1,2} = i_2$, and $x_{1,1} = 0$, and that $\alpha_{1,2} = k_2 = 0$ by Lemma 3.2(3). If $n=2$ then $\alpha=0$ by (2.25), and it follows from (3.17) and (3.18) that $m = nk_4$ as required. Suppose now that $n \ge 4$. Using (2.25), Lemma 3.3(a) and (3.10), we have:

$$\alpha = \sum_{l=2}^{n-1} (2r_{l,l} + r_{l,l+1}) = (n-2)(2r_{2,2} + r_{2,3}). \qquad (3.19)$$

Observe that (3.19) also holds if $n=3$. Applying (3.11) and Lemma 3.2(4), for all $n \ge 3$, we obtain:

$$\alpha = (n-2)(2r_{2,2} + r_{2,3}) = -(n-2)(2r_3 + r_2) = (n-2)k_4,$$

which using (3.17) and (3.18) implies that $m = nk_4$. This completes the proof of the theorem. $\square$

## 4. GENERALISATION TO SEVERAL FACTORS

In this section, we turn our attention to the case of the short exact sequence (1.4) involving mixed braid groups with more than two factors.
4.1. **The case $q = k - 1$.** We start by proving Theorem 1.2 which is a straightforward consequence of Theorem 1.1.

*Proof of Theorem 1.2.* Let $M = \mathbb{T}$ or $\mathbb{K}$, and assume that $q = k - 1$, so the homomorphism in the short exact sequence (1.4) is $p_*: B_{n_1, \dots, n_k}(M) \to B_{n_1}(M)$.

Suppose first that $n_1$ divides $n_i$ for $i = 2, \dots, k$, in other words there exists $l_i \in \mathbb{N}$ such that $n_i = l_i n_1$. In a manner similar to that of the proof of Theorem 1.1, we may construct a cross-section on the level of configuration spaces using the non-vanishing vector field of $M$ as follows. Let $s_{n,l}: D_n(M) \to D_{ln}(M \setminus \{x_1, \dots, x_n\})$ be the map defined by:

$$s_{n,l}(x) = (s_1(x), \dots, s_n(x)), \qquad (4.1)$$

where for $i = 1, \dots, n$, the map $s_i$ is defined by (3.1). Then the map $s: D_{n_1}(M) \to D_{n_1, l_2 n_1, \dots, l_k n_1}(M)$ defined by $s(x) = (x, s_{n_1, l_2}(x), \dots, s_{n_1, l_k}(x))$ for all $x \in D_{n_1}(M)$ is well defined and continuous, and it is a cross-section for $p$. Hence the induced homomorphism $s_*: B_{n_1}(M) \to B_{n_1, l_2 n_1, \dots, l_k n_1}(M)$ is a section for $p_*$.

Conversely, suppose that $s_*: B_{n_1}(M) \to B_{n_1, \dots, n_k}(M)$ is a section for $p_*$. For $i = 2, \dots, k$, let $(p_i)_*: B_{n_1, n_i}(M) \to B_{n_1}(M)$ (resp. $q_i: B_{n_1, \dots, n_k}(M) \to B_{n_1, n_i}(M)$) be the projection obtained by forgetting the second block (resp. all of the blocks with the exception of the 1st and the i-th block). Then $(p_i)_* \circ q_i = p_*$, and it follows that $q_i \circ s_*$ is a section for $(p_i)_*$. So by Theorem 1.1, $n_1$ divides $n_i$ for all $i = 2, \dots, k$ as required. $\square$

We shall make use of the following lemma to prove Theorem 1.3.

**Lemma 4.1.** Let $k \ge 2$, let $n_1, \dots, n_k \in \mathbb{N}$, let $s = \sum_{i=2}^{k-1} n_i$ and $n = n_1 + s$. Let $\iota_1: B_{n_1, \dots, n_k}(M) \to B_{n_1, s, n_k}(M)$ and $\iota_2: B_{n_1, s, n_k}(M) \to B_{n, n_k}(M)$ denote the corresponding inclusions, and let $K = B_{n_k}(M \setminus \{x_1, \dots, x_n\})$. Then $\iota_1$ and $\iota_2$ induce injective homomorphisms $\tilde{\iota}_1: B_{n_1, \dots, n_k}(M)/\Gamma_2(K) \to B_{n_1, s, n_k}(M)/\Gamma_2(K)$ and $\tilde{\iota}_2: B_{n_1, s, n_k}(M)/\Gamma_2(K) \to B_{n, n_k}(M)/\Gamma_2(K)$.

*Proof.* Let $f: B_{n, n_k}(M) \to B_n(M)$ be the homomorphism given geometrically by forgetting the last $n_k$ strings, and let $g: B_{n_1, s, n_k}(M) \to B_{n_1, s}(M)$ and $h: B_{n_1, \dots, n_k}(M) \to B_{n_1, \dots, n_{k-1}}(M)$ denote the restriction of $f$ to $B_{n_1, \dots, n_k}(M)$ and to $B_{n_1, s, n_k}(M)$ respectively. Then $K = \text{Ker}(f) = \text{Ker}(g) = \text{Ker}(h)$, and we have the following commutative diagram of short exact sequences:

<figure><img src="image_5.png" /></figure>

where $\tilde{\tau}_1: B_{n_1, \dots, n_{k-1}}(M) \to B_{n_1, s}(M)$ and $\tilde{\tau}_2: B_{n_1, s}(M) \to B_n(M)$ denote the corresponding inclusions. This gives rise to the following commutative diagram of short exact sequences:

$$ \begin{array}{ccc} 1 & \to & K^{\text{Ab}} \to B_{n_1, \dots, n_k}(M)/\Gamma_2(K) \xrightarrow{\tilde{h}} B_{n_1, \dots, n_{k-1}}(M) \to 1 \\ \hline 1 & \to & K^{\text{Ab}} \to B_{n_1, s, n_k}(M)/\Gamma_2(K) \xrightarrow{\tilde{g}} B_{n_1, s}(M) \to 1 \end{array} \qquad (4.2) $$
where $\hat{l}_1, \hat{l}_2, \hat{f}, \hat{g}$ and $\hat{h}$ are the homomorphisms induced by $l_1, l_2, f, g$ and $h$ respectively. For $i = 1, 2$, the injectivity of $\hat{l}_i$ is a consequence of that of $\bar{l}_i$ and of standard diagram-chasing arguments. $\square$

We now prove Theorem 1.3. The techniques are similar to those of the proof of Theorem 1.1, and we will also use some of the results of Section 3 in conjunction with Lemma 4.1.

**Proof of Theorem 1.3.** Let $M$ be the 2-torus or the Klein bottle, let $p: D_{n_1, \dots, n_k}(M) \to D_{n_1, \dots, n_{k-1}}(M)$ be the projection given by forgetting the last $n_k$ points, and consider the induced homomorphism $p_*: B_{n_1, \dots, n_k}(M) \to B_{n_1, \dots, n_{k-1}}(M)$.

(a) Suppose that there exist $l_1, \dots, l_{k-1} \in \mathbb{N}$ such that $n_k = l_1 n_1 + \dots + l_{k-1} n_{k-1}$. Once more, the existence of a non-vanishing vector field guarantees the existence of a cross-section on the level of configuration spaces. More precisely, let $x \in D_{n_1, \dots, n_{k-1}}(M)$, where $x = (x_{n_1}, \dots, x_{n_{k-1}})$ and $x_{n_i} \in D_{n_i}(M)$ for $i = 1, \dots, k-1$. With the notation of (4.1), let $s: D_{n_1, \dots, n_{k-1}}(M) \to D_{n_1, \dots, n_{k-1}, n_k}(M)$ be the map defined by:

$$s(x) = (x, s_{n_1, l_1}(x_{n_1}), \dots, s_{n_{k-1}, l_{k-1}}(x_{n_{k-1}})) \text{ for all } x \in D_{n_1, \dots, n_{k-1}}(M).$$

Then $s$ is a cross-section for $p$, and the induced homomorphism $s_*: B_{n_1, \dots, n_{k-1}}(M) \to B_{n_1, \dots, n_{k-1}, n_k}(M)$ is a section for $p_*$.

(b) Let $k \ge 2$, and let $n_1, \dots, n_k \in \mathbb{N}$. Suppose that $p_*: B_{n_1, \dots, n_k}(M) \to B_{n_1, \dots, n_{k-1}}(M)$ admits a section $s_*$. If $n_1 = 1$ then it suffices to take $l_1 = n_k$ and $l_2 = \dots = l_{k-1} = 0$. So suppose that $n_1 \ge 2$. We first determine a generating set and some relations of the group $B_{n_1, \dots, n_l}(M)$, where $l \in \{k-1, k\}$. Let $l \ge 1$. Using induction on $l$, applying the methods of [17, Proposition 1, p.139] to the short exact sequence (1.4) with $q = 1$, and arguing as in the proof of Proposition 2.11, one may show that:

$$\{a_i, b_i, n_1 + 1 \le i \le n_1 + \dots + n_l\} \cup \{C_{i,j}, 1 \le i < j, n_1 + 1 \le j \le n_1 + \dots + n_l\} \cup \\ \{a, b\} \cup \left\{ \sigma_i, \text{ where } 1 \le i \le n_1 + \dots + n_l - 1, \text{ and } i \ne \sum_{t=1}^{r} n_t \text{ for } r = 1, \dots, l-1 \right\}$$

is a generating set for $B_{n_1, \dots, n_l}(M)$. If $M = \mathbb{T}$ (resp. $M = \mathbb{K}$), set:

$$S_1 = bab^{-1}a^{-1} \text{ (resp. } S_1 = ba^{-1}b^{-1}a^{-1}\text{), and } S_2 = \sigma_1 \cdots \sigma_{n_1-2} \sigma_{n_1-1}^2 \sigma_{n_1-2} \cdots \sigma_1. \quad (4.3)$$

