2.1 Abelian and toric arrangements

Let $\mathbb {G}$ be a connected abelian Lie group, $\Gamma \cong \mathbb {Z}^d$ a finitely generated free abelian group, and ${T= {\mathrm {Hom}}(\Gamma ,\mathbb {G})\cong \mathbb {G}^d}$ .

Definition 2.1. An abelian arrangement $\mathscr {A}$ is, for some finite set $\mathfrak {X}=\mathfrak {X}(\mathscr {A})\subseteq \Gamma $ , the collection of connected components of the subspaces

$$\begin{align*}H_{\chi} := \{t\in T \mid \chi\in \ker(t) \}\end{align*}$$

with $\chi \in \mathfrak {X}(\mathscr {A})$ .

The complement of $\mathscr {A}$ is denoted by

$$\begin{align*}M(\mathscr{A}) := T\smallsetminus \bigcup_{\chi\in\mathfrak{X}(\mathscr{A})} H_\chi.\end{align*}$$

The poset of layers of $\mathscr {A}$ is the set $\mathscr {P}(\mathscr {A})$ whose elements are the nonempty connected components of intersections $\cap _{\chi \in S}H_\chi $ where $S\subseteq \mathfrak {X}(\mathscr {A})$ , partially ordered by reverse inclusion.

Remark 2.3. Let $\mathscr {A}$ be a toric arrangement and consider $\chi \in \mathfrak {X}(\mathscr {A})$ . Fixing an isomorphism $\Gamma \cong \mathbb {Z}^d$ and the corresponding coordinates on $T\cong \mathbb {C}^\times $ , we have

$$ \begin{align*}H_\chi =\{\mathbf{x}=(x_1,\dots,x_d)\in (\mathbb{C}^\times)^d \mid x_1^{c_1}\cdots x_d^{c_d} = 1\},\end{align*} $$

where $(c_1,\ldots ,c_d)\in \mathbb {Z}^d$ corresponds to $\chi \in \Gamma $ . Let $m:=\gcd (c_1,\ldots ,c_d)$ . Then, $H_\chi $ is connected if and only if $\chi $ is primitive, equivalently, if $m=1$ . In general, the different connected components of $H_{\chi }$ are given by

(2.1) $$ \begin{align} x_1^{{c_1}/{m}}\cdots x_d^{{c_d}/{m}} = \mu, \end{align} $$

where $\mu $ runs over all m-th roots of unity.

Example 2.4. Let $\mathbb {G}=\mathbb {C}^\times $ and $\Gamma =\mathbb {Z}^2$ , so $T\cong (\mathbb {C}^\times )^2$ . The columns $\chi _1$ , $\chi _2$ , and $\chi _3$ of the integer matrix

$$\begin{align*}\begin{pmatrix} 2 & -2 & 0 \\ 0 & 1 & 1 \end{pmatrix}\end{align*}$$

define a toric arrangement $\mathscr {A}=\{H_0,H_1,H_2,H_3\}$ , where $H_0$ and $H_1$ denote the two connected components of $H_{\chi _1}$ , $H_2:=H_{\chi _2}$ , and $H_3:=H_{\chi _3}$ . Considering $\mathbb {R}/\mathbb {Z}\subseteq \mathbb {C}/\mathbb {Z}\cong \mathbb {C}^\times $ , the “real part” of the arrangement is depicted in Figure 1a, and the Hasse diagram for the poset of layers is depicted in Figure 1b.

Figure 1

See Examples 2.4, 2.8, and 3.7, as well as Section 5.

Figure 1 Long description

Panel a shows a square grid representing S super 1 times S super 1. It contains two vertical blue lines at the left and center. Two diagonal pink lines rise from the bottom-left of each vertical section to the top-right. A dark green horizontal line sits at the base. Panel b shows a Hasse diagram for the poset P of A. At the top are two coordinate pairs (1, 1) and (-1, 1). These connect downward to a middle row of four labels: H sub 0 in blue, H sub 2 in pink, H sub 3 in green, and H sub 1 in blue. Specifically, (1, 1) connects to H sub 0, H sub 2, and H sub 3. (-1, 1) connects to H sub 2, H sub 3, and H sub 1. All four middle labels converge at the bottom to a single letter T.

2.2 Supersolvability

A subgroup Y of T is admissible if there is a rank-one direct summand $\Gamma '\subseteq \Gamma $ such that Y is the image of the injection $\epsilon ^*: {\mathrm {Hom}}(\Gamma ',\mathbb {G})\to {\mathrm {Hom}}(\Gamma ,\mathbb {G})$ induced by the projection $\epsilon :\Gamma \to \Gamma '$ . When Y is admissible, the corresponding projection

$$\begin{align*}p:T\to T/Y \cong{\mathrm{Hom}}(\Gamma/\Gamma',\mathbb{G})\end{align*}$$

admits the map induced by the quotient $q:\Gamma \to \Gamma /\Gamma '$ as a section. This allows us to define abelian arrangements

$$\begin{align*}\mathscr{A}_Y:= \{H\in\mathscr{A} \mid H\supseteq Y\} \qquad \mathscr{A}/Y := \{p(H) \mid H\in\mathscr{A}_Y \}\end{align*}$$

in T and $T/Y$ , respectively. Note that $\mathscr {P}(\mathscr {A}_Y)$ is by definition a subposet of $\mathscr {P}(\mathscr {A})$ .

The projection $p:T\to T/Y$ restricts to a map on arrangement complements $\bar {p}:M(\mathscr {A})\to M(\mathscr {A}/Y)$ and induces an isomorphism of posets $\mathscr {P}(\mathscr {A}_Y)\cong \mathscr {P}(\mathscr {A}/Y)$ .

Definition 2.5 [Reference Bibby and Delucchi4, Definitions 2.4.1, 5.1.1].

Let Y be an admissible subgroup of T, and $\mathscr {A}$ an abelian arrangement in T. We say $\mathscr {P}(\mathscr {A}_Y)$ is an M-ideal of $\mathscr {P}(\mathscr {A})$ if, for any two distinct $H_1,H_2\in \mathscr {A}\smallsetminus \mathscr {A}_Y$ and any component X of $H_1\cap H_2$ , there is some $H_3\in \mathscr {A}_Y$ such that $H_3\supseteq X$ . Say $\mathscr {P}(\mathscr {A}_Y)$ is a TM-ideal if, in addition, the intersection $H\cap Y$ is connected for all $H\in \mathscr {A}\smallsetminus \mathscr {A}_Y$ .

Say $\mathscr {A}$ is (strictly) supersolvable if there is a chain

(2.2) $$ \begin{align} \{\hat{0}\} \subset \mathscr{P}(\mathscr{A}_{Y_1})\subset \mathscr{P}(\mathscr{A}_{Y_2}) \subset\cdots\subset \mathscr{P}(\mathscr{A}_{Y_{d-1}}) \subset\mathscr{P}(\mathscr{A}) \end{align} $$

with each $\mathscr {P}(\mathscr {A}_{Y_r})$ a (T)M-ideal of its successor.

For us, the importance of an M-ideal is that it characterizes when the map $\bar {p}:M(\mathscr {A})\to M(\mathscr {A}/Y)$ is a fiber bundle [Reference Bibby and Delucchi4, Theorem A]. Our immediate goal is to show that these bundles are closely related to bundles over configuration spaces. We subsequently specialize to toric arrangements, where this structure has particularly interesting consequences.

2.3 Somewhat ordered configuration spaces

Given a positive integer k and a topological space X, denote the ordered configuration space by

$$\begin{align*}{\mathrm{Conf}}_k(X) = \{(x_1,\dots,x_k)\in X^k \mid x_i\neq x_j \text{ when } i\neq j \}.\end{align*}$$

The symmetric group $\Sigma _{k}$ acts freely on $ {\mathrm {Conf}}_k(X)\subseteq X^k$ by permuting coordinates. The unordered configuration space is the quotient space $ {\mathrm {Conf}}_k(X)/\Sigma _{k}$ , whose elements are regarded as sets (rather than ordered tuples) of distinct points in X. More generally, consider a composition of the integer k, that is, a sequence $\mathbf {k}=(k_1,\dots ,k_m)$ of positive integers satisfying $k=k_1+\cdots +k_m$ . Such a composition determines a subgroup $\Sigma _{\mathbf {k}} := \Sigma _{k_1}\times \dots \times \Sigma _{k_m} \subseteq \Sigma _{k}$ . The somewhat ordered configuration space (or $\mathbf {k}$ -ordered configuration space) is then defined as the quotient

$$\begin{align*}{\mathrm{Conf}}^{\mathbf{k}}(X) := {\mathrm{Conf}}_k(X)/\Sigma_{\mathbf{k}}.\end{align*}$$

An element of $ {\mathrm {Conf}}^{\mathbf {k}}(X)$ can be represented by an ordered tuple $(S_1,\dots ,S_m)$ of pairwise disjoint subsets of X with $|S_i|=k_i$ for each i.

By a classical result of Fadell and Neuwirth [Reference Fadell and Neuwirth22, Theorem 3] (see also [Reference Fadell and Husseini21, Theorem 1.1]), for ordered configuration spaces of a manifold X without boundary, the forgetful map

(2.3) $$ \begin{align} {\mathrm{Conf}}_{k+1}(X)\to{\mathrm{Conf}}_k(X),\quad (x_1,\dots,x_k,x_{k+1}) \mapsto (x_1,\dots,x_k) \end{align} $$

is a fiber bundle, with fiber homeomorphic to X with k points removed. We refer to this as the Fadell-Neuwirth bundle.

2.4 Monodromy and the Artin representation

If $p\colon E {\kern-1pt}\to{\kern-1pt} B$ is a fiber bundle with section s and we choose basepoints $b_0{\kern-1pt}\in{\kern-1pt} B$ and $e_0{\kern-1pt}\in{\kern-1pt} p^{-1}(b_0){\kern-1pt}={\kern-1pt} F\subset E$ , the long exact homotopy sequence of the bundle splits, yielding

Here and below, we use $f_\sharp $ to denote the map on fundamental groups induced by a continuous map $f\colon (X,x_0)\to (Y,f(x_0))$ . With basepoints suppressed, the split exact sequence above realizes the fundamental group $\pi _1(E)$ of the total space as a semidirect product of the fundamental groups of the base and fiber. The semidirect product structure is determined by an action of $\pi _1(B)$ on $\pi _1(F)$ , that is, a homomorphism $\phi \colon \pi _1(B) \to {\mathrm {Aut}}(\pi _1(F))$ . Here, $ {\mathrm {Aut}}(\pi _1(F))$ denotes the group of right automorphisms of $\pi _1(F)$ , with group operation $\alpha \cdot \beta $ given by the composition $\beta \circ \alpha $ for automorphisms $\alpha $ and $\beta $ . The homomorphism $\phi \colon \pi _1(B) \to {\mathrm {Aut}}(\pi _1(F))$ is the monodromy of the bundle $p\colon E \to B$ .

In the case $X=\mathbb {C}$ , the bundle $ {\mathrm {Conf}}^{(k,1)}(\mathbb {C}) \to {\mathrm {Conf}}_k(\mathbb {C})/\Sigma _{k}$ noted above is equivalent (that is, fiberwise homeomorphic) to the bundle denoted $p_k\colon Y^{k+1}\to B^k$ in [Reference Cohen and Suciu12, §2], where $B^k= {\mathrm {Conf}}_k(\mathbb {C})/\Sigma _{k}$ . As noted there, the monodromy of this bundle is the Artin representation $\alpha _k\colon B_k \to {\mathrm {Aut}}(F_k)$ , where $B_k$ is the k-strand Artin (full) braid group and $ {\mathrm {Aut}}(F_k)$ is the group of right automorphisms of the free group $F_k$ . See, for instance, [Reference Hansen24] for a general reference on braids. In terms of the (standard) generators $\sigma _1,\dots ,\sigma _{k-1}$ of $B_k$ and $\mathsf {y}_1^{},\dots ,\mathsf {y}_k^{}$ of $F_k$ , the Artin representation is given by

(2.4) $$ \begin{align} \alpha_k(\sigma_i)(\mathsf{y}_j^{})=\begin{cases} \mathsf{y}_i^{}\mathsf{y}_{i+1}^{}\mathsf{y}_i^{-1}&\text{if }j=i,\\ \mathsf{y}_i^{}&\text{if }j=i+1,\\ \mathsf{y}_j^{}&\text{otherwise.} \end{cases} \end{align} $$

Since the Artin representation is faithful, for a braid $\beta $ , we often abbreviate the automorphism $\alpha _k(\beta )$ simply as $\beta $ . With this convention, the restriction $\hat {\alpha }_k\colon P_k \to {\mathrm {Aut}}(F_k)$ of the Artin representation to the pure braid group $P_k<B_k$ , with generators $a_{i,j}=\sigma _{j-1}^{}\cdots \sigma _{i+1}^{}\sigma _i^2\sigma _{i+1}^{-1}\cdots \sigma _{j-1}^{-1}$ , $1\le i<j\le k$ , is given by

(2.5) $$ \begin{align} a_{i,j}(\mathsf{y}_q^{})=\begin{cases} \mathsf{y}_i^{}\mathsf{y}_j^{} \cdot \mathsf{y}_q^{} \cdot (\mathsf{y}_i^{}\mathsf{y}_j^{})^{-1} &\text{if }q=i\text{ or }q=j,\\ [\mathsf{y}_i^{},\mathsf{y}_j^{}]\cdot \mathsf{y}_q^{} \cdot [\mathsf{y}_i^{},\mathsf{y}_j^{}]^{-1} &\text{if }i<q<j,\\ \mathsf{y}_q^{}&\text{otherwise.} \end{cases} \end{align} $$

One can write $\mathsf {y}_i^{}\mathsf {y}_j^{} \cdot \mathsf {y}_q^{} \cdot (\mathsf {y}_i^{}\mathsf {y}_j^{})^{-1}={[}\mathsf {y}_i^{}\mathsf {y}_j^{},\,\mathsf {y}_q^{}] \cdot \mathsf {y}_q^{}$ and $[\mathsf {y}_i^{},\mathsf {y}_j^{}]\cdot \mathsf {y}_q^{} \cdot [\mathsf {y}_i^{},\mathsf {y}_j^{}]^{-1}={[}[\mathsf {y}_i^{},\mathsf {y}_j^{}],\mathsf {y}_q^{}]\cdot \mathsf {y}_q^{}$ .