As for relation (1) of Proposition 2.11, the surface relation of $B_{n_1, \dots, n_l}(M)$ may be written as:

$$S_2^{-1}S_1 = \prod_{i=n_1+1}^{n_1+\dots+n_l} C_{1,i}C_{2,i}^{-1}. \quad (4.4)$$

In what follows, let $n = \sum_{i=1}^{k-1} n_i$. By (1.4), $\mathrm{Ker}(p_*)$ may be identified with $B_{n_k}(M \setminus \{x_1, \dots, x_n\})$. Let $G = B_{n_1, \dots, n_k}(M)/\Gamma_2(\mathrm{Ker}(p_*))$. Then $p_*$ induces a short exact sequence:

$$1 \to (\mathrm{Ker}(p_*))^{\mathrm{Ab}} \to G \xrightarrow{\hat{p}_*} B_{n_1, \dots, n_{k-1}}(M) \to 1, \quad (4.5)$$

where $\hat{p}_*: G \to B_{n_1, \dots, n_{k-1}}(M)$ is the homomorphism induced by $p_*$. Note that (4.5) is the upper row of (4.2), and $\hat{p}_* = \hat{g}$. Using the hypothesis that $s_*$ is a section for $p_*$, there exists a section $\hat{\phi}: B_{n_1, \dots, n_{k-1}}(M) \to G$ for $\hat{p}_*$ induced by $s_*$. Making use of Proposition 2.12 and the proof of Proposition 2.9, we obtain the following information in $G$:

* using the Artin relations, we see that $\sigma_i = \sigma_{i+1}$ in $G$ for all $n+1 \le i \le n+n_k-2$: we denote the $\Gamma_2(\mathrm{Ker}(p_*))$-coset of $\sigma_i$ by $\sigma_i$.

* for $n+1 \le i < j \le n+n_k$, $C_{i,j} = 1$ in $G$, and $\sigma$ is of order 2 in $G$.

* for $n+1 \le i \le n+n_k-1$, $a_i = a_{i+1}$ and $b_i = b_{i+1}$: we denote the $\Gamma_2(\mathrm{Ker}(p_*))$-cosets of these elements by $x$ and $y$ respectively.

* for $1 \le l \le n$ and $n+1 \le j \le n+n_k-1$, we have $C_{l,j} = C_{l,j+1}$: we denote the coset of these elements by $\rho_l$, where $\rho_1 = 1$ if $M = \mathbb{T}$ and $\rho_1 = x^2$ if $M = \mathbb{K}$. To simplify further the notation in what follows, we set $\rho_{n+1} = 1$.
• $(\mathrm{Ker}(p_*) )^{\mathrm{Ab}}$ is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}^{n+1}$, and the factors of this decomposition are generated by the elements $\sigma, x, y, \rho_2, \dots, \rho_n$. In particular, in $G$, these elements commute pairwise, and the notion of canonical form defined just after equation (2.8) carries over to the situation of the short exact sequence (4.5).

By Lemma 4.1, if $w \in G$ is such that it becomes the trivial element when viewed as an element of $B_{n,n_k}(M)/\Gamma_2(\mathrm{Ker}(p_*))$ then $w$ is itself trivial. In particular, if we take $m = n_k$, then the relations of Proposition 2.12 that exist as expressions in $G$ are also relations in $G$. In particular, the following relations are valid in $G$:

$$
(i) \quad S_2^{-1} S_1 = \begin{cases} \left( \prod_{i=n_1+1}^n C_{1,i} C_{2,i}^{-1} \right) \rho_2^{-n_k} & \text{if } M = \mathbb{T} \\ \left( \prod_{i=n_1+1}^n C_{1,i} C_{2,i}^{-1} \right) \rho_2^{-n_k} x^{2n_k} & \text{if } M = \mathbb{K}. \end{cases}
$$

(ii) For $n_1 + 1 \le i \le n$, we have:

$$
(a) \quad a_i^{-1} y a_i = y \rho_i^{-1} \rho_{i+1}, \quad b_i^{-1} y b_i = \begin{cases} y & \text{if } M = \mathbb{T} \\ y \rho_i \rho_{i+1}^{-1} & \text{if } M = \mathbb{K} \end{cases} \quad \text{and} \quad b_i^{-1} x b_i = \begin{cases} x \rho_i \rho_{i+1}^{-1} & \text{if } M = \mathbb{T} \\ x \rho_i^{-1} \rho_{i+1} & \text{if } M = \mathbb{K}. \end{cases}
$$

(b) for $1 \le q \le n$, $a_i^{-1}\rho_q a_i = \rho_q$, and:

$$
b_i^{-1} \rho_q b_i = \begin{cases} \rho_q & \text{if } M = \mathbb{T}, \text{ or if } M = \mathbb{K} \text{ and } i < q \\ \rho_i^{-2} \rho_{i+1}^2 \rho_q & \text{if } M = \mathbb{K} \text{ and } i \ge q. \end{cases}
$$

(iii) $a$ commutes with $\sigma, x$ and $\rho_q$, where $2 \le q \le n$, and $\sigma$ commutes with $b_j$ for $n_1 + 1 \le j \le n$.
If $M = \mathbb{T}$, then $a^{-1}ya = y\rho_2$, $aya^{-1} = y\rho_2^{-1}$, $b^{-1}xb = x\rho_2^{-1}$, $bxb^{-1} = x\rho_2$, and $b$ commutes with $y$. If
$M = \mathbb{K}$ then $a^{-1}ya = yx^{-2}\rho_2$, $aya^{-1} = yx^2\rho_2^{-1}$, $b^{-1}xb = x^{-1}\rho_2$, $bxb^{-1} = x^{-1}\rho_2^{-1}$, $b^{-1}yb = yx^2\rho_2^{-1}$ and
$byb^{-1} = yx^{-2}\rho_2$.

Relation (i) follows from (4.4), and relations (ii)(a) (resp. relations (ii)(b)) follow from relations (2), (6)
and (7) (resp. relations (3) and (8)) of Theorem 2.1, using the above information about G, and not-
ably the fact that the $\Gamma_2(\mathrm{Ker}(p_*))$-cosets of $a_j, b_j$ and $C_{i,j}$ are $x, y$ and $\rho_i$ respectively. Relations (iii)
are consequences of Proposition 2.12 and Remarks 2.13(c). One may check that if $M = \mathbb{K}$ and $i \ge q$
then $b_i\rho_q b_i^{-1} = \rho_i^{-2}\rho_{i+1}^2\rho_q$.

To complete the proof of part (b), we follow the strategy of the proof of Theorem 1.1 by studying
the images of some of the relations of $B_{n_1, \dots, n_{k-1}}(M)$ under the homomorphism $\hat{\phi}$. We may write the
images of the elements $a, b$ and $\sigma_i$, where $1 \le i \le n-1$ and $i \ne \sum_{t=1}^r n_t$ for $r=1, \dots, k-2$, in the
form of equation (3.2), where $n$ is taken to be equal to $\sum_{i=1}^{k-1} n_i$. Similarly, for $n_1+1 \le j \le n$, we
set:

$$
\hat{\phi}(b_j) = b_j \cdot x^{t_j} y^{p_j} \sigma^{s_{j,0}} \rho_2^{s_{j,2}} \cdots \rho_n^{s_{j,n}},
$$

where $t_j, p_j, s_{j,2}, \dots, s_{j,n} \in \mathbb{Z}$, and $s_{j,0}$ is defined modulo 2.

With appropriate restrictions on $i$ and $j$, the conclusions of Lemmas 2.15 and 3.2 are also valid
here. More precisely, set $t = n_1$, $s = \sum_{i=2}^{k-1} n_i$, so $t + s = n$, and $m = n_k$ as in the statement of
Lemma 4.1, and let $\Gamma' = \{ \sum_{t=1}^r n_t \mid r = 1, \dots, k-2 \}$. It follows from that lemma that if $w$ is an
element of $G$ for which either $\hat{l}_1(w)$ is a relator in $B_{n_1, s, n_k}(M)/\Gamma_2(K)$ or $\hat{l}_2 \circ \hat{l}_1(w)$ is a relator in
$B_{n, n_k}(M)/\Gamma_2(K)$ then $w$ is a relator in $G$. In particular, in the current setting:

(a) for all $t \ge 4$ (resp. $t \ge 3$), the conclusion of Lemma 2.15(a) (resp. Lemma 2.15(b)) holds for all $i, j \in \{1, \dots, n-1\} \setminus \Gamma'$ (resp. for all $1 \le i \le t-2$).

(b) the conclusion of Lemma 3.2 holds.

(c) the conclusion of Lemma 3.3 remains valid when $n$ is replaced by $t = n_1$ and $m$ is replaced by $\sum_{i=2}^{k} n_i$.
Taking $l = k - 1$, let us study the surface relation (4.4) of $B_{n_1, \dots, n_{k-1}}(M)$ using (4.3). By (2.24) and (2.26), the canonical form of $\hat{\phi}(S_2^{-1}S_1)$ is given by:

$$ \hat{\phi}(S_2^{-1}S_1) = S_2^{-1}S_1 \cdot w' \rho_2^{\alpha+\delta}, \quad (4.6) $$

where $w'$ is a word in $x, y, \sigma, \rho_3, \dots, \rho_n$. Using (2.25), (3.10) and Lemmas 3.3(a) and 3.2(4), it follows that $\alpha = (n_1 - 2)k_4$. By (3.18) and Lemma 3.2(3), we have $\delta = 2k_4$, and it follows that the exponent of $\rho_2$ in the canonical form of $\hat{\phi}(S_2^{-1}S_1)$ given in (4.6) is equal to $n_1k_4$.

To compute the exponent of $\rho_2$ in the canonical form of $\hat{\phi}(\prod_{j=n_1+1}^n C_{1,j}C_{2,j}^{-1})$, we first study $C_{1,j}C_{2,j}^{-1}$ in $P_n(M)$ for $j = n_1 + 1, \dots, n$. For such values of $j$, taking $i = 1$ in relation (2) of Theorem 2.1 and recalling that $a = a_1$, we see that $a^{-1}b_ja = b_ja_jC_{1,j}^{-1}C_{2,j}a_j^{-1}$, and thus:

$$ C_{2,j}^{-1}C_{1,j} = a_j^{-1}a^{-1}b_j^{-1}ab_ja_j, \quad (4.7) $$

using the fact that $a$ and $a_j$ commute by relation (1) of Theorem 2.1. On the other hand, taking $i = j = 1$ and $k = j$ in relation (3) of Theorem 2.1, we see that $a^{-1}C_{1,j}a = a_jC_{2,j}^{-1}C_{1,j}a_j^{-1}C_{2,j}$, and using the fact that $a$ commutes with $C_{2,j}$ by the same relation, we obtain $C_{1,j}C_{2,j}^{-1} = aa_jC_{2,j}^{-1}C_{1,j}a_j^{-1}a^{-1}$. Substituting (4.7) in this equation, in $P_n(M)$ it follows that:

$$ C_{1,j}C_{2,j}^{-1} = b_j^{-1}ab_ja^{-1}. \quad (4.8) $$

Let us compute $\hat{\phi}(b_j^{-1}ab_ja^{-1})$. Note that $j \ge 3$ since $n_1 \ge 2$.

* If $M = \mathbb{T}$, using relations (ii)(a), (ii)(b) and (iii) above, we have:

$$ \begin{aligned} \hat{\phi}(b_j^{-1}ab_ja^{-1}) &= y^{-p_j}x^{-t_j}b_j^{-1}ax^{k_1}y^{k_2}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1} \\ &= b_j^{-1}y^{-p_j}x^{-t_j}w_{j,j+1}ax^{k_1}y^{k_2}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1} \\ &= b_j^{-1}ax^{k_1-t_j}y^{k_2-p_j}\rho_2^{-p_j}w_{j,j+1}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1} = b_j^{-1}ab_j\rho_2^{-p_j}w'_{j,j+1}a^{-1} \\ &= b_j^{-1}ab_ja^{-1}\rho_2^{-p_j}w'_{j,j+1}, \end{aligned} \quad (4.9) $$

where $w_{j,j+1}$ and $w'_{j,j+1}$ are words in $\rho_j$ and $\rho_{j+1}$.

* If $M = \mathbb{K}$, since $j \ge 3$, for all $k \le j$, we have $b_j^{-1}\rho_k b_j = v\rho_k$, where $v$ is a word in $\rho_j$ and $\rho_{j+1}$. Then by relations (ii)(a), (ii)(b) and (iii) above, we obtain:

$$ \begin{aligned} \hat{\phi}(b_j^{-1}ab_ja^{-1}) &= y^{-p_j}x^{-t_j}b_j^{-1}ax^{k_1}y^{k_2}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1}w''_{j,j+1} \\ &= b_j^{-1}y^{-p_j}x^{-t_j}w_{j,j+1}ax^{k_1}y^{k_2}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1}w''_{j,j+1} \\ &= b_j^{-1}ax^{k_1-t_j}y^{k_2-p_j}\rho_2^{-p_j}w_{j,j+1}b_jx^{t_j-k_1}y^{p_j-k_2}a^{-1}w''_{j,j+1} \\ &= b_j^{-1}ab_j\rho_2^{-p_j}w_{j,j+1}a^{-1}w''_{j,j+1} = b_j^{-1}ab_ja^{-1}\rho_2^{-p_j}w'_{j,j+1}, \end{aligned} \quad (4.10) $$

where $w_{j,j+1}$, $w'_{j,j+1}$ and $w''_{j,j+1}$ are words in $\rho_j$ and $\rho_{j+1}$.

Using (4.4) with $l = k - 1$ for both $M = \mathbb{T}$ and $M = \mathbb{K}$, and (4.8)-(4.10), we see that:

$$ \begin{aligned} \hat{\phi}(S_2^{-1}S_1) &= \hat{\phi} \left( \prod_{j=n_1+1}^{n} C_{1,j}C_{2,j}^{-1} \right) = \hat{\phi} \left( \prod_{j=n_1+1}^{n} b_j^{-1}ab_ja^{-1} \right) = \prod_{j=n_1+1}^{n} b_j^{-1}ab_ja^{-1}\rho_2^{-p_j}w'_{j,j+1} \\ &= \left( \prod_{j=n_1+1}^{n} C_{1,j}C_{2,j}^{-1}w'_{j,j+1} \right) \rho_2^{-\sum_{j=n_1+1}^{n} p_j}, \end{aligned} \quad (4.11) $$

where to obtain the last equality, we have used also relation (ii)(b). When we put (4.11) in canonical form, relations (ii)(a), (ii)(b) and (iii) imply that no new terms in $\rho_2$ are introduced during this
process. It follows from relation (i), (4.6) and (4.11) that $n_1k_4 - n_k = -\sum_{j=n_1+1}^{n} p_j$, hence:

$$n_k = n_1k_4 + \sum_{j=n_1+1}^{n} p_j. \qquad (4.12)$$

To complete the proof, it remains to compute the terms $p_j$ in (4.12) for $j = n_1 + 1, \dots, n$. Let $\Gamma = \{\sum_{t=1}^r n_t | r = 1, \dots, k-1\}$. We claim that $l_{i,2} = 0$ for all $1 \le i \le n-1$ and $i \notin \Gamma$. To see this, first note that relations (2)–(8) of Proposition 2.12 and relations (1)–(7) of Theorem 2.3 hold in our setting, with the exception of those relations involving $\sigma_i$ or $\sigma_j$, where $1 \le i, j \le n-1$ and $\{i, j\} \cap \Gamma \ne \emptyset$. If $i=1$ then by considering the relation $b^{-1}\sigma_1 a = \sigma_1 a \sigma_1 b^{-1}\sigma_1$ and arguing in a manner similar to that of the proof of Lemma 3.2(2), we see that $l_{1,2} = 0$. Now suppose that $2 \le i \le n-1$ and that $i \notin \Gamma$. Using relations (2)–(3) of Theorem 2.1, for $i < j \le n$ we have $a_i^{-1}b_ja_i = b_ja_jC_{i,j}^{-1}C_{i+1,j}a_j^{-1} = b_ja_i^{-1}C_{i+1,j}C_{i,j}^{-1}a_i$, and thus $C_{i,j}C_{i+1,j}^{-1} = b_j^{-1}a_ib_ja_i^{-1}$. Taking $j = i+1$ in this relation and using the fact that $C_{i,i+1} = \sigma_i^2$, it follows that $\sigma_i^2 = b_{i+1}^{-1}a_ib_{i+1}a_i^{-1}$, and using the equality:

$$b_{i+1} = \sigma_i^{-1}b_i\sigma_i^{-1} \qquad (4.13)$$

obtained via relation (5) of Proposition 2.7, we obtain:

$$b_i^{-1}\sigma_i a_i = \sigma_i a_i \sigma_i b_i^{-1}\sigma_i. \qquad (4.14)$$

Let $q_i$ denote the exponent of $y$ in the canonical form of $\hat{\phi}(a_i)$. Since the exponent of $y$ is the same on both sides of each of the relations (ii)(a), (ii)(b) and (iii), it follows that the exponent of $y$ in the canonical form of $\hat{\phi}(b_i^{-1}\sigma_i a_i)$ (resp. of $\hat{\phi}(\sigma_i a_i \sigma_i b_i^{-1}\sigma_i)$) is equal to $l_{i,2} - p_i + q_i$ (resp. to $3l_{i,2} - p_i + q_i$). Using the fact that (4.14) also holds when viewed as a relation in $G$, we deduce that $l_{i,2} = 0$, which proves the claim. In a similar manner, if $\sum_{t=1}^r n_t < i < \sum_{t=1}^{r+1} n_t$, where $r = 1, \dots, k-2$, then computing the exponent of $y$ in the image by $\hat{\phi}$ of (4.13), and using the fact that this equality also holds in $G$, we obtain $p_{i+1} = p_i + 2l_{i,2}$, and thus $p_i = p_{i+1}$ since $l_{i,2} = 0$. So there exists $\alpha_{n_{r+1}} \in \mathbb{Z}$ such that $p_i = \alpha_{n_{r+1}}$ for all $\sum_{t=1}^r n_t < i \le \sum_{t=1}^{r+1} n_t$ and $r = 1, \dots, k-2$. We deduce from (4.12) that $n_k = n_1k_4 + n_2\alpha_{n_2} + \dots + n_{k-1}\alpha_{n_{k-1}}$, and this completes the proof of the theorem. $\square$

### 4.2. The case $k=3$, $q=1$ and $n_3=1$. In Theorem 1.3(a), we obtained a geometric section on the level of configuration spaces by adding new distinct points in accordance with the relation between $n_k$ and $n_1, \dots, n_{k-1}$ using the non-vanishing vector field on $\mathbb{T}$ and $\mathbb{K}$. However, the algebraic techniques used to prove the relation of Theorem 1.3(b) leave open the possibility that some of the coefficients of $n_1, \dots, n_{k-1}$ in that relation be negative, and it is not clear how to interpret this geometrically. In this section, we study the case where $M = \mathbb{T}$ or $M = \mathbb{K}$, $k=3$, $q=1$ and $n_3=1$, which is the situation of Theorem 1.4. In this case, if $n_1, n_2 \ge 2$ are coprime then there exist $l_1, l_2 \in \mathbb{Z}$ such that $n_3 = l_1n_1 + l_2n_2$, and one of $l_1$ and $l_2$ must be negative. As we shall see, there does not exist a section in this case. This gives some evidence to support the conjecture that the converse of Theorem 1.3(a) is true, namely that a section on the algebraic level is induced by a geometric section via the non-vanishing vector field, or in other words, the coefficients of $n_1, \dots, n_{k-1}$ in the statement of Theorem 1.3(b) must in fact be non negative.