Observe that pure braid automorphisms are IA-automorphisms of the free group $F_k$ , inducing the identity on the abelianization. Also, as noted in [Reference Cohen and Suciu12, §2], the restriction $\hat {\alpha }_k$ of the Artin representation to $P_k=\pi _1( {\mathrm {Conf}}_k(\mathbb {C}))$ is the monodromy of the Fadell-Neuwirth bundle $ {\mathrm {Conf}}_{k+1}(\mathbb {C})\to {\mathrm {Conf}}_k(\mathbb {C})$ .

2.5 Abelian arrangement bundles as pullbacks

Let $\mathbb {G}$ be a connected abelian Lie group. Let $\mathscr {A}$ be an essential abelian arrangement in $T\cong \mathbb {G}^d$ and Y an admissible subgroup of T such that $\mathscr {P}(\mathscr {A}_Y)$ is an M-ideal of $\mathscr {P}(\mathscr {A})$ . Then, the projection $p:T\to T/Y$ restricts to a map $\bar {p}:M(\mathscr {A})\to M(\mathscr {A}/Y)$ . We prove that the restriction $\bar {p}$ is a pullback of a configuration space bundle from Proposition 2.9, building on special cases seen in [Reference Cohen8, Theorem 1.1.5] and [Reference Bibby and Delucchi4, Theorem 3.5.1].

Theorem 2.12. Let $\mathscr {A}$ be an abelian arrangement in $T\cong \mathbb {G}^d$ , and suppose that $\mathscr {P}(\mathscr {A}_Y)$ is a corank-one M-ideal of $\mathscr {P}(\mathscr {A})$ . There exist a composition $\mathbf {k}$ and a continuous map $g:M(\mathscr {A}/Y) \to {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {G})$ such that $\bar {p}:M(\mathscr {A})\to M(\mathscr {A}/Y)$ is the pullback of $\pi : {\mathrm {Conf}}^{(\mathbf {k},1)}(\mathbb {G})\to {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {G})$ along g, as in Figure 2.

Figure 2

Pullback diagram of Theorem 2.12.

Proof. Write $\mathscr {A}\smallsetminus \mathscr {A}_Y = \{H_1,\dots ,H_l\}$ . We think of $T\cong \mathbb {G}\times (T/Y)$ , and for $q=(x,t)\in T$ we let $[q]_1=x$ denote the first coordinate. By [Reference Bibby and Delucchi4, Corollary 3.3.2], for each i, the restriction of p to $H_i\subseteq T$ is a covering map $p_i\colon H_i\to T/Y$ . As such, the number $k_i:=|p_i^{-1}(t)|$ is independent of the choice of $t\in T/Y$ . The sequence $\mathbf {k}=(k_1,\dots ,k_l)$ is the composition we will use.

Define the function $g:M(\mathscr {A}/Y)\to {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {G})$ by

$$\begin{align*}g(t) = ([p_1^{-1}(t)]_1,\dots,[p_l^{-1}(t)]_1).\end{align*}$$

This is well defined since, by [Reference Bibby and Delucchi4, Lemma 3.2.4, Proposition 3.2.5], one has $[p_i^{-1}(t)]_1\cap [p_j^{-1}(t)]_1=\emptyset $ for every $t\in M(\mathscr {A}/Y)$ and $i\neq j$ .

In order to prove that g is continuous, take an open set $U\subseteq {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {G})$ and consider $t\in g^{-1}(U)$ . We will construct an open neighborhood of t in $M(\mathscr {A}/Y)$ contained in $g^{-1}(U)$ . Since $ {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {G})$ has the quotient topology from $ {\mathrm {Conf}}_k(\mathbb {G})$ , which has the subspace topology from $\mathbb {G}^k$ , we can choose small open sets $U_{ij}\subseteq \mathbb {G}$ , for $1\leq i\leq l$ and $1\leq j\leq k_i$ , so that

$$\begin{align*}U_{11}\times \dots\times U_{lk_l} \subseteq{\mathrm{Conf}}_k(\mathbb{G})\end{align*}$$

is a representative of an open neighborhood of $g(t)$ in U. For all $i,j$ , the set $ (U_{ij}\times T/Y) \cap H_i $ is open in $H_i$ . Since covering maps are open, for every $i,j$ , the set

$$ \begin{align*}V_{ij}:= p_{i}((U_{ij}\times T/Y) \cap H_i) \cap M(\mathscr{A}/Y) \end{align*} $$

is an open neighborhood of t in $M(\mathscr {A}/Y)$ with $g(V_{ij})\subseteq U$ . Thus, $\bigcap _{ij}V_{ij}$ is the desired open neighborhood of t in $g^{-1}(U)$ .

To complete the diagram (2), the map h is defined on $M(\mathscr {A})\subseteq T\cong \mathbb {G}\times (T/Y)$ via $h(t,x)=(g(t),x)$ . The check that this square satisfies the universal property of a pullback is routine (as in the proof of [Reference Bibby and Delucchi4, Theorem 5.3.1]).

3.1 Toric arrangement bundles

In the case that $\mathbb {G}=\mathbb {C}^\times $ , there is a close relationship between toric arrangements and configurations of points in the plane. This, in turn, has several particularly nice consequences.

Proof. From Theorem 2.12, we have a composition $\mathbf {k}$ and a map $g:M(\mathscr {A}/Y)\to {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {C}^\times )$ along which we can pull back the bundle $ {\mathrm {Conf}}^{(\mathbf {k},1)}(\mathbb {C}^\times )\to {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {C}^\times )$ to the bundle $\bar {p}$ , as in the left-hand square of Figure 3. We further have a continuous map $z\colon {\mathrm {Conf}}^{\mathbf {k}}(\mathbb {C}^\times )\to {\mathrm {Conf}}^{(\mathbf {k},1)}(\mathbb {C})$ , given by $(S_1,\dots ,S_m)\mapsto (S_1,\dots ,S_m,0)$ , making the middle square of Figure 3 a pullback diagram. Letting $n=k+1$ , from Remark 2.11, we also have a map $w \colon {\mathrm {Conf}}^{(\mathbf {k},1)}(\mathbb {C}) \to {\mathrm {Conf}}^{(n)}(\mathbb {C}) = {\mathrm {Conf}}_n(\mathbb {C})/\Sigma _{n}$ making the right-hand square a pullback.

Figure 3

Pullback diagram of Theorem 3.1.

Parts (1) and (2) of the theorem follow with $\mathbf {n}=(\mathbf {k},1)$ , $f=z\circ g$ , and $\mathbf {a}=w\circ z \circ g$ .

For part (3), as noted in Remark 2.14, if $\mathscr {P}(\mathscr {A}_Y)$ is a TM-ideal, the composition of Theorem 2.12 is the trivial composition $\mathbf {k}=(1,1,\dots ,1)$ . Consequently, $\mathbf {n}=(\mathbf {k},1)$ is trivial as well, and $ {\mathrm {Conf}}^{(\mathbf {k},1)}(\mathbb {C})$ is the ordered configuration space $ {\mathrm {Conf}}_n(\mathbb {C})$ . Setting $\mathbf {b}=z \circ g$ in this instance completes the proof.

3.2 Polynomials

Parts (2) and (3) of Theorem 3.1 bring to the fore the relationship between (strictly) supersolvable toric arrangements and Hansen’s theory of polynomial coverings [Reference Hansen24] and the associated braid bundles of [Reference Cohen and Suciu12]. In these situations, choices of the pullback maps $\mathbf {a}$ and $\mathbf {b}$ may be obtained directly from the characters defining the toric arrangement.

Now let $\mathscr {A}$ be an essential supersolvable toric arrangement in $(\mathbb {C}^\times )^{d+1}$ , with $\mathscr {P}(\mathscr {A}_Y)$ a corank-one M-ideal of $\mathscr {P}(\mathscr {A})$ . Write $\mathscr {A}\smallsetminus \mathscr {A}_Y=\{H_1,\dots ,H_l\}$ . Choosing coordinates $(x_1,\dots ,x_d,y)=(\mathbf {x},y)$ appropriately, for every $j=1,\ldots l$ the hypersurface $H_j$ is defined by

$$\begin{align*}\begin{aligned} H_j&=\{(x_1,\dots,x_d,y) \in (\mathbb{C}^\times)^{d+1} \mid y^{m_{j,0}} - \mu_j x_1^{m_{j,1}}x_2^{m_{j,2}} \cdots x_d^{m_{j,d}}=0\}\\ &=\{(\mathbf{x},y) \in (\mathbb{C}^\times)^{d+1} \mid y^{m_{j,0}} - \mu_j\mathbf{x}^{\mathbf{m}_j}=0\}, \end{aligned} \end{align*}$$

where $m_{j,0}\in \mathbb {Z}_{>0}$ is a positive integer, $\mathbf {m}_j=(m_{j,1},\dots ,m_{j,d}) \in \mathbb {Z}^d$ , and $\mu _j$ is a root of unity (cf. Remark 2.3). Since $\mathscr {A}$ is supersolvable over $\mathscr {B}=\mathscr {A}/Y$ , the map $f\colon M(\mathscr {B}) \times \mathbb {C} \to \mathbb {C}$ given by

$$\begin{align*}f(\mathbf{x},y) = y \prod_{j=1}^l(y^{m_{j,0}} - \mu_j\mathbf{x}^{\mathbf{m}_j}) = y^n + \sum_{i=1}^n a_i(\mathbf{x}) y^{n-i} \end{align*}$$

is a simple Weierstrass polynomial on $M(\mathscr {B})$ in the sense of [Reference Hansen24]: the coefficient maps $a_i\colon M(\mathscr {B}) \to \mathbb {C}$ are continuous, and, for each $\mathbf {x}\in M(\mathscr {B})$ , the polynomial $f(\mathbf {x},y) \in \mathbb {C}[y]$ has distinct roots. Note that the coefficient map $a_n$ is the constant function $a_n(\mathbf {x})=0$ . Sending $\mathbf {x}$ to the set of roots of the polynomial $f(\mathbf {x},y) = y^n + \sum _{i=1}^n a_i(\mathbf {x}) y^{n-i}$ defines a map $\mathbf {a} \colon M(\mathscr {B}) \to {\mathrm {Conf}}_n(\mathbb {C})/\Sigma _{n}$ . Identifying the unordered configuration space with the complement of the discriminant in $\mathbb {C}^n$ via Remark 3.4, this corresponds to sending $\mathbf {x}$ to (the coefficients of) the polynomial $f(\mathbf {x},y)$ . Following Hansen [Reference Hansen24], we call $\mathbf {a} \colon M(\mathscr {B}) \to {\mathrm {Conf}}_n(\mathbb {C})/\Sigma _{n}$ the coefficient map.

If, moreover, $\mathscr {P}(\mathscr {A}_Y)$ is a TM-ideal, then, factoring and reindexing as needed, we can assume that $m_{j,0}=1$ for each j. In this instance, the simple Weierstrass polynomial f is completely solvable, meaning it factors as

$$\begin{align*}f(\mathbf{x},y) = y \prod_{j=1}^l(y - \mu_j\mathbf{x}^{\mathbf{m}_j}) = \prod_{i=1}^n (y-b_i(\mathbf{x})), \end{align*}$$

where $\mu _j$ is some root of unity, with continuous root maps $b_i\colon M(\mathscr {B}) \to \mathbb {C}$ . Since the roots are distinct, this defines a root map $\mathbf {b} \colon M(\mathscr {B}) \to {\mathrm {Conf}}_n(\mathbb {C})$ , given by $\mathbf {x} \mapsto (b_1(\mathbf {x}),\dots ,b_n(\mathbf {x}))$ . Note that $n=l+1$ in this instance.