Let $M$ be the 2-torus or the Klein bottle, let $k=3$, $q=1$ and $n_3=1$, let $n_1 = t$, $n_2 = s$, where $t, s \ge 2$, and let $n = t + s$. We study the projection $p_*: B_{t,s,1}(M) \to B_{t,s}(M)$. In order to prove Theorem 1.4, namely that $p_*$ does not admit a section, the idea is to assume on the contrary that there exists a section $\phi: B_{t,s}(M) \to B_{t,s,1}(M)$, and to study the induced homomorphism of certain quotients of the two groups, for example by $\Gamma_l(P_n(M))$ and $\Gamma_l(P_{n+1}(M))$ respectively. If $l=2$, it turns out that this induced homomorphism admits a section, and so with our methods, we need to take a larger value of $l$. As we shall see, $l=3$ will be sufficient. We will make use of the framework of Section 2.1 and the commutative diagram (2.4), where we take $s \ne 0$, $m=1$, $H = \Gamma_3(P_{n+1}(M))$, $H' = \Gamma_3(P_n(M))$ and $X = \{n+1\}$. In order to apply the results of that section, we must first check that conditions (I)–(IV) are satisfied. Since $m=1$, relation (I) follows from
**Proposition 2.5.** and relation (III) holds trivially because $\sigma = 1$. To check that relations (II) and (IV) hold in our setting, we first give some information about the quotient groups $B_{t,s}(M)/\Gamma_3(P_n(M))$ and $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$. If $u$ and $v$ are two elements of a group, let $[u, v] = uvu^{-1}v^{-1}$ denote their commutator.

**Proposition 4.2.** Let $M$ be the 2-torus or the Klein bottle. Then $C_{i,j} \in \Gamma_2(P_n(M))$ for all $1 \le i < j \le n$.

*Proof.* Let $1 \le i < j \le n$. By relation (2) of Theorem 2.1, we have:

$$C_{i,j}^{-1}C_{i+1,j} = a_j^{-1}[b_j^{-1}, a_i^{-1}]a_j \in \Gamma_2(P_n(M)). \quad (4.15)$$

Taking $i = j - 1$, we see that $C_{j-1,j} \in \Gamma_2(P_n(M))$, and it follows by reverse induction on $i$ and (4.15) that $C_{i,j} \in \Gamma_2(P_n(M))$ for all $i = 1, \dots, j - 1$. $\square$

**Remark 4.3.** Let $1 \le i < j \le n$ (resp. $1 \le i < j \le n + 1$). By Proposition 4.2, the element $C_{i,j}$ of $B_{t,s}(M)/\Gamma_3(P_{t+s}(M))$ (resp. of $B_{t,s,1}(M)/\Gamma_3(P_{t+s+1}(M))$) commutes with the $\Gamma_3(P_{t+s}(M))$-coset (resp. the $\Gamma_3(P_{t+s+1}(M))$-coset) of every element of $P_{t+s}(M)$ (resp. of $P_{t+s+1}(M)$).

**Proposition 4.4.** Let $M$ be either the 2-torus or the Klein bottle. The following relations are valid in $B_{t,s}(M)/\Gamma_3(P_n(M))$ and $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$:

$$(1) \quad a_i a_j = a_j a_i \quad \text{for } i, j = 1, t+1, \dots, n.$$

$$(2) \quad b_j b_i b_j^{-1} = \begin{cases} b_i & \text{if } M = \mathbb{T} \text{ and } i, j = 1, t+1, \dots, n \\ a_j^{-1} b_i a_j & \text{if } M = \mathbb{K}, \text{ and either } i = 1, j = t+1, \dots, n \text{ or } t+1 \le i < j \le n. \end{cases}$$

$$(3) \quad a_n \sigma_i = \sigma_i a_n \quad \text{and} \quad b_n \sigma_i = \sigma_i b_n \quad \text{for } i = 1, \dots, t-1, t+1, \dots, n-2.$$

$$(4) \quad a_1 \sigma_i = \sigma_i a_1 \quad \text{and} \quad b_1 \sigma_i = \sigma_i b_1 \quad \text{for } i = t+1, \dots, n-1.$$

$$(5) \quad \sigma_i \sigma_j = \sigma_j \sigma_i \quad \text{for } 1 \le i, j \le n-1, \text{ where } |i-j| \ge 2 \text{ and } i, j \ne t.$$

$$(6) \quad \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \quad \text{for } i = 1, \dots, t-2, t+1, \dots, n-2.$$

$$(7) \quad \sigma_i a_i \sigma_i^{-1} b_i = b_i \sigma_i a_i \sigma_i \quad \text{for } i = 1, t+1.$$

$$(8) \quad \sigma_i^{-1} a_{i+1} = a_i \sigma_i \quad \text{and} \quad \sigma_i b_{i+1} = b_i \sigma_i^{-1} \quad \text{for } i = t+1, \dots, n-1.$$

$$(9) \quad \text{for } j = t+1, \dots, n, C_{1,j}C_{2,j}^{-1} = \begin{cases} a_j^{-1} b_1^{-1} a_j b_1 & \text{if } M = \mathbb{T} \\ (a_j^{-1} b_1^{-1} a_j b_1)^{-1} & \text{if } M = \mathbb{K}. \end{cases} \quad \text{This relation also holds for } j = n+1 \text{ in } B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M)).$$

*Proof.* With the exception of relation (2) in the case $M = \mathbb{K}$, relation (7) if $i = t+1$ and relation (9), all of the relations given in the statement appear in the presentation of $B_{t,s+1}(M)$ in Proposition 2.11, where we view $B_{t,s,1}(M)$ as a subgroup of $B_{t,s+1}(M)$, and so are valid in the given quotients.

If $M = \mathbb{K}$, by relation (6) of Theorem 2.1, we have $C_{i,j}C_{i+1,j}^{-1} = b_{j-1}b_i^{-1}b_jb_i$, and relation (2) may be obtained by substituting this equality in relation (7) of Theorem 2.1. To prove relation (7) for $i = t+1$, by relation (2) of Theorem 2.1 and Remark 4.3, we see that $b_{t+2}C_{t+1,t+2}a_{t+1} = a_{t+1}b_{t+2}$. In this equality, we then replace $C_{t+1,t+2}$ by $\sigma_{t+1}^2$, and $b_{t+2}$ by $\sigma_{t+1}^{-1}b_{t+1}\sigma_{t+1}^{-1}$ using relation (8), and this yields the given relation. Finally, to prove relation (9) we use relation (7) of Theorem 2.1 and Remark 4.3. $\square$

We now list the equations that we will use presently to put certain relations of the quotient group $B_{t,s}(M)/\Gamma_3(P_n(M))$ in canonical form.

**Proposition 4.5.** We have the following relations in $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$:

$$(1) \quad C_{l,j} \text{ commutes with } a_i \text{ and } b_i \text{ for all } i, l, j \text{ for all } 1 \le l < j \le n+1 \text{ and } i = 1, t+1, \dots, n+1.$$

$$(2) \quad \text{for } i = 1, t+1, \dots, n, b_{n+1}a_i = a_i b_{n+1} C_{i,n+1}^{-1} C_{i+1,n+1}, \text{ and:}$$

$$a_{n+1}b_i = \begin{cases} b_i a_{n+1} C_{i,n+1} C_{i+1,n+1}^{-1} & \text{if } M = \mathbb{T} \\ b_i a_{n+1} (C_{i,n+1} C_{i+1,n+1}^{-1})^{-1} & \text{if } M = \mathbb{K}. \end{cases}$$
$$ (3) \quad b_{n+1}a_{n+1} = \begin{cases} a_{n+1}b_{n+1}C_{1,n+1} & \text{if } M = \mathbb{T} \\ a_{n+1}^{-1}b_{n+1}C_{1,n+1} & \text{if } M = \mathbb{K}. \end{cases} $$

$$ (4) \quad a_{n+1}a_i = a_i a_{n+1} \quad \text{for } i = 1, \dots, n. $$

$$ (5) \quad \text{for } i = 1, \dots, n, b_{n+1}b_i = \begin{cases} b_i b_{n+1} & \text{if } M = \mathbb{T} \\ b_i b_{n+1} C_{i,n+1} C_{i+1,n+1}^{-1} & \text{if } M = \mathbb{K}. \end{cases} $$

$$ (6) \quad \text{for } 1 \le i \le n-1, \text{ where } i \ne t, \text{ and } 1 \le l < n+1: $$

$$ \sigma_i^{-1} C_{l,n+1} \sigma_i = \begin{cases} C_{l,n+1} & \text{if } l \neq i+1 \\ C_{l-1,n+1} C_{l,n+1}^{-1} C_{l+1,n+1} & \text{if } l = i+1. \end{cases} $$

**Proof.** Relation (1) follows from Remark 4.3, relation (2) is a consequence of relations (2) and (7) of Theorem 2.1 with $j = n + 1$ and Remark 4.3, relations (3), (4) and (5) may be deduced from relation (5) with $i = n+1$, relation (1) with $j = n+1$, and relation (6) with $j = n+1$ of Theorem 2.1 respectively, and relation (6) is a consequence of relation (6) of Proposition 2.7.