These considerations yield the following versions of parts (2) and (3) of Theorem 3.1, which may be checked directly. Recall from §2.4 that the monodromy of a bundle $p\colon E \to B$ , with fiber F and a (fixed) section, is the homomorphism from $\pi _1(B)$ to $ {\mathrm {Aut}}(\pi _1(F))$ , the group of (right) automorphisms of $\pi _1(F)$ , giving the action of the fundamental group of the base on that of the fiber. Also recall the Artin representation discussed in §2.4 and that the homomorphism on fundamental groups induced by a map f is denoted by $f_\sharp $ .

Example 3.7. Recall the toric arrangement $\mathscr {A}$ from Example 2.4, which, by Example 2.8, has a TM-ideal $\mathscr {P}(\mathscr {A}_{H_0})$ and an M-ideal $\mathscr {P}(\mathscr {A}_{H_3})$ .

The quotient by $H_0$ then induces a fiber bundle $\bar {p}\colon M(\mathscr {A})\to M(\mathscr {A}/H_0)=\mathbb {C}\smallsetminus \{0,-1,1\}$ , which can be pulled back from an ordered, or unordered, configuration space bundle:

In particular, the root map $\mathbf {b}\colon \mathbb {C}\smallsetminus \{0,-1,1\} \to {\mathrm {Conf}}_3(\mathbb {C})$ is given by $\mathbf {b}(x)=(0,x^2,1)$ . The induced homomorphism $\mathbf {b}_\sharp \colon F_3 \to P_3$ may be obtained from the calculations of §5 below. If $F_3=\langle \mathsf {x}_0,\mathsf {x}_1,\mathsf {x}_2\rangle $ , where $\mathsf {x}_0,\mathsf {x}_1,\mathsf {x}_2$ are represented by based loops $\gamma _0$ , $\gamma _1$ , and $\gamma _2$ about $0$ , $-1$ , and $1$ , respectively (see §5.1 for explicit formulas), we have

(3.1) $$ \begin{align} \mathbf{b}_\sharp(\mathsf{x}_0)=a_{1,2}^2,\quad \mathbf{b}_\sharp(\mathsf{x}_1)=a_{1,2}^{}a_{2,3}^{}a_{1,2}^{-1},\quad \mathbf{b}_\sharp(\mathsf{x}_2)=a_{2,3}^{}. \end{align} $$

The quotient by $H_3$ induces a fiber bundle $\bar {p}\colon M(\mathscr {A})\to M(\mathscr {A}/H_3)=\mathbb {C}\smallsetminus \{0,1\}$ , which can be pulled back from a configuration space bundle through g, $z\circ g$ , or $w\circ z\circ g$ , as below:

The coefficient map $\mathbf {a} \colon \mathbb {C}\smallsetminus \{0,1\} \to {\mathrm {Conf}}_5(\mathbb {C})/\Sigma _{5}$ is given by (the roots of) $\mathbf {a}(y)=x(x^2-1)(x^2-y) \in \mathbb {C}[x]$ . Let $\pi _1(\mathbb {C}\smallsetminus \{0,1\})=F_2=\langle \mathsf {u}_0,\mathsf {u}_1\rangle $ , where $\mathsf {u}_j$ is represented by a counterclockwise circular path $\mu _j$ of radius $1/2$ based at $1/2$ and centered at j. Lifts of the loops $\mathbf {a} \circ \mu _j$ in $ {\mathrm {Conf}}_5(\mathbb {C})$ are depicted in Figure 4. These paths may be used to check that the homomorphism $\mathbf {a}_\sharp \colon F_2 \to B_5$ induced by $\mathbf {a}$ is given by $\mathbf {a}_\sharp (\mathsf {u}_0)=\sigma _2\sigma _3\sigma _2$ and $\mathbf {a}_\sharp (\mathsf {u}_1)=\sigma _1^2\sigma _4^2$ .

Figure 4

Paths in $ {\mathrm {Conf}}_5(\mathbb {C})$ representing $\mathbf {a}_\sharp (\mathsf {u}_0)$ and $\mathbf {a}_\sharp (\mathsf {u}_1)$ .

Figure 4 Long description

The braid strands are labelled left-to-right by the numbers minus 1, minus 1 over square root of 2, 0, 1 over square root of 2, 1. The left-hand side braid has the first, third and fifth strands vertical, the second strand ends at the fourth position crossing in front of the third and fourth strands, the fourth strand ends ta the second position crossing in the back of the third and fourth strands. The right-hand side braid has the first, third and fifth strands vertical, the second strand winds around the first and returns to its original position, and the fourth winds around the fifth and returns to its original position.

3.3 Fundamental group

For a supersolvable toric arrangement $\mathscr {A}$ , the results of §§3.1–3.2 may easily be used to see that the complement $M(\mathscr {A})$ is a $K(G,1)$ -space, where $G=G(\mathscr {A})=\pi _1(M(\mathscr {A}))$ , as shown in [Reference Bibby and Delucchi4, Corollary B]. These results have further topological and group-theoretic implications. We begin with several properties of the group G.

An iterated semidirect product of finitely generated free groups $G=F_{n_r} \rtimes \big (F_{n_{r-1}} (\rtimes \dots \rtimes (F_{n_2} \rtimes F_{n_1}\big )$ is said to be an almost-direct product if the action of the group $\rtimes _{i=1}^{j} F_{n_i}$ on $H_1(F_{n_k};\mathbb {Z})$ is trivial for each j and k with $1\le j<k \le r$ . That is, each of the homomorphisms $\rtimes _{i=1}^{k-1} F_{n_i} \to {\mathrm {Aut}}(F_{n_k})$ determining the iterated semidirect product structure of G has image contained in the subgroup of IA-automorphisms, consisting of automorphisms that induce the identity on the abelianization of $F_{n_k}$ .

Proof. We proceed by induction on the rank of $\mathscr {A}$ . As the base case is clear, assume that the rank of $\mathscr {A}$ is greater than one. Since $\mathscr {A}$ is supersolvable, we have, by Proposition 3.6, a supersolvable arrangement $\mathscr {B}$ and a fiber bundle $M(\mathscr {A})\to M(\mathscr {B})$ whose fiber F is homeomorphic to $\mathbb {C}$ with finitely many points removed. The associated long exact sequence on homotopy groups reduces to a short exact sequence on the fundamental groups

$$\begin{align*}1 \longrightarrow \pi_1(F) \longrightarrow \pi_1(M(\mathscr{A})) \longrightarrow \pi_1(M(\mathscr{B})) \longrightarrow 1. \end{align*}$$

This short exact sequence splits by Corollary 3.2, implying that $\pi _1(M(\mathscr {A}))$ is a semidirect product of $\pi _1(M(\mathscr {B}))$ (an iterated semidirect product of free groups with actions given by braid automorphisms, by induction) and $\pi _1(F)$ (a free group). By Proposition 3.6 (1), the monodromy of the bundle ${M(\mathscr {A})\to M(\mathscr {B})}$ factors through a braid group, so $\pi _1(M(\mathscr {B}))$ acts on $\pi _1(F)$ by braid automorphisms.

If we moreover have that $\mathscr {A}$ is strictly supersolvable, then we can choose $\mathscr {B}$ to also be strictly supersolvable and the fiber bundle $M(\mathscr {A}) \to M(\mathscr {B})$ is pulled back from the ordered configuration space bundle, via Theorem 3.1 (3). By induction, $\pi _1(M(\mathscr {B}))$ is an almost-direct product of free groups with actions given by pure braid automorphisms. By Proposition 3.6 (2), the monodromy of the bundle $M(\mathscr {A})\to M(\mathscr {B})$ factors through a pure braid group, so $\pi _1(M(\mathscr {B}))$ acts on $\pi _1(F)$ by pure braid automorphisms, which as noted in §2.4 act trivially on homology.

Recall that a discrete group is said to be linear if it admits a faithful, finite-dimensional linear representation (over some field). From the work of Bigelow [Reference Bigelow5] and Krammer [Reference Krammer27], it is known that the Artin braid group is linear. This, together with the above, can be used to establish the linearity of supersolvable toric arrangement groups. The proof given below follows that of [Reference Cohen, Cohen and Prassidis10], where supersolvable hyperplane arrangement groups were shown to be linear.

Proof. We again proceed by induction on the rank of $\mathscr {A}$ . The base case is clear, as the fundamental group of the complement of a rank-one toric arrangement is a finitely generated free group.

For $\mathscr {A}$ supersolvable, as above, we have a corank-one supersolvable arrangement $\mathscr {B}$ and a fiber bundle $M(\mathscr {A})\to M(\mathscr {B})$ . By Theorem 3.1 (2), this bundle may be realized as a pullback of the bundle $ {\mathrm {Conf}}^{(n,1)}(\mathbb {C}) \to {\mathrm {Conf}}_n(\mathbb {C})/\Sigma _{n}$ over the unordered configuration space, with fiber $\mathbb {C}\smallsetminus \{n\ \text {points}\}$ . This yields a commutative diagram of fundamental groups with split short exact rows

realizing the group $\pi _1(M(\mathscr {A}))$ as a pullback.

The fundamental group $\pi _1( {\mathrm {Conf}}_n(\mathbb {C})/\Sigma _{n})$ is the n-strand Artin braid group $B_n$ , which, as noted above, is linear. The group $\pi _1( {\mathrm {Conf}}^{(n,1)}(\mathbb {C}))$ may be realized as the subgroup of the $(n+1)$ -strand braid group $B_{n+1}$ for which the endpoint of the last strand is fixed, so is also linear. Assuming inductively that $\pi _1(M(\mathscr {B}))$ is linear, it follows that the pullback $\pi _1(M(\mathscr {A}))$ is also linear, as it is a subgroup of the product $\pi _1( {\mathrm {Conf}}^{(n,1)}(\mathbb {C}))\times \pi _1(M(\mathscr {B}))$ of linear groups.

Remark 3.10. If $\mathscr {A}=\mathscr {A}_r$ is supersolvable of rank r, then, from (2.2), we have an increasing chain of M-ideals $\mathscr {P}(\mathscr {A}_{j})$ , $1\le j <r$ , corresponding supersolvable arrangements $\mathscr {A}_j:= \mathscr {A}_{Y_j}$ , so that $\mathscr {A}_j=\mathscr {A}_{j+1}/Y_j$ , and coefficient maps $\mathbf {a}_j \colon M(\mathscr {A}_{j-1}) \to {\mathrm {Conf}}_{n_{j}}(\mathbb {C})/\Sigma _{n_j}$ for $j\geq 2$ . Setting $n_1=1+|\mathscr {A}_1|$ , the fundamental group $G=G(\mathscr {A})=\pi _1(M(\mathscr {A}))$ is an iterated semidirect product of free groups $G=\rtimes _{j=1}^{r} F_{n_j}$ , acting upon one another by braid automorphisms by Proposition 3.6 (1).

For each j, $2\le j \le r$ , let $\phi _j={\alpha }_{n_j} \circ (\mathbf {a}_j)_\sharp \colon \pi _1(M(\mathscr {A}_{j-1})) \to {\mathrm {Aut}}(F_{n_j})$ . If $F_{n_j}=\langle \mathsf {y}_{p,j}^{}\ 1\le p \le n_j\rangle $ , the group G has generators $\mathsf {y}_{p,j}$ , $1\le p \le n_j$ , $1\le j \le r$ , and relations

(3.2) $$ \begin{align} \mathsf{y}_{p,i}^{-1} \mathsf{y}_{q,j}^{} \mathsf{y}_{p,i}^{}=\phi_j(\mathsf{y}_{p,i}^{})(\mathsf{y}_{q,j}^{}),\ 1\le p \le n_i, 1\le q \le n_j, 1\le i < j \le r. \end{align} $$

If, further, $\mathscr {A}$ is strictly supersolvable, we have corresponding root maps $\mathbf {b}_j \colon M(\mathscr {A}_{j-1}) \to {\mathrm {Conf}}_{n_{j}}(\mathbb {C})$ , and the homomorphism $\phi _j\colon \pi _1(M(\mathscr {A}_{j-1})) \to {\mathrm {Aut}}(F_{n_j})$ may be expressed as $\phi _j=\hat {\alpha }_{n_j} \circ (\mathbf {b}_j)_\sharp $ . In this instance, G is an almost-direct product of free groups, acting upon one another by pure braid automorphisms, by Proposition 3.6 (2).

Since $\phi _j$ is the composition of the Artin representation and the homomorphism induced by the coefficient or root map, determining the latter yields an explicit presentation of the group $G=\pi _1(M(\mathscr {A}))$ . We illustrate this with our running example next. See §5.1 and §7.1 for further illustrations.

Example 3.11. Recall the toric arrangement $\mathscr {A}$ from Example 2.4 and the associated fiber bundles from Example 3.7.

For the first of these bundles, $\bar {p}\colon M(\mathscr {A})\to M(\mathscr {A}/H_0)=\mathbb {C}\smallsetminus \{0,-1,1\}$ , the action of the fundamental group of the base $F_3=\langle \mathsf {x}_0,\mathsf {x}_1,\mathsf {x}_2\rangle $ on that of the fiber $F_3=\langle \mathsf {y}_1,\mathsf {y}_2,\mathsf {y}_3\rangle $ is the composition $\phi =\hat \alpha _3 \circ \mathbf {b}_\sharp $ of the root-map-induced homomorphism $\mathbf {b}_\sharp \colon F_3 \to P_3$ and the Artin representation $\hat \alpha _3\colon P_3 \to {\mathrm {Aut}}(F_3)$ . Computing with the expression for $\mathbf {b}_\sharp $ from (3.1) in Example 3.7 and with the Artin representation given in (2.5) yields a presentation of $\pi _1(M(\mathscr {A}))$ with generators $\mathsf {x}_0,\mathsf {x}_1,\mathsf {x}_2,\mathsf {y}_1,\mathsf {y}_2,\mathsf {y}_3$ and relations

$$\begin{align*}u^{-1}v u = \phi(u)(v)=w(u,v) \cdot v\cdot w(u,v)^{-1}\end{align*}$$

for $u\in \{\mathsf {x}_0,\mathsf {x}_1,\mathsf {x}_2\}$ , $v\in \{\mathsf {y}_1,\mathsf {y}_2,\mathsf {y}_3\}$ , and

$$\begin{align*}\begin{matrix} \begin{aligned} w(\mathsf{x}_0^{},\mathsf{y}_1^{})&=(\mathsf{y}_1^{}\mathsf{y}_2^{})^{2}, \\ w(\mathsf{x}_0^{},\mathsf{y}_2^{})&= \mathsf{y}_1^{}\mathsf{y}_2^{}\mathsf{y}_1^{},\\ w(\mathsf{x}_0^{},\mathsf{y}_3^{})&=1, \end{aligned} & \begin{aligned} w(\mathsf{x}_1^{},\mathsf{y}_1^{})&=[\mathsf{y}_1^{}\mathsf{y}_2^{}\mathsf{y}_3^{} \mathsf{y}_2^{-1} \mathsf{y}_1^{-1},\,\mathsf{y}_2], \\ w(\mathsf{x}_1^{},\mathsf{y}_2^{})&= \mathsf{y}_1^{}\mathsf{y}_2^{}\mathsf{y}_3^{} \mathsf{y}_2^{-1} \mathsf{y}_1^{-1},\\ w(\mathsf{x}_1^{},\mathsf{y}_3^{})&=[\mathsf{y}_2^{-1},\mathsf{y}_1^{-1}]\cdot \mathsf{y}_2, \end{aligned} & \begin{aligned} w(\mathsf{x}_2^{},\mathsf{y}_1^{})&=1, \\ w(\mathsf{x}_2^{},\mathsf{y}_2^{})&= \mathsf{y}_2^{}\mathsf{y}_3^{},\\ w(\mathsf{x}_2^{},\mathsf{y}_3^{})&=\mathsf{y}_2^{}. \end{aligned} \end{matrix}\end{align*}$$

For the second bundle, $\bar {p}\colon M(\mathscr {A})\to M(\mathscr {A}/H_3)=\mathbb {C}\smallsetminus \{0,1\}$ , the action of the fundamental group of the base $F_2=\langle \mathsf {u}_0,\mathsf {u}_1\rangle $ on that of the fiber $F_5=\langle \mathsf {v}_1,\dots ,\mathsf {v}_5\rangle $ is the composition $\psi =\alpha _5 \circ \mathbf {a}_\sharp $ of the coefficient-map-induced homomorphism $\mathbf {a}_\sharp \colon F_2 \to B_5$ and the Artin representation $\alpha _5\colon B_5 \to {\mathrm {Aut}}(F_5)$ . Computing with the expression for $\mathbf {a}_\sharp $ given in Example 3.7 and (2.4) yields a presentation of $\pi _1(M(\mathscr {A}))$ with generators $\mathsf {u}_0,\mathsf {u}_1,\mathsf {v}_1,\mathsf {v}_2,\mathsf {v}_3,\mathsf {v}_4,\mathsf {v}_5$ and relations

$$\begin{align*}\begin{matrix} \begin{aligned} \mathsf{u}_0^{-1}\mathsf{v}_1^{}\mathsf{u}_0^{}&=\mathsf{v}_1^{},\\ \mathsf{u}_0^{-1}\mathsf{v}_2^{}\mathsf{u}_0^{}&= \mathsf{v}_2^{}\mathsf{v}_3^{}\mathsf{v}_4^{}\mathsf{v}_3^{-1}\mathsf{v}_2^{-1},\\ \mathsf{u}_0^{-1}\mathsf{v}_3^{}\mathsf{u}_0^{}&=\mathsf{v}_2^{}\mathsf{v}_3^{}\mathsf{v}_2^{-1}, \end{aligned} & \begin{aligned} \mathsf{u}_0^{-1}\mathsf{v}_4^{}\mathsf{u}_0^{}&=\mathsf{v}_2^{},\\ \mathsf{u}_0^{-1}\mathsf{v}_5^{}\mathsf{u}_0^{}&=\mathsf{v}_5,\\ \ & \end{aligned} & \begin{aligned} \mathsf{u}_1^{-1}\mathsf{v}_1^{}\mathsf{u}_1^{}&=\mathsf{v}_1^{}\mathsf{v}_2^{}\mathsf{v}_1\mathsf{v}_2^{-1}\mathsf{v}_1^{-1}, \\ \mathsf{u}_1^{-1}\mathsf{v}_2^{}\mathsf{u}_1^{}&= \mathsf{v}_1^{}\mathsf{v}_2^{}\mathsf{v}_1^{-1},\\ \mathsf{u}_1^{-1}\mathsf{v}_3^{}\mathsf{u}_1^{}&=\mathsf{v}_3^{}, \end{aligned} & \begin{aligned} \mathsf{u}_1^{-1}\mathsf{v}_4^{}\mathsf{u}_1^{}&=\mathsf{v}_4^{}\mathsf{v}_5^{}\mathsf{v}_4^{}\mathsf{v}_5^{-1}\mathsf{v}_4^{-1},\\ \mathsf{u}_1^{-1}\mathsf{v}_5^{}\mathsf{u}_1^{}&=\mathsf{v}_4^{}\mathsf{v}_5^{}\mathsf{v}_4^{-1}.\\ \ & \end{aligned} \end{matrix}\end{align*}$$

Rewriting these relations using $\mathsf {v}_2^{}=\mathsf {u}_0^{-1}\mathsf {v}_4^{}\mathsf {u}_0^{}$ yields a presentation with six generators and nine relations. Letting $\mathsf {w}=\mathsf {y}_1^{}\mathsf {y}_2^{}\mathsf {y}_3^{}\mathsf {y}_2^{-1}\mathsf {y}_1^{-1}$ , one can check that the correspondence between this presentation and that arising from the first bundle is given by

$$\begin{align*}\mathsf{v}_1^{} \mapsto \mathsf{x}_1 [\mathsf{w},\,\mathsf{y}_2], \quad \mathsf{v}_3^{} \mapsto \mathsf{x}_0^{}, \quad \mathsf{v}_5^{} \mapsto \mathsf{x}_2^{}, \quad \mathsf{u}_0^{} \mapsto \mathsf{y}_1, \quad \mathsf{v}_4 \mapsto \mathsf{y}_1^{}\mathsf{y}_2^{}\mathsf{y}_1^{-1}, \quad \mathsf{u}_1 \mapsto \mathsf{w}. \end{align*}$$

4.1 Lower central series Lie algebra

We first investigate the (integral) lower central series (LCS) Lie algebra of the fundamental group $G(\mathscr {A})=\pi _1(M(\mathscr {A}))$ of the complement of a strictly supersolvable toric arrangement $\mathscr {A}$ . We denote this Lie algebra by $\mathfrak {h}(G(\mathscr {A}))$ , or, more briefly, $\mathfrak {h}(\mathscr {A})$ . We begin with a brief discussion of the LCS Lie algebra of the pure braid group, sometimes referred to as the universal Yang-Baxter Lie algebra, which will play a prominent role in what follows.

Example 4.1. The structure of the LCS Lie algebra of the pure braid group $P_n=\pi _1( {\mathrm {Conf}}_n(\mathbb {C}))$ was determined by Kohno [Reference Kohno26]. The Lie algebra $\mathfrak {h}(P_n)$ is generated by $A_{i,j}=[a_{i,j}]$ , $1\le i<j \le n$ , the homology classes of the generators $a_{i,j}$ of $P_n$ , and has relations

$$\begin{align*}[A_{i,j},A_{k,l}]=0 \ \text{for}\ i,j,k,l\ \text{distinct, and}\ [A_{q,k},A_{i,j}+A_{i,k}+A_{j,k}]=0\ \text{for}\ q=i,j. \end{align*}$$

From this description, it follows that $\mathfrak {h}(P_{n+1})$ is the semidirect product of $\mathfrak {h}(P_n)$ by $\mathbb {L}[n]$ (the free Lie algebra generated by $A_{i,n+1}$ , $1\le i\le n$ ) determined by the Lie homomorphism $\theta _n\colon \mathfrak {h}(P_n) \to {\mathrm {Der}}(\mathbb {L}[n])$ given by $\theta _n(A_{i,j})= {\mathrm {ad}}(A_{i,j})$ . From the relations above, the adjoint action of $\mathfrak {h}(P_n)$ on $\mathbb {L}[n]$ is given by

(4.2) $$ \begin{align} \begin{aligned} \theta_n(A_{i,j})(A_{q,n+1})&={\mathrm{ad}}(A_{i,j})(A_{q,n+1})\\ &=[A_{i,j},A_{q,n+1}]=\begin{cases} [A_{q,n+1},A_{i,n+1}+A_{j,n+1}]&\text{if }q=i,j,\\ 0&\text{otherwise.} \end{cases} \end{aligned} \end{align} $$

We will refer to (4.2), resp., the underlying relations, as the infinitesimal pure braid relations.

Proof. For $\mathscr {A}$ strictly supersolvable of rank r, the fundamental group $G(\mathscr {A})=\rtimes _{i=1}^{r} F_{n_i}$ is an almost-direct product of free groups. The LCS Lie algebra of the free group $F_n$ is the free Lie algebra $\mathbb {L}[n]$ . From the almost-direct product structure of $G(\mathscr {A})$ , we have an isomorphism of abelian groups $\mathfrak {h}(\mathscr {A}) \cong \mathbb {L}[n_1]\oplus \dots \oplus \mathbb {L}[n_r]$ as in [Reference Falk and Randell23, Theorem 3.1] (see also [Reference Cohen, Cohen and Xicoténcatl11, Theorem 4.4]).

We must show that the sequence of homological root homomorphisms, together with (4.2), determines the iterated semidirect product structure of the Lie algebra $\mathfrak {h}(\mathscr {A})$ . This is accomplished by induction on the rank r of $\mathscr {A}$ . In the base case $r=1$ , there is nothing to prove, as the fundamental group $G(\mathscr {A})$ is a finitely generated free group, the lower central series Lie algebra $\mathfrak {h}(\mathscr {A})$ is a free Lie algebra, and the sequence of homological root homomorphisms is vacuous.

For the general case, write $\mathscr {B}=\mathscr {A}_{r-1}$ , $n=n_r$ , and denote the root map $\mathbf {b}_r$ simply by $\mathbf {b}$ . By induction, the LCS Lie algebra $\mathfrak {h}(\mathscr {B})$ of $G(\mathscr {B})$ is an iterated semidirect product of free Lie algebras determined by (4.2) and the (truncated) sequence of homological root homomorphisms $\big ((\mathbf {b}_2)_*,\dots ,(\mathbf {b}_{r-1})_*)$ . Since the bundle $\bar {p}:M(\mathscr {A})\to M(\mathscr {B})$ is the pullback of the bundle $\pi \colon {\mathrm {Conf}}_{n+1}(\mathbb {C}) \to {\mathrm {Conf}}_n(\mathbb {C})$ along the root map $\mathbf {b}=\mathbf {b}_r \colon M(\mathscr {B}) \to {\mathrm {Conf}}_n(\mathbb {C})$ , we have commutative diagrams of fundamental groups and associated LCS Lie algebras:

Since the Fadell-Neuwirth bundle admits a section, the bundle $M(\mathscr {A})\to M(\mathscr {B})$ does as well, and the rows of both diagrams are split exact.