To obtain relation (6), we view $B_{t,s,1}(M)$ as a subgroup of $B_{t,s+1}(M)$ and we make use of the presentation of $B_{t,s+1}(M)$ given in Proposition 2.11. Let $1 \le l < n+1$. If $t+1 \le i \le t+s-1$ then relation (6) is obtained using relation (6) of Proposition 2.7, which is one of the relations of Type I of Proposition 2.11, and if $1 \le i \le t-1$, relation (6) follows from relation (9) of Proposition 2.11. $\square$

As we mentioned just before Proposition 4.2, relations (I) and (III) of Section 2.1 are satisfied in our setting. Relation (II) follows from Propositions 4.4 and 4.5, the surface relation is a consequence of Proposition 2.11, and relation (IV) follows from Remark 4.3. We may thus make use of the results of Section 2.1.

**Proposition 4.6.** Let $z_1, z_2 \in B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$ be such that $z_1 = a_{n+1}^{p_1} b_{n+1}^{p_2} C_{1,n+1}^{p_3} \cdots C_{n,n+1}^{p_{n+2}}$ and $z_2 = a_{n+1}^{q_1} b_{n+1}^{q_2} C_{1,n+1}^{q_3} \cdots C_{n,n+1}^{q_{n+2}}$, where $p_1, \dots, p_{n+2}, q_1, \dots, q_{n+2} \in \mathbb{Z}$. Suppose that $z_1 = z_2$.

(a) If $M = \mathbb{T}$, then $p_i = q_i$ for all $i = 1, \dots, n+2$.

(b) If $M = \mathbb{K}$, then $p_1 \equiv q_1 \bmod 4, p_2 = q_2$, and $p_i \equiv q_i \bmod 2$ for all $i = 3, \dots, n+2$.

In particular, if $M = \mathbb{T}$ or $\mathbb{K}$ then $p_i \equiv q_i \bmod 2$ for all $i = 1, \dots, n+2$.

**Proof.** Let $z_1$ and $z_2$ be as defined in the statement, and suppose that $z_1 = z_2$. Note that $z_i \in P_{n+1}(M)/\Gamma_3(P_{n+1}(M))$ for $i = 1, 2$. Let $\rho: P_{n+1}(M)/\Gamma_3(P_{n+1}(M)) \to \pi_1(M)/\Gamma_3(\pi_1(M))$ be the homomorphism induced by the homomorphism from $P_{n+1}(M)$ to $\pi_1(M)$ that geometrically forgets all but the last string. If $M = \mathbb{T}$, $\Gamma_3(\pi_1(\mathbb{T}))$ is trivial because $\pi_1(\mathbb{T}) \cong \mathbb{Z}^2$ is Abelian. If $M = \mathbb{K}$, then $\pi_1(\mathbb{K}) \cong \langle a_1 \rangle \times \langle b_1 \rangle$ where both factors are infinite cyclic and the action is the non-trivial one, $\Gamma_2(\pi_1(\mathbb{K})) = \langle a_1^2 \rangle$ and $\Gamma_3(\pi_1(\mathbb{K})) = \langle a_1^4 \rangle$ by [16, page 19]. Thus $\pi_1(M)/\Gamma_3(\pi_1(M))$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$ (resp. to $\mathbb{Z}_4 \times \mathbb{Z}$) if $M = \mathbb{T}$ (resp. if $M = \mathbb{K}$). Since $a_1^{p_1} b_1^{p_2} = \rho(z_1) = \rho(z_2) = a_1^{q_1} b_1^{q_2}$ in $\pi_1(M)/\Gamma_3(\pi_1(M))$, it follows that $p_i = q_i$ for $i = 1, 2$ (resp. $p_1 \equiv q_1 \bmod 4$ and $p_2 = q_2$) if $M = \mathbb{T}$ (resp. if $M = \mathbb{K}$).

For $i = 1, \dots, n$, consider the homomorphism $\rho_i: P_{n+1}(M)/\Gamma_3(P_{n+1}(M)) \to P_2(M)/\Gamma_3(P_2(M))$ induced by the homomorphism from $P_{n+1}(M)$ to $P_2(M)$ that geometrically forgets all but the $i$th string and the last string. Then $a_2^{p_1} b_2^{p_2} C_{1,2}^{p_{i+2}} = \rho_i(z_1) = \rho_i(z_2) = a_2^{q_1} b_2^{q_2} C_{1,2}^{q_{i+2}}$.

* If $M = \mathbb{T}$, from above, we have $p_i = q_i$ for $i = 1, 2$, and so $C_{1,2}^{p_{i+2}} = C_{1,2}^{q_{i+2}}$. Now $P_2(\mathbb{T}) \cong \pi_1(\mathbb{T} \setminus \{x_1\}) \times \mathbb{Z}^2$ by [4, Lemma 17], and using the fact that $\pi_1(\mathbb{T} \setminus \{x_1\})$ is the free group generated by $a_2$ and $b_2$, it follows that $\Gamma_i(P_2(\mathbb{T})) \cong \Gamma_i(\pi_1(\mathbb{T} \setminus \{x_1\})) = \Gamma_i(\langle a_2, b_2 \rangle)$ for all $i \ge 2$. Further, $C_{1,2} = [b_2^{-1}, a_2^{-1}]$ by taking $i = n = 2$ in relation (5) of Theorem 2.1, and by [18, page 337, Theorem 5.12], the coset of this element generates the infinite cyclic group $\Gamma_2(\langle a_2, b_2 \rangle)/\Gamma_3(\langle a_2, b_2 \rangle)$. Using the fact that $P_2(\mathbb{T})/\Gamma_3(P_2(\mathbb{T}))$ is torsion free [4, Theorem 4], it follows that $p_{i+2} = q_{i+2}$ for all $i = 1, \dots, n$, and this proves part (a).
• Let $M = \mathbb{K}$. If $X$ is a subset of a group $G$, let $\langle\langle X\rangle\rangle_G$ denote the normal closure of $X$ in $G$. First recall that by [16, equation (5.8)], $P_2(\mathbb{K}) \cong \pi_1(\mathbb{K} \setminus \{x_1\}) \rtimes \pi_1(\mathbb{K})$, where $\pi_1(\mathbb{K} \setminus \{x_1\})$ is the free group generated by $a_2$ and $b_2$. By [16, Theorem 5.4], for all $m \ge 2$, we have:

$$ \Gamma_m(P_2(\mathbb{K})) = \langle\langle a_2^{2m-1}, x^{2m-i} : x \in \Gamma_i(\pi_1(\mathbb{K} \setminus \{x_1\})), 2 \le i \le m \rangle\rangle_{\pi_1(\mathbb{K}\setminus\{x_1\})} \rtimes \langle\langle (a_1a_2)^{2m-1} \rangle\rangle. \quad (4.16) $$

If $m = 2, 3$, we see from (4.16) that $a_2^2 \in \Gamma_2(P_2(\mathbb{K}))$ and $a_4^2 \in \Gamma_3(P_3(\mathbb{K}))$. From this and the first two paragraphs of this proof we conclude that $C_{1,2}^{p_i+2} = C_{1,2}^{q_i+2}$ in $P_2(\mathbb{K})/\Gamma_3(P_2(\mathbb{K}))$. Taking $i = n = 2$ in relation (5) of Theorem 2.1, we obtain $C_{1,2} = [b_2^{-1}, a_2]a_2^2$, so $C_{1,2} \in \Gamma_2(P_2(\mathbb{K}))$, and thus $C_{1,2}^2 \in \Gamma_3(P_2(\mathbb{K}))$ by (4.16). So to prove the result in this case, it suffices to show that $C_{1,2} \notin \Gamma_3(P_2(\mathbb{K}))$. Suppose on the contrary that $C_{1,2} \in \Gamma_3(P_2(\mathbb{K}))$. Then using (4.16), we have:

$$ [b_2^{-1}, a_2]a_2^2 = C_{1,2} = \left( \prod_{i=1}^{l} \alpha_i a_2^{4\epsilon_i} \alpha_i^{-1} \cdot x_i \right) (a_1 a_2)^{4p}, \quad (4.17) $$

where $p \in \mathbb{Z}$, and for $1 \le i \le l$, $\alpha_i \in \pi_1(\mathbb{K} \setminus \{x_1\})$, $x_i \in \Gamma_2(\pi_1(\mathbb{K} \setminus \{x_1\}))$, and $\epsilon_i \in \mathbb{Z}$. Taking the image of $C_{1,2}$ under the projection from $P_2(\mathbb{K})$ onto $P_1(\mathbb{K})$ given by forgetting the second string, it follows from (4.17) that $a_1^{4p} = 1$ in $P_1(\mathbb{K})$, and so $p = 0$. So (4.17) is a relation in the free group $\pi_1(\mathbb{K} \setminus \{x_1\})$ generated by $a_2$ and $b_2$, and projecting this equation into the Abelianisation $\pi_1(\mathbb{K} \setminus \{x_1\})/\Gamma_2(\pi_1(\mathbb{K} \setminus \{x_1\}))$ which is a free Abelian group, we obtain $a_2^2 = a_2^{4\sum_{i=1}^l \epsilon_i}$, which yields a contradiction. We conclude that $C_{1,2} \notin \Gamma_3(P_2(\mathbb{K}))$, and since $C_{1,2}^2 \in \Gamma_3(P_2(\mathbb{K}))$, it follows from the fact that $C_{1,2}^{p_i+2} = C_{1,2}^{q_i+2}$ in $P_2(\mathbb{K})/\Gamma_3(P_2(\mathbb{K}))$ that $p_{i+2} \equiv q_{i+2} \mod 2$, which proves part (b) for $M = \mathbb{K}$. The last part of the statement then follows easily. $\square$