Using the fact that $\mathfrak {h}(P_{n+1})$ is the semidirect product of $\mathfrak {h}(P_n)$ by $\mathbb {L}[n]$ determined by the Lie homomorphism $\theta _n\colon \mathfrak {h}(P_n) \to {\mathrm {Der}}(\mathbb {L}[n])$ given in (4.2), the right-hand pullback diagram of Lie algebras implies that $\mathfrak {h}(\mathscr {A})$ is the semidirect product of $\mathfrak {h}(\mathscr {B})$ by $\mathbb {L}[n]$ determined by the composite $\theta _n \circ \mathbf {b}_* \colon \mathfrak {h}(\mathscr {B}) \to {\mathrm {Der}}(\mathbb {L}[n])$ . This completes the proof.

Remark 4.3. The preceding result may be used to obtain a presentation for the LCS Lie algebra $\mathfrak {h}(\mathscr {A})$ . Denote the homology classes of the generators $y_{p,j}$ of $G(\mathscr {A})$ by $e_{p,j}$ . From the proof of Theorem 4.2, these classes satisfy $[e_{p,i},e_{q,j}] = {\mathrm {ad}}\left ((\mathbf {b}_j)_*(e_{p,i})\right )(e_{q,j})$ in $\mathfrak {h}(\mathscr {A})$ , and all relations in $\mathfrak {h}(\mathscr {A})$ are consequences of these. Thus, $\mathfrak {h}(\mathscr {A})$ is the quotient of the free Lie algebra generated by $\{e_{p,j} \mid 1\le j \le r,\ 1\le p\le n_j\}$ by the Lie ideal generated by

$$\begin{align*}[e_{p,i},e_{q,j}] - {\mathrm{ad}}\left((\mathbf{b}_j)_*(e_{p,i})\right)(e_{q,j}),\ 1\le i<j \le r,\ 1\le p\le n_i,\ 1\le q\le n_j. \end{align*}$$

4.2 Cohomology ring

We now turn our attention to the cohomology ring of the complement of a strictly supersolvable toric arrangement.

Proof. From §§3.1–3.3, if $\mathscr {A}$ is strictly supersolvable of rank r, the fundamental group of the complement $G(\mathscr {A})=\pi _1(M(\mathscr {A}))=\rtimes _{i=1}^{r} F_{n_i}$ is an almost-direct product of free groups (Corollary 3.8), and the complement $M(\mathscr {A})$ is a $K(G(\mathscr {A}),1)$ -space. Consequently, results of [Reference Cohen9] may be used to determine the structure of $H^*(M(\mathscr {A});\mathbb {Z})=H^*(G(\mathscr {A});\mathbb {Z})$ . We phrase the proof in terms of homology and cohomology of groups. For the remainder of the paper, we suppress coefficients when considering (co)homology with integer coefficients, for example, $H^*(G(\mathscr {A}))=H^*(G(\mathscr {A});\mathbb {Z})$ .

Since $G(\mathscr {A})$ is an almost-direct product of free groups, the integral homology groups are free abelian, with ranks given by

$$\begin{align*}\sum_{i=1}^r \mathrm{rank}\,H_i(G(\mathscr{A})) \cdot t^i = \prod_{j=1}^r (1+n_jt). \end{align*}$$

See [Reference Cohen and Suciu13, Corollary 3.2]. In particular, $H_1(G(\mathscr {A}))$ has rank $N=n_1+\dots +n_r$ , and $H_2(G(\mathscr {A}))$ has rank $\sum _{i<j} n_i n_j$ . Note that $N=r+|\mathscr {A}|$ , and observe that $\mathrm {rank}\,H_2(G(\mathscr {A}))$ is equal to the number of relations recorded in (3.2). As shown in [Reference Cohen9, §2], the abelianization map $\mathfrak {a}\colon G(\mathscr {A}) \to \mathbb {Z}^N$ induces isomorphisms $a_*\colon H_1(G(\mathscr {A}))\to H_1(\mathbb {Z}^N)$ , $a^*\colon H^1(\mathbb {Z}^N) \to H^1(G(\mathscr {A}))$ , a monomorphism $\mathfrak {a}_*\colon H_2(G(\mathscr {A})) \to H_2(\mathbb {Z}^N)$ , and an epimorphism $\mathfrak {a}^*\colon H^2(\mathbb {Z}^N) \to H^2(G(\mathscr {A}))$ .

Denote the integral exterior algebra $H^*(\mathbb {Z}^N)$ by $\mathsf {E}$ , and let $\mathsf {I}$ be the ideal in $\mathsf {E}$ generated by $\ker \big (\mathfrak {a}^*\colon H^2(\mathbb {Z}^N)\to H^2(G(\mathscr {A}))\big )$ . Similarly, let $\mathsf {E}_{\mathbb {Q}}=H^*(\mathbb {Z}^N;\mathbb {Q})$ and let $\mathsf {I}_{\mathbb {Q}}$ be the ideal in $\mathsf {E}_{\mathbb {Q}}$ corresponding to $\mathsf {I}\subset \mathsf {E}$ . By [Reference Cohen9, Theorem 3.1], we have $H^*(G(\mathscr {A});\mathbb {Q}) \cong {\mathsf {E}_{\mathbb {Q}}}/{\mathsf {I}_{\mathbb {Q}}}$ . Since $H^*(G(\mathscr {A}))$ is torsion-free (and is finitely generated), it follows that $H^*(G(\mathscr {A})) \cong {\mathsf {E}}/{\mathsf {I}}$ . As the map $\mathfrak {a}^*\colon H^2(\mathbb {Z}^N)\to H^2(G(\mathscr {A}))$ is dual to $\mathfrak {a}_*\colon H_2(G(\mathscr {A})) \to H_2(\mathbb {Z}^N)$ , to prove the theorem, it suffices to show that the latter is determined by the sequence of homological root homomorphisms and the infinitesimal pure braid relations.

Recall that $G(\mathscr {A})=\rtimes _{j=1}^{r} F_{n_j}$ has generators $\mathsf {y}_{p,j}$ , $1\le p \le n_j$ , $1\le j \le r$ , and relations given by (3.2). Write a representative relation of this kind as $\mathsf {x}^{-1} \mathsf {y} \mathsf {x}=\phi (\mathsf {x})(\mathsf {y})$ , where $\mathsf {x}=\mathsf {y}_{p,i}$ , $\mathsf {y}=\mathsf {y}_{q,j}$ , $i<j$ , and $\phi $ is the composition of the (faithful) Artin representation and the homomorphism induced by the root map $\mathbf {b}_j$ . The pure braid $\mathsf {z}=(\mathbf {b}_j)_\sharp (\mathsf {x})$ acts by conjugation, so we have $\phi (\mathsf {x})(\mathsf {y})=\mathsf {z}^{-1}\mathsf {y} \mathsf {z} = \mathsf {w} \mathsf {y} \mathsf {w}^{-1}$ , where $\mathsf {w},\mathsf {y} \in F_{n_j}$ . Since $\mathsf {y}\mathsf {x}=\mathsf {x}\mathsf {y}[\mathsf {y}^{-1},\mathsf {w}]=\mathsf {x}\mathsf {y}\cdot \mathsf {y}^{-1} [\mathsf {w},\mathsf {y}]{\mathsf {y}} = \mathsf {x}\mathsf {y}\cdot \big [\mathsf {y}^{-1},[\mathsf {w},\mathsf {y}]\big ] \cdot [\mathsf {w},\mathsf {y}]$ , this representative relation can be rewritten in the form

(4.3) $$ \begin{align} \mathsf{y}\mathsf{x} = \mathsf{x}\mathsf{y}\cdot \big[\mathsf{y}^{-1},[\mathsf{w},\mathsf{y}]\big] \cdot [\mathsf{w},\mathsf{y}], \end{align} $$

as in [Reference Cohen9, Proposition 2.2].

The homology classes $e_{p,j}$ of the generators $\mathsf {y}_{p,j}$ of $G(\mathscr {A})$ form a basis for $H_1(G(\mathscr {A}))=H_1(\mathbb {Z}^N)$ . Identifying $H_2(\mathbb {Z}^N)=\mathbb {Z}^{\binom {N}{2}}$ with the second graded piece of the exterior algebra $\mathsf {E}$ , this group has basis $e_{p,i}e_{q,j}$ , where $1\le i\le j \le r$ , $1\le p \le n_i$ , $1\le q \le n_j$ , and $p<q$ if $i=j$ . Since, as noted above, the (free abelian) group $H_2(G(\mathscr {A}))$ has rank $\sum _{i<j} n_in_j$ , this group has generators in correspondence with (the reformulations (4.3) of) the relations (3.2). If $\mathbf {r}$ denotes the (generator corresponding to the) representative relation (4.3) above, with $\mathsf {x}=\mathsf {y}_{p,i}$ and $\mathsf {y}=\mathsf {y}_{q,j}$ , it follows from [Reference Cohen9, §2] that $\mathfrak {a}_*(\mathbf {r}) = e_{p,i}e_{q,j} + {\mathsf {W}} e_{q,j}$ , where ${\mathsf {W}}=\mathfrak {a}(\mathsf {w})$ is the image of $\mathsf {w}$ under the abelianization map.

The relation (4.3) also gives rise to a relation in the LCS Lie algebra $\mathfrak {h}(\mathscr {A})$ of $G(\mathscr {A})$ , as in [Reference Cohen8, Lemma 2.3.4]. Rewriting the relation $\mathbf {r}$ as

$$\begin{align*}1_{G(\mathscr{A})} = [\mathsf{x}\mathsf{w},\mathsf{y}]=\left[\mathsf{x},[\mathsf{w},\mathsf{y}]\right] \cdot [\mathsf{w},\mathsf{y}] \cdot [\mathsf{x},\mathsf{y}], \end{align*}$$

we have $0=[{\mathsf {W}},e_{q,j}] + [e_{p,i},e_{q,j}] = [e_{p,i}+{\mathsf {W}},e_{q,j}]$ in $\mathfrak {h}(\mathscr {A})$ . Since these (defining) relations in $\mathfrak {h}(\mathscr {A})$ are determined by the sequence of homological root homomorphisms and the infinitesimal pure braid relations, by Theorem 4.2, so is the map $\mathfrak {a}_*\colon H_2(G(\mathscr {A})) \to H_2(\mathbb {Z}^N)$ , as required.

4.3 Topological complexity

Let X be a path-connected topological space with the homotopy type of a finite cell complex, and let $X^I$ denote the space of all continuous paths $\gamma \colon I=[0,1] \to X$ .

The homotopy-type invariant $\mathsf {TC}(X)$ , introduced by Farber [Reference Farber20], is motivated by the motion-planning problem from robotics. This notion may be extended to a discrete group G by defining $\mathsf {TC}(G)$ to be the topological complexity of an Eilenberg-Mac Lane space of type $K(G,1)$ .

5.1 Fundamental group

By Proposition 3.6 (2), the bundle $M(\mathscr {C}_{n,m}) \to M(\mathscr {B})$ is equivalent to the pullback of the Fadell-Neuwirth bundle $ {\mathrm {Conf}}_4(\mathbb {C}) \to {\mathrm {Conf}}_3(\mathbb {C})$ along the map $\mathbf {b}\colon M(\mathscr {B}) \to {\mathrm {Conf}}_3(\mathbb {C})$ given by $\mathbf {b}(x)=(0,x^m,1)$ . We determine the map on fundamental groups induced by $\mathbf {b}$ . With $\mu =\exp (2\pi \boldsymbol {\iota }/n)$ , where $\boldsymbol {\iota }=\sqrt {-1}$ , the fundamental group of $M(\mathscr {B})=\mathbb {C}\smallsetminus \{0,1,\mu ,\dots ,\mu ^{n-1}\}$ is free on $n+1$ generators. Fix $\epsilon>0$ small, and fix the basepoint $\ast =1-\epsilon $ in $M(\mathscr {B})$ . Let $\ell (t)=1-\epsilon \exp (2 \pi \boldsymbol {\iota } t)$ , $0\le t \le 1$ , be a loop about $1$ based at $\ast $ , and for $1\le j \le n$ , let $f_j(t)=(1-\epsilon )\exp (2 \pi \boldsymbol {\iota } t)$ , $0\le t \le j/n$ , be the circular arc from $*$ to $\mu ^j(1-\epsilon )$ . Note that $\mu ^j \ell (t)$ is a loop about $\mu ^j$ based at $\mu ^j(1-\epsilon )$ , and that $f_n(t)$ is a loop about $0$ based at $\ast $ . Loops based at $\ast $ representing the generators of $\pi _1(M(\mathscr {B}))=F_{n+1}$ are then given by $\gamma _0(t)=f_n(t)$ , $\gamma _j(t)=f_j(t) \centerdot \mu ^j\ell (t) \centerdot \bar f_j(t)$ for $1 \le j \le n-1$ , where $\bar f_j(t):=f_j(1-t)$ denotes the reverse path, $f(t) \centerdot g(t)$ denotes concatenation, and $\gamma _n(t)=\ell (t)$ . The case $n=6$ , $m=3$ is illustrated in Figure 5.

Figure 5

Loops in $M(\mathscr {B})$ when $n=6$ , $m=3$ .

The fundamental group of $ {\mathrm {Conf}}_3(\mathbb {C})$ (with basepoint $\mathbf {b}(\ast )=(0,r,1)$ , where $0<r=(1-\epsilon )^m<1$ ) is the three-strand pure braid group $P_3=\langle a_{1,2}^{},a_{1,3}^{},a_{2,3}^{}\mid a_{1,2}^{}a_{1,3}^{}a_{2,3}^{}= a_{1,3}^{}a_{2,3}^{}a_{1,2}^{}=a_{2,3}^{}a_{1,2}^{}a_{1,3}^{}\rangle $ . The pure braids $a_{1,2}^{}$ and $a_{2,3}^{}$ may be represented by loops $(0,r\exp (2\pi \boldsymbol {\iota } \theta ),1)$ and $(0,1-(1-r)\exp (2\pi \boldsymbol {\iota } \theta ),1)$ , $0\le \theta \le 1$ , respectively. (An explicit representative of $a_{1,3}^{}$ will not be needed in the following calculations.) The map $\mathbf {b}_\sharp \colon \pi _1(M(\mathscr {B})) \to P_3$ induced by $\mathbf {b}$ is given by

(5.1) $$ \begin{align} \begin{aligned} \mathbf{b}_\sharp([\gamma_0])&=[\mathbf{b}\circ f_n]= [(0,r\exp(2\pi \boldsymbol{\iota} m t),1)] = a_{1,2}^m,\\ \mathbf{b}_\sharp([\gamma_n])&= [\mathbf{b}\circ \ell]= [(0,(1-\epsilon\exp(2 \pi \boldsymbol{\iota} t))^m,1)] = [(0,1-r\exp(2\pi \boldsymbol{\iota} t),1)] = a_{2,3}^{}, \end{aligned} \end{align} $$

and, recalling that $n=km$ , for $1 \le j < n$ ,

(5.2) $$ \begin{align} \mathbf{b}_\sharp([\gamma_j])= \begin{cases} a_{1,2}^q a_{2,3}^{}a_{1,2}^{-q}&\text{if }j=qk\text{, so that }\mu^j\text{ is an }m\text{-th root of unity,} \\ 1 &\text{otherwise}. \end{cases} \end{align} $$

A few details of these calculations follow. It will be enough to take $\epsilon < \min \{\frac {1}{2},\sin (\pi /2mn)\}$ .

For the second equation in (5.1), we show that for every t the segment $S(t)$ between $\ell (t)^m$ and $(1-r\exp (2\pi \boldsymbol {\iota } t))$ is contained in $\mathbb {C}\smallsetminus \{0,1\}$ , so that the map $H\colon I\times I \to {\mathrm {Conf}}_3(\mathbb {C}), (s,t) \mapsto (0,h(s,t),1)$ with $h(s,t)= s(\ell (t))^m + (s-1)(1-r\exp (2\pi \boldsymbol {\iota } t))$ is a well-defined path homotopy. A straightforward check shows that if $t=0$ , then $S(t)\subseteq ]0,1[$ , and if $t=\frac {1}{2}$ , then $S(t)\subseteq ]1,\infty [$ . If $0<t<\frac {1}{2}$ , then $\Im (1-r\exp (2\pi \boldsymbol {\iota } t))<0$ and the condition on $\epsilon $ implies $ 0>\arg (\ell ( t))> -\frac {\pi }{m} $ so that $\Im (\ell (t)^m)<0$ as well, and thus, by convexity, $S(t)\subseteq \mathbb {R} + \boldsymbol {\iota } \mathbb {R}_{<0}\subseteq \mathbb {C}\smallsetminus \{0,1\}$ . The case $\frac {1}{2} < t < 1$ is analogous.

For (5.2), let $1 \le j \le n$ and consider two cases. First, if $j=kq$ for some integer q, let $\theta =mt$ and observe that the path $\mathbf {b}\circ f_j(t)$ , $0\le t \le j/n$ , is in fact the loop $(0,r \exp (2 \pi \boldsymbol {\iota } \theta ),1)$ , $0 \le \theta \le q$ , which represents $a_{1,2}^q$ . For such j, note also that $\mathbf {b}\circ (\mu ^j \ell (t))=\mathbf {b}\circ \ell (t)$ represents $a_{2,3}^{}$ . If, on the other hand, k does not divide j, using the condition on $\epsilon $ , one shows that the loop $(\mu ^j \ell (t))^m$ is contained in a “sector” U of amplitude $<\pi /n$ around the nontrivial n-th root of unity $\mu ^{jm}\neq 1$ . Since U is contractible and misses $0$ and $1$ , any path contained in it is nullhomotopic in $\mathbb {C}\smallsetminus \{0,1\}$ . This shows that $\mathbf {b}\circ \ell (t)$ is nullhomotopic in $ {\mathrm {Conf}}_3(\mathbb {C})$ .

Since the bundle $M(\mathscr {C}_{n,m}) \to M(\mathscr {B})$ is equivalent to the pullback along $\mathbf {b}$ of the bundle $ {\mathrm {Conf}}_4(\mathbb {C}) \to {\mathrm {Conf}}_3(\mathbb {C})$ , with fiber $\mathbb {C}\smallsetminus \{\text {three points}\}$ , the fundamental group $\pi _1(M(\mathscr {C}_{n,m})) \cong F_3 \rtimes _{\alpha } F_{n+1}$ is the semidirect product of free groups determined by the homomorphism $\phi =\hat \alpha _3 \circ \mathbf {b}_\sharp \colon F_{n+1} \to {\mathrm {Aut}}(F_3)$ , where $\hat \alpha _3\colon P_3 \to {\mathrm {Aut}}(F_3)$ is (the restriction of) the Artin representation. Denoting the generators of $F_{n+1}$ by $\mathsf {x}_i^{}=[\gamma _i]$ , $0\le i \le n$ , and those of $F_3$ by $\mathsf {y}_1^{},\mathsf {y}_2^{},\mathsf {y}_3^{}$ , the group $\pi _1(M(\mathscr {C}_{n,m}))$ has presentation

(5.3) $$ \begin{align} \pi_1(M(\mathscr{C}_{n,m})) = \langle \mathsf{x}_0^{}, \mathsf{x}_1^{}, \dots, \mathsf{x}_n^{}, \mathsf{y}_1^{},\mathsf{y}_2^{},\mathsf{y}_3^{} \mid \mathsf{x}_i^{-1}\mathsf{y}_j^{}\mathsf{x}_i^{} = \phi(\mathsf{x}_i^{})(\mathsf{y}_j^{}),\ 0\le i \le n,\ 1 \le j \le 3 \rangle. \end{align} $$

This may be made explicit using the Artin representation; see §2.4. For $q \in \mathbb {Z}$ , one checks that $a_{1,2}^q(\mathsf {y}_i^{})=(\mathsf {y}_1^{}\mathsf {y}_2^{})^q\mathsf {y}_i^{}(\mathsf {y}_1^{}\mathsf {y}_2^{})^{-q}=[(\mathsf {y}_1^{}\mathsf {y}_2^{})^q,\,\mathsf {y}_i^{}]\cdot \mathsf {y}_i^{}$ for $i=1,2$ , and $a_{1,2}^q(\mathsf {y}_3^{})=\mathsf {y}_3^{}$ . Using this, one can show that

$$\begin{align*}(a_{1,2}^qa_{2,3}^{}a_{1,2}^{-q})(\mathsf{y}_i)=\begin{cases} [\mathsf{w}_q^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q},\, \mathsf{y}_1^{}]\cdot \mathsf{y}_1^{}&\text{if }i=1,\\ [\mathsf{w}_q^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q}\mathsf{y}_2^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{q}\mathsf{y}_3^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q},\, \mathsf{y}_2^{}]\cdot \mathsf{y}_2^{}&\text{if }i=2, \\ [(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q}\mathsf{y}_2^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{q},\mathsf{y}_3^{}] \cdot \mathsf{y}_3^{}&\text{if }i=3,\\ \end{cases} \end{align*}$$

where

$$\begin{align*}\begin{aligned} \mathsf{w}_q&=(a_{2,3}^{}a_{1,2}^{-q})\big((\mathsf{y}_1^{}\mathsf{y}_2^{})^q\big)=a_{1,2}^{-q}\big(a_{2,3}^{}\big((\mathsf{y}_1^{}\mathsf{y}_2^{})^q\big)\big)\\ &= \big(\mathsf{y}_1^{}\mathsf{y}_2^{}\mathsf{y}_3^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q}\mathsf{y}_2^{}(\mathsf{y}_1^{}\mathsf{y}_2^{})^q\mathsf{y}_3^{-1}(\mathsf{y}_1^{}\mathsf{y}_2^{})^{-q}\mathsf{y}_2^{-1}(\mathsf{y}_1^{}\mathsf{y}_2^{})^q\big)^q. \end{aligned} \end{align*}$$

Rewriting (5.1) and (5.2) as

(5.4) $$ \begin{align} \phi(\mathsf{x}_0^{})=a_{1,2}^m,\quad\phi(\mathsf{x}_n^{})=a_{2,3}^{},\quad \phi(\mathsf{x}_j^{})=\begin{cases} a_{1,2}^qa_{2,3}^{}a_{1,2}^{-q}&\text{when }1\le j=qk \le n-1,\\ 1&\text{when }1\le j \neq qk \le n-1,\end{cases} \end{align} $$

the calculations above may be used to express $\mathsf {x}_i^{-1}\mathsf {y}_j^{}\mathsf {x}_i^{} = \phi (\mathsf {x}_i^{})(\mathsf {y}_j^{})$ in terms of the generators $\mathsf {x}_i^{}, \mathsf {y}_j^{}$ .

Example 5.1. Consider the case $m=3$ , $n=km=6$ . Loops in the base of the strictly supersolvable bundle $M(\mathscr {C}_{6,3})\to M(\mathscr {B})$ are depicted in Figure 5. The discussion above yields a presentation for the group $\pi _1(M(\mathscr {C}_{6,3}))$ with generators $\mathsf {x}_0^{},\mathsf {x}_1^{},\dots ,\mathsf {x}_6^{},\mathsf {y}_1^{},\mathsf {y}_2^{},\mathsf {y}_3^{}$ and relations $\mathsf {x}_i^{-1}\mathsf {y}_j^{}\mathsf {x}_i^{}=w_{i,j}^{} \mathsf {y}_j^{}w_{i,j}^{-1}$ . Note that $w_{i,j}^{} \mathsf {y}_j^{}w_{i,j}^{-1}=[w_{i,j},\,\mathsf {y}_j^{}]\cdot \mathsf {y}_j^{}$ . For $i=1,3,5$ , we have $w_{i,j}=1$ . For other values of i, writing $u^v=v^{-1}uv$ , we have

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l} w_{0,1}=(\mathsf{y}_1^{}\mathsf{y}_2^{})^3,\quad & w_{0,2}=(\mathsf{y}_1^{}\mathsf{y}_2^{})^3,\quad & w_{0,3}=1,\\ w_{2,1}=\mathsf{w}_1^{}(\mathsf{y}_1\mathsf{y}_2)^{-1},&w_{2,2}=\mathsf{y}_3^{(\mathsf{y}_1\mathsf{y}_2)^{-1}},&w_{2,3}=\mathsf{y}_2^{(\mathsf{y}_1\mathsf{y}_2)},\\ w_{4,1}=\mathsf{w}_2^{}(\mathsf{y}_1\mathsf{y}_2)^{-2},\quad & w_{4,2}=\mathsf{w}_2^{}\mathsf{y}_2^{(\mathsf{y}_1\mathsf{y}_2)^2}\mathsf{y}_3^{}(\mathsf{y}_1\mathsf{y}_2)^{-2},\quad & w_{4,3}=\mathsf{y}_2^{(\mathsf{y}_1\mathsf{y}_2)^2},\\ w_{6,1}=1,&w_{6,2}=\mathsf{y}_2^{}\mathsf{y}_3^{},&w_{6,3}=\mathsf{y}_2^{}\mathsf{y}_3^{}. \end{array} \end{align*}$$

5.2 Lower central series Lie algebra

Passing to (integral) homology, let $X_j=[\mathsf {x}_j]$ , $0 \le j \le n$ , and $A_{i,j}=[a_{i,j}]$ , $1\le i<j\le 3$ , denote the generators of $H_1(M(\mathscr {B}))\cong \mathbb {Z}^{n+1}$ and $H_1( {\mathrm {Conf}}_3(\mathbb {C}))\cong \mathbb {Z}^{3}$ , respectively. From (5.4), the homological root homomorphism $\mathbf {b}_*\colon H_1(M(\mathscr {B})) \to H_1( {\mathrm {Conf}}_3(\mathbb {C}))$ is then given by

$$\begin{align*}\mathbf{b}_*(X_0)=m A_{1,2}, \quad \mathbf{b}_*(X_j)=\begin{cases}A_{2,3}&\text{if }j=qk\text{, so that }\xi^j\text{ is an }m\text{-th root of unity,} \\ 0 &\text{otherwise}. \end{cases} \end{align*}$$