We now come back to the section $\hat{\phi}: B_{t,s}(M)/\Gamma_3(P_n(M)) \to B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$ for the induced homomorphism $\hat{p}_*: B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M)) \to B_{t,s}(M)/\Gamma_3(P_n(M))$. It may be defined on the following elements of $B_{t,s}(M)/\Gamma_3(P_n(M))$ by:

$$ \left\{ \begin{array}{ll} \hat{\phi}(\sigma_i) = \sigma_i \cdot a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} & \text{for } i=1, \dots, t-1, t+1, \dots, n-1 \\ \hat{\phi}(a_i) = a_i \cdot a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} C_{1,n+1}^{x_{i,1}} \cdots C_{n,n+1}^{x_{i,n}} & \text{for } i=1, t+1, t+2, \dots, n \\ \hat{\phi}(b_i) = b_i \cdot a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{y_{i,1}} \cdots C_{n,n+1}^{y_{i,n}} & \text{for } i=1, t+1, t+2, \dots, n, \end{array} \right. \quad (4.18) $$

where $s_{i,j}, r_{i,j}, \alpha_{i,j}, \beta_{i,j}, x_{i,j}, y_{i,j} \in \mathbb{Z}$ for the relevant values of $i$ and $j$. Since we are working with mixed braid groups, $\sigma_t$ is not an element of $B_{t,s}(M)$. If $M = \mathbb{K}$, by Proposition 4.6, any conclusion about the coefficients will be modulo 2. So for both $\mathbb{T}$ and $\mathbb{K}$, the computations that follow will be carried out with coefficients in $\mathbb{Z}_2$, in accordance with the last part of the statement of that proposition.

**Lemma 4.7.** With the above notation, we have:

(a) for $i = 1, t+1, \dots, n$ and $k = 1, 2$, $\alpha_{i,k} \equiv 0 \pmod{2}$ and $\beta_{i,k} \equiv 0 \pmod{2}$.

(b) for $i = 1, \dots, t-1, t+1, \dots, n-1$ and $k = 1, 2$, $s_{i,k} \equiv 0 \pmod{2}$.

(c) for $i = 1, t+1, \dots, n$ and $k = 1, 2$.

**Proof.** We start by supposing that $i = t+1, \dots, n$. We will study the coefficients of $a_{n+1}$ and $b_{n+1}$ in the image of relation (8) of Proposition 4.4 by $\hat{\phi}$. Using relations (1), (2), (6) of Proposition 4.5, (3) of Proposition 4.4 and Remark 4.3, we have:

$$ \begin{aligned} \hat{\phi}(\sigma_i^{-1} a_{i+1}) &= C_{n,n+1}^{-r_{i,n}} \cdots C_{1,n+1}^{-r_{i,1}} b_{n+1}^{-s_{i,2}} a_{n+1}^{-s_{i,1}} \sigma_i^{-1} a_{i+1} \cdot a_{n+1}^{\alpha_{i+1,1}} b_{n+1}^{\alpha_{i+1,2}} C_{1,n+1}^{x_{i+1,1}} \cdots C_{n,n+1}^{x_{i+1,n}} \\ &= \sigma_i^{-1} a_{i+1} b_{n+1}^{-s_{i,2}} a_{n+1}^{\alpha_{i+1,1}-s_{i,1}} b_{n+1}^{\alpha_{i+1,2}} w, \end{aligned} $$

and

$$ \hat{\phi}(a_i \sigma_i) = a_i \sigma_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\alpha_{i,2}} a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} w', $$
where $w, w'$ are words in the $C_{j,n+1}$, $j = 1, \dots, n$. Since $\hat{\phi}(\sigma_i^{-1}a_{i+1}) = \hat{\phi}(a_i\sigma_i)$ and $\sigma_i^{-1}a_{i+1} = a_i\sigma_i$ in $B_{t,s,1}(\mathbb{T})/\Gamma_3(P_{n+1}(\mathbb{T}))$, to be able to compare the coefficients of $a_{n+1}$ and $b_{n+1}$, we need to put $b_{n+1}^{-s_{i,2}}a_{n+1}^{\alpha_{i+1,1}-s_{i,1}}b_{n+1}^{\alpha_{i+1,2}}$ and $a_{n+1}^{\alpha_{i,1}}b_{n+1}^{\alpha_{i,2}}a_{n+1}^{s_{i,1}}b_{n+1}^{s_{i,2}}$ in canonical form. To do so, it suffices to conjugate $a_{n+1}^{\alpha_{i+1,1}-s_{i,1}}$ by $b_{n+1}^{-s_{i,2}}$, and $a_{n+1}^{s_{i,1}}$ by $b_{n+1}^{\alpha_{i,2}}$, which we do using (3) of Proposition 4.5. If $M = \mathbb{K}$, this may alter the sign of the exponent of $a_{n+1}$, but modulo 2, this exponent remains the same. By comparing the coefficients of $a_{n+1}$ and $b_{n+1}$, it follows from relations (1) and (3) of Proposition 4.5 and Proposition 4.6 that for $i = t+1, \dots, n-1$ and $k = 1, 2$:

$$ \alpha_{i+1,k} \equiv \alpha_{i,k} \mod 2 \qquad (4.19) $$

$$ \beta_{i+1,k} \equiv \beta_{i,k} \mod 2. \qquad (4.20) $$

Applying induction for $i = t+1, \dots, n$, to prove the result, it suffices to show that $\alpha_{1,k} \equiv \alpha_{n,k} \equiv 0$ and $\beta_{1,k} \equiv \beta_{n,k} \equiv 0$, for $k=1,2$. We now analyse the image of relation (1) of Proposition 4.4 by $\hat{\phi}$. Since we will be comparing coefficients modulo 2, it will be convenient not to take into account the signs of certain exponents. Using relations (1)–(4) of Proposition 4.5, for $i, j = 1, t+1, \dots, n$, we have:

$$ \begin{aligned} \hat{\phi}(a_ia_j) &= a_ia_{n+1}^{\alpha_{i,1}}b_{n+1}^{\alpha_{i,2}}a_ja_{n+1}^{\alpha_{j,1}}b_{n+1}^{\alpha_{j,2}}w = a_ia_ja_{n+1}^{\alpha_{i,1}}b_{n+1}^{\alpha_{i,2}}C_{j,n+1}^{\alpha_{i,2}}C_{j+1,n+1}^{\alpha_{i,2}}a_{n+1}^{\alpha_{j,1}}b_{n+1}^{\alpha_{j,2}}w \\ &= a_ia_ja_{n+1}^{\alpha_{i,1}+\alpha_{j,1}}b_{n+1}^{\alpha_{j,2}+\alpha_{i,2}}C_{1,n+1}^{\alpha_{i,2}\alpha_{j,1}}C_{j,n+1}^{\alpha_{i,2}}C_{j+1,n+1}^{\alpha_{i,2}}w, \end{aligned} \qquad (4.21) $$

where $w = \prod_{k=1}^n C_{k,n+1}^{x_{i,k}+x_{j,k}}$. In a similar manner, we obtain:

$$ \hat{\phi}(a_j a_i) = a_j a_i a_{n+1}^{\alpha_{i,1}+\alpha_{j,1}} b_{n+1}^{\alpha_{j,2}+\alpha_{i,2}} C_{1,n+1}^{\alpha_{i,2}\alpha_{i,1}} C_{i,n+1}^{\alpha_{j,2}} C_{i+1,n+1}^{\alpha_{j,2}} w. \qquad (4.22) $$

First let $i=1$ and $j=t+1$ in equations (4.21) and (4.22). Since $\hat{\phi}(a_ia_j) = \hat{\phi}(a_ja_i)$ and $a_ia_j = a_ja_i$ in $B_{t,s,1}(\mathbb{T})/\Gamma_3(P_{n+1}(\mathbb{T}))$ by relation (4) of Proposition 4.5, we obtain $C_{1,n+1}^{\alpha_{1,2}\alpha_{t+1,1}}C_{t+1,n+1}^{\alpha_{i,2}}C_{t+2,n+1}^{\alpha_{i,2}} = C_{1,n+1}^{\alpha_{t+1,2}\alpha_{i,1}}C_{1,n+1}^{\alpha_{t+1,2}}C_{2,n+1}^{\alpha_{t+1,2}}$. Comparing the coefficients of $C_{t+2,n+1}$ and using Proposition 4.6, we conclude that:

$$ \alpha_{1,2} \equiv 0 \mod 2. \qquad (4.23) $$

Now take $i = t+1$ and $j = n$ in equations (4.21) and (4.22). In a similar way, we obtain $C_{1,n+1}^{\alpha_{t+1,2}\alpha_{i,1}}C_{n,n+1}^{\alpha_{t+1,2}} = C_{1,n+1}^{\alpha_{n,2}\alpha_{t+1,1}}C_{t+1,n+1}^{\alpha_{i,2}}C_{t+2,n+1}^{\alpha_{i,2}}$. If $s > 2$ (resp. $s = 2$) then comparing the coefficients of $C_{n,n+1}$ (resp. of $C_{n-1,n+1}$) and using Proposition 4.6, we see that:

$$ \alpha_{t+1,2} \equiv 0 \mod 2 \text{ (resp. } \alpha_{n,2} \equiv 0 \mod 2). \qquad (4.24) $$