By Theorem 4.2, the integral lower central series (LCS) Lie algebra $\mathfrak {h}(\mathscr {C}_{n,m})$ of $\pi _1(M(\mathscr {C}_{n,m})$ is the semidirect product of the free Lie algebra $\mathbb {L}[n+1]$ (generated by $X_j$ , $0\le j\le n$ ) by the free Lie algebra $\mathbb {L}[3]$ (generated by $Y_i=A_{i,4}$ , $1\le i \le 3$ ) determined by the Lie homomorphism $\Theta =\theta _3 \circ \mathbf {b}_* \colon \mathbb {L}[n+1] \to {\mathrm {Der}}(\mathbb {L}[3])$ , where $\theta _3(A_{i,j})= {\mathrm {ad}}(A_{i,j})$ . Using the above description of the map $\mathbf {b}_*$ , we have ${[}X_0,Y_i]=m {\mathrm {ad}}(A_{1,2})(Y_i)$ , ${[}X_j,Y_i]= {\mathrm {ad}}(A_{2,3})(Y_i)$ if $j=qk$ , and ${[}X_j,Y_i]=0$ if $j\neq qk$ , yielding

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l} [X_0,Y_1]=m[Y_1,Y_2],\quad & [X_0,Y_2]=m[Y_2,Y_1],\quad & [X_0,Y_3]=0,\\ {[}X_j,Y_1]=0,\quad & [X_j,Y_2]=[Y_2,Y_3],\quad & [X_j,Y_3]=[Y_3,Y_2], \quad \text{if }j=kq. \end{array} \end{align*}$$

Consequently, the LCS Lie algebra $\mathfrak {h}$ may be realized as the quotient of the free Lie algebra $\mathbb {L}[n+4]$ (generated by $X_j$ , $0\le j\le n$ , and $Y_1,Y_2,Y_3$ ) by the Lie ideal $\mathcal J$ generated by

$$\begin{align*}[X_0,Y_1]-m[Y_1,Y_2],\ [X_0,Y_2]+m[Y_1,Y_2],\ [X_0,Y_3],\ [X_j,Y_1],\ [X_j,Y_2] - c_j[Y_2,Y_3],\ [X_j,Y_3] + c_j[Y_2,Y_3], \end{align*}$$

where $1\le j \le n$ , $c_j=1$ if $j=kq$ , and $c_j=0$ if $j \neq kq$ .

Remark 5.2. The space $M(\mathscr {C}_{n,m})$ is a $K(G,1)$ -space for the almost-direct product of free groups $G=\pi _1(M(\mathscr {C}_{n,m}))=F_3 \rtimes _{\phi } F_{n+1}$ . The relations in the presentation (5.3), resp., the generators of the Lie ideal $\mathcal J$ above, are in correspondence with a basis $\{r_{i,j}, 0 \le i \le n, 1\le j\le 3\}$ for $H_2(G)=H_2(M(\mathscr {C}_{n,m})) \cong \mathbb {Z}^{3n+3}$ . Furthermore, the injective map $\mathfrak {a}_* \colon H_2(G) \to H_2(\mathbb {Z}^{n+4})$ induced by the abelianization $\mathfrak {a} \colon G \to \mathbb {Z}^{n+4}$ may be determined from the generators of $\mathcal J$ as in the proof of Theorem 4.4.

Fixing generators $e_j$ , $0\le j \le n$ , and $f_1,f_1,f_3$ for $H_1(\mathbb {Z}^{n+4})$ , the group $H_2(\mathbb {Z}^{n+4})$ may be identified with the second graded piece of the exterior algebra $\bigwedge H_1(\mathbb {Z}^{n+4})$ , generated by $e_ie_j, e_if_k, f_kf_l$ ( $i<j$ , $k<l$ ). The map $\mathfrak {a}_* \colon H_2(G) \to H_2(\mathbb {Z}^{n+4})$ is then given by

(5.5) $$ \begin{align} \begin{array}{l@{\quad}l@{\quad}l} \mathfrak{a}_*(r_{0,1})=e_0f_1-mf_1f_2,\quad &\mathfrak{a}_*(r_{0,2})=e_0f_2+mf_1f_2,\quad &\mathfrak{a}_*(r_{0,3})=e_0f_3,\\ \mathfrak{a}_*(r_{j,1)}=e_jf_1,\quad &\mathfrak{a}_*(r_{j,2})=e_jf_2-c_jf_2f_3,\quad &\mathfrak{a}_*(r_{j,3})=e_jf_3+c_jf_2f_3, \end{array} \end{align} $$

where, as above, $1\le j \le n$ , $c_j=1$ if $j=kq$ , and $c_j=0$ if $j \neq kq$ .

Example 5.3. We continue with the case $m=3$ , $n=6$ . The LCS Lie algebra of $G=\pi _1(M(\mathscr {C}_{6,3}))$ is the quotient of the free Lie algebra $\mathbb {L}[10]$ , generated by $X_0,X_1,\dots ,X_6,Y_1,Y_2,Y_3$ , by the Lie ideal ${\mathcal J}$ generated by

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} [X_0,Y_1]-3[Y_1,Y_2],\quad &[X_0,Y_2]+3[Y_1,Y_2],\quad &[X_0,Y_3],\\ {[}X_j,Y_1],\quad &[X_j,Y_2],\quad &[X_j,Y_3],\quad &\text{for }j=1,3,5,\\ {[}X_j,Y_1],\quad &[X_j,Y_2]-[Y_2,Y_3],\quad &[X_j,Y_3]+[Y_2,Y_3],\quad &\text{for }j=2,4,6.\\ \end{array} \end{align*}$$

The map $\mathfrak {a}_2 \colon H_2(G) \to H_2(\mathbb {Z}^{10})$ is given by

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \mathfrak{a}(r_{0,1})=e_0f_1-3f_1f_2,\quad &\mathfrak{a}(r_{0,2})=e_0f_2+3f_1f_2,\quad &\mathfrak{a}(r_{0,3})=e_0f_3,\\ \mathfrak{a}(r_{j,1})=e_jf_1,\quad &\mathfrak{a}(r_{j,2})=e_jf_2,\quad &\mathfrak{a}(r_{j,3})=e_jf_3,\quad &\text{for }j=1,3,5,\\ \mathfrak{a}(r_{j,1})=e_jf_1,\quad &\mathfrak{a}(r_{j,2})=e_jf_2-f_2f_3,\quad &\mathfrak{a}(r_{j,3})=e_jf_3+f_2f_3,\quad &\text{for }j=2,4,6.\\ \end{array} \end{align*}$$

5.3 Cohomology ring

As discussed in the proof of Theorem 4.4, the integral cohomology ring of the $K(G,1)$ -space $M(\mathscr {C}_{n,m})$ is isomorphic to $\mathsf {E}/\mathsf {I}$ , where $\mathsf {E}=\bigwedge H^1(\mathbb {Z}^{n+4})$ and $\mathsf {I}=\langle \ker (\mathfrak {a}^*\colon H^2(\mathbb {Z}^{n+4}) \to H^2(G)\rangle $ . Denote the generators of $H^1(\mathbb {Z}^{n+4})= {\mathrm {Hom}}(H^1(\mathbb {Z}^{n+4}),\mathbb {Z})$ by the same symbols as those of $H_1(\mathbb {Z}^{n+4})$ , so that $\mathsf {E}$ is the exterior algebra on $e_0,e_1,\dots ,e_n,f_1,f_2,f_3$ . Calculating with (5.5) and recalling that $n=km$ , we obtain $H^*(M(\mathscr {C}_{n,m})) \cong \mathsf {E}/\mathsf {I}$ , where

$$\begin{align*}\mathsf{I} = \ker(\mathfrak{a}_2^*) = \left\langle e_ie_j, 0\le i<j\le n,\ f_1f_2+me_0(f_1-f_2),\ f_1f_3,\ f_2f_3+\sum_{q=1}^m e_{kq}(f_2-f_3)\right\rangle. \end{align*}$$

We compare our presentation of the cohomology ring for toric arrangements with others in the literature. One presentation, from [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6, Theorem 7.4] and [Reference Bazzocchi, Pagaria and Pismataro2, Theorem 5.9], is for the integral (hence also rational) cohomology, and the other, from [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6, Theorem 6.13], is a different presentation for the rational cohomology. Below, we describe each of these presentations in the case of $M(\mathscr {C}_{n,m})$ , after some simplification by removing redundant generators, and provide explicit isomorphisms to our presentation.

First, consider the exterior $\mathbb {Z}$ -algebra $\mathsf {E}_1$ on generators $z_1$ , $z_2$ , $w_1$ , $w_2$ , and $w_{0,j}$ for $j=1,\dots ,n$ , and the ideal

$$\begin{align*}\mathsf{I}_1=\left\langle \begin{array}{l@{\quad}l@{\quad}l} (-mz_1+z_2)w_1, & z_2w_2, \qquad & z_1w_{0,j} \text{ for } 0\leq j\leq n, \\ w_{0,i}w_{0,j} \text{ for } 1\leq i<j\leq n, \ & & w_1w_2+mz_1w_2+\sum_{q=1}^m w_{0,qk}(w_1-w_2) \end{array} \right\rangle.\end{align*}$$

As in [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6, Theorem 7.4], we view the generators of $\mathsf {E}_1$ as represented by the differential forms below, with $\mathsf {I}_1$ capturing precisely the relations satisfied by these forms, so that the quotient $\mathsf {E}_1/\mathsf {I}_1$ is isomorphic to $H^*(M(\mathscr {C}_{n,m}))$ . With $\boldsymbol {\iota }=\sqrt {-1}$ , write

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l} z_1 = \frac{1}{2\pi\boldsymbol{\iota}}\mathrm{dlog}(x), \quad & z_2 = \frac{1}{2\pi\boldsymbol{\iota}}\mathrm{dlog}(y), \\ w_1 = \frac{1}{2\pi\boldsymbol{\iota}}\mathrm{dlog}(1-x^{-m}y), \quad & w_2 = \frac{1}{2\pi\boldsymbol{\iota}}\mathrm{dlog}(1-y), \quad & w_{0,j} = \frac{1}{2\pi\boldsymbol{\iota}}\mathrm{dlog}(1-\zeta^j x) \text{ for } 1\leq j\leq n. \end{array}\end{align*}$$

Comparing this presentation to the one we derived above, there is an explicit isomorphism from $\mathsf {E}_1/\mathsf {I}_1$ to $\mathsf {E}/\mathsf {I}$ given by

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} z_1 \mapsto e_0, \qquad & z_2 \mapsto f_1, \qquad \\ w_1 \mapsto -me_0+f_2, \qquad & w_2 \mapsto f_3, \qquad & w_{0,j} \mapsto e_j & \text{ for } 1\leq j \leq n.\\ \end{array}\end{align*}$$

Alternatively, using rational coefficients, consider the exterior $\mathbb {Q}$ -algebra $\mathsf {E}_2$ on generators $\psi _0$ , $\psi _1$ , $\psi _2$ , $\overline {\omega }_1$ , $\overline {\omega }_2$ , and $\overline {\omega }_{0,j}$ for $1\leq j\leq n$ , and the ideal

$$\begin{align*}\mathsf{I}_2=\left\langle \begin{array}{l@{\quad}l@{\quad}l} \psi_0-k\psi_1+k\psi_2, & \psi_1\overline{\omega}_1, & \psi_2\overline{\omega}_2, \\ \psi_0\overline{\omega}_{0,j} \text{ for } 1\leq j \leq n, \ & \overline{\omega}_{0,i}\overline{\omega}_{0,j} \text{ for } 1\leq i<j\leq n, \ & \overline{\omega}_1\overline{\omega}_2-\psi_1\psi_2+\sum_{q=1}^m\overline{\omega}_{0,qk}(\overline{\omega}_1-\overline{\omega}_2) \end{array} \right\rangle.\end{align*}$$

By [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6, Theorem 6.13], the quotient $\mathsf {E}_2/\mathsf {I}_2$ is isomorphic to $H^*(M(\mathscr {C}_{n,m});\mathbb {Q})$ . The generators are represented by differential forms distinct from those above (see [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6, Definition 2.20]). Comparing with our presentation, there is an explicit isomorphism from $\mathsf {E}_2/\mathsf {I}_2$ to the rationalization of $\mathsf {E}_1/\mathsf {I}_1$ (and hence also to that of $\mathsf {E}/\mathsf {I}$ ) given by

$$\begin{align*}\begin{array}{l@{\quad}l@{\quad}l} \psi_0\mapsto nz_1, \qquad & \psi_1\mapsto -mz_1+z_2,\qquad & \psi_2\mapsto z_2, \\ \overline{\omega}_1\mapsto 2w_1+mz_1-z_2, \quad & \overline{\omega}_2 \mapsto 2w_2-z_2, \quad & \overline{\omega}_{0,j} \mapsto 2w_{0,j}-z_1 \text{ for } 1\leq j\leq n. \end{array}\end{align*}$$