We deduce from (4.19), (4.23) and (4.24) that for $i = 1, t+1, \dots, n$:

$$ \alpha_{i,2} \equiv 0 \mod 2. \qquad (4.25) $$

We now consider relation (2) of Proposition 4.4. If $M = \mathbb{T}$, then arguing as above, for $i = 1, t+1, \dots, n$, we obtain:

$$ \beta_{i,1} \equiv 0 \mod 2. \qquad (4.26) $$

If $M = \mathbb{K}$, for either $i = 1, j = t+1, \dots, n$ or $t+1 \le i < j \le n$, using relations (1)–(3) and (5) of Proposition 4.5, we have:

$$ \begin{aligned} \hat{\phi}(b_j b_i b_j^{-1}) &= b_j a_{n+1}^{\beta_{j,1}} b_{n+1}^{\beta_{j,2}} b_i a_{n+1}^{\beta_{i,1}-\beta_{j,1}} b_{n+1}^{\beta_{i,2}-\beta_{j,2}} C_{1,n+1}^{\beta_{j,1}(\beta_{i,2}-\beta_{j,2})} b_j^{-1} w \\ &= b_j b_i a_{n+1}^{\beta_{j,1}} b_{n+1}^{\beta_{j,2}} C_{i,n+1}^{\beta_{j,1}+\beta_{j,2}} C_{i+1,n+1}^{\beta_{i,1}-\beta_{j,1}} b_{n+1}^{\beta_{i,2}-\beta_{j,2}} C_{1,n+1}^{\beta_{j,1}(\beta_{i,2}-\beta_{j,2})} b_j^{-1} w \\ &= b_j b_i a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{\beta_{j,2}(\beta_{i,1}-\beta_{j,1})} C_{1,n+1}^{\beta_{j,1}(\beta_{i,2}-\beta_{j,2})} C_{i,n+1}^{\beta_{j,1}+\beta_{j,2}} C_{i+1,n+1}^{\beta_{j,1}+\beta_{j,2}} b_j^{-1} w \\ &= b_j b_i b_j^{-1} a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{\beta_{j,2}\beta_{i,1}+\beta_{i,2}\beta_{j,1}} C_{i,n+1}^{\beta_{j,1}+\beta_{j,2}} C_{i+1,n+1}^{\beta_{j,1}+\beta_{j,2}} C_{j,n+1}^{\beta_{i,1}+\beta_{i,2}} C_{j+1,n+1}^{\beta_{i,1}+\beta_{i,2}} w, \end{aligned} \qquad (4.27) $$

where $w = \prod_{k=1}^n C_{k,n+1}^{y_{i,k}}$. Also, applying (1)–(4) of Proposition 4.5 and (4.25), we see that:

$$ \begin{aligned} \hat{\phi}(a_j^{-1} b_i a_j) &= a_{n+1}^{-\alpha_{i,1}} a_j^{-1} b_i a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} a_j a_{n+1}^{\alpha_{j,1}} w = a_j^{-1} b_i a_{n+1}^{\beta_{i,1}-\alpha_{j,1}} C_{i,n+1}^{\alpha_{j,1}} C_{i+1,n+1}^{\alpha_{j,1}} b_{n+1}^{\beta_{i,2}} a_j a_{n+1}^{\alpha_{j,1}} w \\ &= a_j^{-1} b_i a_j a_{n+1}^{\beta_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{\beta_{i,2}\alpha_{j,1}} C_{j,n+1}^{\beta_{i,2}} C_{j+1,n+1}^{\beta_{i,2}} C_{i,n+1}^{\alpha_{j,1}} C_{i+1,n+1}^{\alpha_{j,1}} w. \end{aligned} \qquad (4.28) $$
Making use of relation (2) of Proposition 4.4 in $B_{t,s,1}(\mathbb{T})/\Gamma_3(P_{n+1}(\mathbb{T}))$, and comparing the coefficients of $C_{j,n+1}$ for the given values of $j$, in (4.27) and (4.28) using Proposition 4.6, it follows that $\beta_{i,1} \equiv 0 \mod 2$ for $i = 1, t+1, \dots, n-1$, and applying (4.19), we also deduce the result for $i = n$. Therefore, for $i = 1, t+1, \dots, n$, we obtain:

$$ \beta_{i,1} \equiv 0 \mod 2. \qquad (4.29) $$

It follows from (4.25), (4.26) and (4.29) that $\alpha_{i,2} \equiv \beta_{i,1} \equiv 0 \mod 2$ for $i = 1, t+1, \dots, n$, which proves half of part (a) of the statement. Before showing that the other congruences of part (a) hold, we first prove part (b). To do so, we start by studying the image of relations (4) of Proposition 4.4 by $\hat{\phi}$. Using (4.23), relations (1)–(4) and (6) of Proposition 4.5, and relation (3) of Proposition 4.4, for $i = t+1, \dots, n-1$, we have:

$$ \begin{aligned} \hat{\phi}(a_1\sigma_i) &= a_1 a_{n+1}^{\alpha_{1,1}} C_{1,n+1}^{x_{1,1}} \cdots C_{n,n+1}^{x_{1,n}} \sigma_i a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} \\ &= a_1 \sigma_i a_{n+1}^{\alpha_{1,1}+s_{i,1}} b_{n+1}^{s_{i,2}} w C_{i,n+1}^{x_{1,i+1}} C_{i+2,n+1}^{x_{1,i+1}}, \end{aligned} \qquad (4.30) $$

where $w = \prod_{k=1}^{n} C_{k,n+1}^{x_{1,k}+r_{i,k}}$ and

$$ \begin{aligned} \hat{\phi}(\sigma_i a_1) &= \sigma_i a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} a_1 a_{n+1}^{\alpha_{1,1}} C_{1,n+1}^{x_{1,1}} \cdots C_{n,n+1}^{x_{1,n}} \\ &= \sigma_i a_1 a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} C_{2,n+1}^{s_{i,2}} a_{n+1}^{\alpha_{1,1}} C_{1,n+1}^{r_{i,1}+x_{1,1}} \cdots C_{n,n+1}^{r_{i,n}+x_{1,n}} \\ &= \sigma_i a_1 a_{n+1}^{s_{i,1}+\alpha_{1,1}} b_{n+1}^{s_{i,2}} w C_{1,n+1}^{x_{1,2}\alpha_{1,1}+s_{i,2}} C_{2,n+1}^{s_{i,2}}. \end{aligned} \qquad (4.31) $$

Since $i \ge 3$, using the fact that $a_1\sigma_i = \sigma_i a_1$ in $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$ by relation (4) of Proposition 4.4, and comparing the coefficients of $C_{2,n+1}$ in (4.30) and (4.31) and making use of Proposition 4.6, for $i = t+1, \dots, n-1$, it follows that:

$$ s_{i,2} \equiv 0 \mod 2. \qquad (4.32) $$

In a similar manner, analysing the image by $\hat{\phi}$ of the relation $b_1\sigma_i = b_1\sigma_i$ for $i = t+1, \dots, n-1$, using (4.32), and comparing the coefficients of $C_{2,n+1}$, we see that $s_{i,1} \equiv 0 \mod 2$.

Now suppose that $i = 1, \dots, t-1$. Analysing the image by $\hat{\phi}$ of the relation $a_n\sigma_i = \sigma_i a_n$ (resp. $b_n\sigma_i = \sigma_i b_n$), we have:

$$ \begin{aligned} \hat{\phi}(a_n\sigma_i) &= a_n a_{n+1}^{\alpha_{n,1}} C_{1,n+1}^{x_{n,1}} \cdots C_{n,n+1}^{x_{n,n}} \sigma_i a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} \\ &= a_n \sigma_i a_{n+1}^{\alpha_{n,1}+s_{i,1}} b_{n+1}^{s_{i,2}} w C_{i,n+1}^{x_{n,i+1}} C_{i+2,n+1}^{x_{n,i+1}}, \end{aligned} \qquad (4.33) $$

where $w = \prod_{k=1}^{n} C_{k,n+1}^{x_{n,k}+r_{i,k}}$ and

$$ \begin{aligned} \hat{\phi}(\sigma_i a_n) &= \sigma_i a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{1,n+1}^{r_{i,1}} \cdots C_{n,n+1}^{r_{i,n}} a_n a_{n+1}^{\alpha_{n,1}} C_{1,n+1}^{x_{n,1}} \cdots C_{n,n+1}^{x_{n,n}} \\ &= \sigma_i a_n a_{n+1}^{s_{i,1}} b_{n+1}^{s_{i,2}} C_{n,n+1}^{r_{i,1}} a_{n+1}^{\alpha_{n,1}} C_{1,n+1}^{r_{i,1}+x_{n,1}} \cdots C_{n,n+1}^{r_{i,n}+x_{n,n}} \\ &= \sigma_i a_n a_{n+1}^{s_{i,1}+\alpha_{n,1}} b_{n+1}^{s_{i,2}} w C_{1,n+1}^{x_{1,2}\alpha_{n,1}} C_{n,n+1}^{s_{i,2}}. \end{aligned} \qquad (4.34) $$

Since $i \le t-1$ and $t+1 < n$, we have $i+2 < n$, and comparing the coefficient of $C_{n,n+1}$ in (4.33) and (4.34), we conclude that $s_{i,2} \equiv 0 \mod 2$. In a similar manner, we obtain $s_{i,1} \equiv 0 \mod 2$ using the relation $b_n\sigma_i = \sigma_i b_n$. In summary, for $i = 1, \dots, t-1$, we have:

$$ s_{i,1} \equiv s_{i,2} \equiv 0 \mod 2, \qquad (4.35) $$

which proves part (b) of the statement.

We now return to the proof of the outstanding cases of part (a), as well as that of part (c).

We first study relation (7) of Proposition 4.4. Let $i \in \{1, t+1\}$. Setting $w = \prod_{k=1}^{n} C_{k,n+1}^{x_{i,k}+y_{i,k}}$,
using (1)–(4) and (6) of Proposition 4.5, relation (3) of Proposition 4.4 in $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$, and equations (4.25), (4.26), (4.29) and (4.35), we have:

$$ 
\begin{aligned}
\hat{\phi}(\sigma_i a_i \sigma_i^{-1} b_i) &= \sigma_i a_i a_{n+1}^{\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}} \sigma_i^{-1} b_i b_{n+1}^{\beta_{i,2}} C_{i+1,n+1}^{y_{i,i+1}} w \\
&= \sigma_i a_i \sigma_i^{-1} a_{n+1}^{\alpha_{i,1}} C_{i,n+1}^{x_{i,i+1}} C_{i+1,n+1}^{x_{i,i+1}} C_{i+2,n+1}^{x_{i,i+1}} b_i b_{n+1}^{\beta_{i,2}} C_{i+1,n+1}^{y_{i,i+1}} w \\
&= \sigma_i a_i \sigma_i^{-1} b_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\beta_{i,2}} C_{i,n+1}^{x_{i,i+1}+\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}+\alpha_{i,1}+y_{i,i+1}} C_{i+2,n+1}^{x_{i,i+1}} w
\end{aligned}
$$

and

$$ 
\begin{aligned}
\hat{\phi}(b_i \sigma_i a_i \sigma_i) &= b_i b_{n+1}^{\beta_{i,2}} C_{i+1,n+1}^{y_{i,i+1}} \sigma_i C_{i+1,n+1}^{r_{i,i+1}} a_i a_{n+1}^{\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}} \sigma_i C_{i+1,n+1}^{r_{i,i+1}} w \\
&= b_i \sigma_i b_{n+1}^{\beta_{i,2}} C_{i,n+1}^{y_{i,i+1}} C_{i+1,n+1}^{y_{i,i+1}+r_{i,i+1}} C_{i+2,n+1}^{y_{i,i+1}} a_i a_{n+1}^{\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}} \sigma_i C_{i+1,n+1}^{r_{i,i+1}} w \\
&= b_i \sigma_i a_i b_{n+1}^{\beta_{i,2}} C_{i,n+1}^{y_{i,i+1}+\beta_{i,2}} C_{i+1,n+1}^{y_{i,i+1}+r_{i,i+1}+\beta_{i,2}} C_{i+2,n+1}^{y_{i,i+1}} a_{n+1}^{\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}} \sigma_i C_{i+1,n+1}^{r_{i,i+1}} w \\
&= b_i \sigma_i a_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{\alpha_{i,1}\beta_{i,2}} C_{i,n+1}^{y_{i,i+1}+\beta_{i,2}} C_{i+1,n+1}^{y_{i,i+1}+r_{i,i+1}+\beta_{i,2}+x_{i,i+1}} C_{i+2,n+1}^{y_{i,i+1}} \sigma_i C_{i+1,n+1}^{r_{i,i+1}} w \\
&= b_i \sigma_i a_i \sigma_i a_{n+1}^{\alpha_{i,1}} b_{n+1}^{\beta_{i,2}} C_{1,n+1}^{\alpha_{i,1}\beta_{i,2}} C_{i,n+1}^{r_{i,i+1}+x_{i,i+1}} C_{i+1,n+1}^{y_{i,i+1}+\beta_{i,2}+x_{i,i+1}} C_{i+2,n+1}^{r_{i,i+1}+\beta_{i,2}+x_{i,i+1}} w.
\end{aligned}
$$

Using the equalities $\sigma_i a_i \sigma_i^{-1} b_i = b_i \sigma_i a_i \sigma_i$ in $B_{t,s,1}(M)/\Gamma_3(P_{n+1}(M))$ by relation (7) of Proposition 4.4 and $\hat{\phi}(\sigma_i a_i \sigma_i^{-1} b_i) = \hat{\phi}(b_i \sigma_i a_i \sigma_i)$, and applying Proposition 4.6, we see that:

$$ C_{i,n+1}^{x_{i,i+1}+\alpha_{i,1}} C_{i+1,n+1}^{x_{i,i+1}+\alpha_{i,1}+y_{i,i+1}} C_{i+2,n+1}^{x_{i,i+1}} = C_{1,n+1}^{\alpha_{i,1}\beta_{i,2}} C_{i,n+1}^{r_{i,i+1}+x_{i,i+1}} C_{i+1,n+1}^{y_{i,i+1}+\beta_{i,2}+x_{i,i+1}} C_{i+2,n+1}^{r_{i,i+1}+\beta_{i,2}+x_{i,i+1}}. \quad (4.36) $$

Comparing certain coefficients of (4.36) modulo 2, we obtain the following congruences:

(i) for the coefficient of $C_{i+1,n+1}$, $\alpha_{i,1} \equiv \beta_{i,2} \mod 2$.

(ii) if $i = 1$ (resp. $i = t+1$), for the coefficient of $C_{3,n+1}$ (resp. of $C_{t+1,n+1}$), we have $r_{i,i+1} \equiv \beta_{i,2} \mod 2$, where we use (i) in the case $i = t+1$.

(iii) for the coefficient of $C_{1,n+1}$, $\alpha_{i,1}^2 \equiv 0 \mod 2$ using (i) and (ii).

It follows that $r_{i,i+1} \equiv \alpha_{i,1} \equiv \beta_{i,2} \equiv 0 \mod 2$ for $i \in \{1, t+1\}$. We thus obtain part (c), and applying (4.19), (4.20) and induction on $t+1 \le i \le n$, the remaining congruences of part (a) follow. $\square$

Part (a) of the following lemma is the analogue in our setting of Lemma 3.3(a).

**Lemma 4.8.** Let $M = \mathbb{T}$ or $\mathbb{K}$. Using the notation of (4.18), we have:

(a) if $t \ge 4$ then $r_{j,k} \equiv 0 \mod 2$ for all $1 \le j \le t-1$ and $k = 2, \dots, j-1, j+3, \dots, t$.

(b) if $t \ge 3$ then $r_{i+1,i+2} \equiv r_{i,i+1} \equiv r_{1,2} \equiv 0 \mod 2$ if $1 \le i \le t-2$.

*Proof.* Let $M = \mathbb{T}$ or $\mathbb{K}$.

(a) Let $t \ge 4$. Let $1 \le i, j \le t-1$ be such that $|i-j| \ge 2$. Applying relation (3) of Proposition 4.5 to (2.34) in the case $m = 1$, we have:

$$ C_{1,n+1}^{\gamma} = C_{i,n+1}^{r_{j,i+1}} C_{i+1,n+1}^{-2r_{j,i+1}} C_{i+2,n+1}^{r_{j,i+1}} C_{j,n+1}^{-r_{i,j+1}} C_{j+1,n+1}^{2r_{i,j+1}} C_{j+2,n+1}^{-r_{i,j+1}}, $$

which is in canonical form (possibly up to permutation of some of the factors). Comparing the coefficients of $C_{i+1,n+1}$ (resp. $C_{j+1,n+1}$) if $i < j$ (resp. $i > j$) and using Proposition 4.6, we see that $r_{j,i+1} \equiv 0 \mod 2$ (resp. $r_{i,j+1} \equiv 0 \mod 2$). So for all $1 \le j \le t-1$, $r_{j,k} \equiv 0 \mod 2$ for all $k = 2, \dots, j-1$ (resp. for all $k = j+3, \dots, t$) as required.

To prove part (b), let $t \ge 3$, and let $1 \le i \le t-2$. Then comparing the coefficients of $C_{i+1,n+1}$ in equation (2.35) and using relation (3) of Proposition 4.5 and the lift of relation (2) of Theorem 2.3, we see that $\rho \equiv 0 \mod 2$ by Proposition 4.6 and it follows that $r_{i+1,i+2} \equiv r_{i,i+1} \mod 2$. Part (b) is then a consequence of Lemma 4.7(c) and induction on $i$. $\square$

We end this paper by proving Theorem 1.4.
Proof of Theorem 1.4. Let $M$ be the 2-torus or the Klein bottle, and let $t, s \ge 2$. Suppose on the contrary that the projection $p_*: B_{t,s,1}(M) \to B_{t,s}(M)$ admits a section. As we showed earlier in this section, we may make use of the framework of Section 2.1, and so we use the notation defined there. Taking $m = 1$ in (2.33) and applying Proposition 4.6, we see that:

$$ \gamma + 1 \equiv \alpha + \delta \mod 2 \qquad (4.37) $$

To obtain a contradiction to (4.37), it suffices to prove that $\alpha, \gamma$ and $\delta$ are even. Applying Lemma 4.8(b) to (2.25), we see that $\alpha$ is even. By (2.27), we have $\delta \equiv \beta_{1,2} + \alpha_{1,1} \mod 2$, and from (2.28) and (2.30), $\gamma = -\sum_{i=t+1}^n \alpha_{i,1}$ (resp. $\gamma = \sum_{i=t+1}^n (\alpha_{i,2} - \alpha_{i,1})$ if $M = \mathbb{T}$ (resp. if $M = \mathbb{K}$). It follows from Lemma 4.7(a) that $\delta$ and $\gamma$ are even too. This contradicts (4.37), which proves Theorem 1.4. $\square$

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DEPARTAMENTO DE MATEMÁTICA - IME-USP, RUA DO MATÃO 1010, CEP: 05508-090 - SÃO PAULO - SP - BRAZIL

Email address: dlgoncal@ime.usp.br

NORMANDIE UNIV., UNICAEN, CNRS, LMNO, 14000 CAEN, FRANCE

Email address: john.guaschi@unicaen.fr

UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO, UFES, DEPARTAMENTO DE MATEMÁTICA, 29075-910, VITÓRIA, ESPÍRITO SANTO, BRAZIL

Email address: carolina.pereiro@ufes.br