6.1 Homology generators

Assume without loss of generality that each hypersurface $H\in \mathscr {B}$ is connected and is given, as in (2.1), by the equation $x_1^{m_{H,1}}\cdots x_d^{m_{H,d}}=\mu _H$ , where $\mu _H$ is a root of unity and $\mathbf {m}_H=(m_{H,1},\dots ,m_{H,d})\in \mathbb {Z}^d$ . We abbreviate this defining equation as $\mathbf {x}^{\mathbf {m_H}}=\mu _H$ , and write $f_H(\mathbf {x})=\mathbf {x}^{\mathbf {m_H}}-\mu _H$ . The first integral (de Rham) cohomology group $H^1(M(\mathscr {B}))$ of the complement of $\mathscr {B}$ is free abelian of rank $N=d+|\mathscr {B}|$ . A basis is given by the set of differential forms

(6.2) $$ \begin{align} \{\zeta_i \mid 1\le i \le d\}\cup\{\omega_H \mid H\in \mathscr{B}\}, \end{align} $$

where $\zeta _i=\frac {1}{2\pi \boldsymbol {\iota }}\mathrm {dlog}\, x_i$ and $\omega _H=\frac {1}{2\pi \boldsymbol {\iota }}\mathrm {dlog}\, f_H=\frac {1}{2\pi \boldsymbol {\iota }}\mathrm {dlog}(\mathbf {x}^{\mathbf {m}_H}-\mu _H)$ ; see [Reference Callegaro, D’Adderio, Delucchi, Migliorini and Pagaria6] and the references therein (and recall that $\boldsymbol {\iota }=\sqrt {-1}$ ). We exhibit a dual basis for the first integral homology group $H_1(M(\mathscr {B}))$ .

The complement $M(\mathscr {B})$ may be realized as $\mathbb {C}^d\smallsetminus V$ , where V is the variety in $\mathbb {C}^d$ determined by the toric arrangement $\mathscr {B}$ , together with the coordinate hyperplanes. Let $L=\{\mathbf {x}_0+z\cdot \mathbf {v}\mid z\in \mathbb {C}\}$ , where $\mathbf {x}_0\in M(\mathscr {B})$ is the basepoint noted above and $\mathbf {v}\in \mathbb {C}^d$ , be a complex line in $\mathbb {C}^d$ that is transverse to V. The intersection $L\cap V$ is a collection of points: the d points $\mathbf {q}_i$ where L meets the coordinate hyperplane $\{x_i=0\}$ , and, for each $H\in \mathscr {B}$ , the points $\mathbf {q}_{H,1},\dots \mathbf {q}_{H,r_H}$ where L meets H.

For $\epsilon>0$ sufficiently small, each of the disks $D_\epsilon (\mathbf {q}_i)$ , resp., $D_\epsilon (\mathbf {q}_{H,j})$ , in L of radius $\epsilon $ centered at the aforementioned points meets the variety V at only $\mathbf {q}_i$ , resp., $\mathbf {q}_{H,j}$ . Let $\xi _i$ and $\xi _{H,j}$ denote the (oriented) boundary circles of the disks $D_\epsilon (\mathbf {q}_i)$ and $D_\epsilon (\mathbf {q}_{H,j})$ ,

(6.3) $$ \begin{align} \xi_{i}(t) = \mathbf{q}_i+\epsilon \exp(2 \pi \boldsymbol{\iota} t)\cdot\mathbf{v},\quad \xi_{H,j}(t) = \mathbf{q}_{H,j}+\epsilon \exp(2 \pi \boldsymbol{\iota} t)\cdot\mathbf{v},\quad 0\le t \le 1. \end{align} $$

Viewing these circles in $M(\mathscr {B})$ via the inclusion $L\cap M(\mathscr {B}) \hookrightarrow M(\mathscr {B})$ , denote their homology classes in $H_1(M(\mathscr {B}))$ by $X_i=[\xi _i]$ and $X_{H,j}=[\xi _{H,j}]$ . As we will see below, for $H\in \mathscr {B}$ , the classes $X_{H,j}$ and $X_{H,k}$ coincide for $1\le j,k\le r_H$ . So write $X_H=X_{H,1}$ for $H\in \mathscr {B}$ .

Proposition 6.1. The first homology group $H_1(M(\mathscr {B})) \cong \mathbb {Z}^{d+|\mathscr {B}|}$ has basis

(6.4) $$ \begin{align} \{X_i \mid 1\le i \le d\} \cup \{X_H \mid H \in \mathscr{B}\}, \end{align} $$

dual to the basis (6.2) of $H^1(M(\mathscr {B}))$ .

Proof. We use the (perfect) Kronecker pairing $H^1(M(\mathscr {B}))\otimes H_1(M(\mathscr {B})) \to \mathbb {Z}$ , given, in terms of (the cohomology class of) a differential form $\omega $ and a homology class $X=[\xi ]$ , by integration, $\langle \omega ,X\rangle =\int _\xi \omega $ .

Restricting the functions $f_H$ giving the hypersurfaces in $\mathscr {B}$ to the line $L=\{\mathbf {x}_0+z\cdot \mathbf {v}\}$ yields functions of one complex variable $\phi _{H}(z)=f_H(\mathbf {x}_0+z\cdot \mathbf {v})$ and $\psi _{H}(z)=\phi ^{\prime }_{H}(z)/\phi _{H}(z)$ . Similarly, restricting the coordinate functions $x_i$ yields $\phi _i(z)=x_i(\mathbf {x}_0+z\cdot \mathbf {v})$ and $\psi _i(z)=\phi ^{\prime }_i(z)/\phi _i(z)$ . Express the points where L meets $\{x_i=0\}$ as $\mathbf {q}_i=\mathbf {x}_0+z_i\cdot \mathbf {v}$ , and the points where L meets $H\in \mathscr {B}$ as $\mathbf {q}_{H,j}=\mathbf {x}_0+z_{H,j}\cdot \mathbf {v}$ , $1\le j\le r_H$ . Note that $\phi _i(z_i)=0$ , that $\phi _H(z_{H,j})=0$ for each j, $1\le j\le r_H$ , and that these are the only zeros of these functions. Writing $\mathbf {x}_0=(x_{0,1},\dots ,x_{0,d})$ and $\mathbf {v}=(v_{1},\dots ,v_{d})$ , we see immediately that the function $\psi _i(z)=v_i/(x_{0,i}+v_iz)$ has a simple pole at $z_i=-x_{0,i}/v_i$ , with residue $1$ . Since $\lim _{z\to z_{H,j}} (z-z_{H,j})\cdot \psi _{H}(z)=1$ by L’Hospital’s rule, the function $\psi _{H}(z)$ has a simple pole at $z_{H,j}$ , with residue $1$ , for each j, $1\le j\le r_H$ .

With these observations, one can use the residue theorem to show that the set (6.4) is a basis for $H_1(M(\mathscr {B}))$ , dual to the basis (6.2) for $H^1(M(\mathscr {B}))$ . First, consider the Kronecker pairing $\langle \omega _K,X_{H,j}\rangle $ for $H,K\in \mathscr {B}$ . We have

$$\begin{align*}\langle \omega_K,X_{H,j}\rangle=\int_{\xi_{H,j}} \omega_K=\int_{\xi_{H,j}}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog} f_K(\mathbf{x})={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\int_{\gamma_{H,j}} \psi_K(z), \end{align*}$$

where $\gamma _{H,j}$ is the circle of radius $\epsilon $ centered at $z_{H,j}$ , oriented counterclockwise. If $K\neq H$ , then, by our choice of $\epsilon $ , none of the poles $z_{K,l}$ of $\psi _K(z)$ lie within this circle, and $\langle \omega _K,X_{H,j}\rangle =\frac {1}{2\pi \boldsymbol {\iota }}\int _{\gamma _{H,j}} \psi _K(z)=0$ . Similarly, we have $\langle \zeta _i,X_{H,j}\rangle =\frac {1}{2\pi \boldsymbol {\iota }}\int _{\gamma _{H,j}} \psi _i(z)=0$ , since the pole $z_i$ of $\psi _i(z)$ does not lie within this circle either.

If $K=H$ , then

$$\begin{align*}\langle \omega_H,X_{H,j}\rangle={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\int_{\gamma_{H,j}} \psi_H(z)=1 \end{align*}$$

by the residue theorem, since $z_{H,j}$ is the only pole of $\psi _H(z)$ within the circle $\gamma _{H,j}$ . Thus, for ${1\le j,k\le r_H}$ , the homology classes $X_H=X_{H,j}$ and $X_{H,k}$ are two expressions of the dual of the cohomology class $\omega _H$ , hence are equal. In particular, $X_H=X_{H,1}$ is the dual of $\omega _H$ .

Similar considerations yield

$$\begin{align*}\langle \zeta_i,X_{j}\rangle={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\int_{\gamma_{j}} \psi_i(z)=\delta_{i,j} \quad\text{and}\quad \langle \zeta_i,X_H\rangle=\langle \zeta_i,X_{H,1}\rangle={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\int_{\gamma_{H,1}} \psi_i(z)=0. \end{align*}$$

If $i=j$ , a quick calculation yields $\langle \zeta _i,X_{j}\rangle =1$ . The remaining cases follow from the fact that the pole $z_i$ of $\psi _i(z)$ does not lie within either the circle $\gamma _{H,1}$ or, for $i\neq j$ , the circle $\gamma _j$ of radius $\epsilon $ centered at $z_j$ . Consequently, the homology class $X_i$ is the dual of the cohomology class $\zeta _i$ .

6.2 The homological root homomorphism

We now determine the map

$$\begin{align*}\mathbf{b}_*\colon H_1(M(\mathscr{B}))\to H_1({\mathrm{Conf}}_n(\mathbb{C})) \end{align*}$$

in homology induced by the root map (6.1). Recall that this map is given explicitly by $\mathbf {b}(\mathbf {x})=(b_1(\mathbf {x}),b_2(\mathbf {x}),\dots ,b_n(\mathbf {x}))$ , with $b_1(\mathbf {x})=0$ and $b_{j+1}(\mathbf {x})=\mu _{j}\mathbf {x}^{\mathbf {m}_{j}}=\mu _jx_1^{m_{j,1}}\cdots x_d^{m_{j,d}}$ for $1\le j\le l$ , where $l=|\mathscr {A}|-|\mathscr {B}|$ and $n=l+1$ . Our goal is to find an explicit description of $\mathbf {b}_\ast $ in terms of the generators of $H_1(M(\mathscr {B}))$ and $H_1( {\mathrm {Conf}}_n(\mathbb {C}))$ given in Proposition 6.1 and Remark 6.2.

Remark 6.3. Let $\pi \colon (\mathbb {C}^\times )^{d+1} \to (\mathbb {C}^\times )^d$ , $\pi (\mathbf {x},y)=\mathbf {x}$ , be the projection map that forgets the last coordinate. For $H_i,H_j \in \mathscr {A}\smallsetminus \mathscr {A}_Y$ and any connected component L of $H_i\cap H_j$ , Lemma 2.7 implies that $\pi (L)$ is a layer of $\mathscr {B}$ of dimension equal to the dimension of L. Therefore, $\pi (L)$ is a hypersurface of $\mathscr {B}$ .

Explicitly, if $H_i$ and $H_j$ are given by $y=\mu _i\mathbf {x}^{\mathbf {m}_i}$ and $y=\mu _j\mathbf {x}^{\mathbf {m}_j}$ , let $\mathbf {a}=(\mathbf {m}_j-\mathbf {m}_i)/m(i,j)$ , where $m(i,j)=\gcd (m_{j,1}-m_{i,1},\dots ,m_{j,d}-m_{i,d})$ . The intersection $H_i\cap H_j$ is then given by the equation $(\mathbf {x}^{\mathbf {a}})^{m(i,j)}-\mu _i\mu _j^{-1}=0$ , with connected components given by $\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q=0$ , where $\lambda =|\mu _i\mu _j^{-1}|^{1/m(i,j)}$ , $\eta =\exp (2\pi \boldsymbol {\iota }/m(i,j))$ , and $1\le q \le m(i,j)$ . Thus, $\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q=0$ is an equation of a hypersurface H of $\mathscr {B}$ . Since H is given by the vanishing of $f_H(\mathbf {x})=\mathbf {x}^{\mathbf {m}_H}-\mu _H$ , we have either $f_H(\mathbf {x})=\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q$ (when $\mathbf {m}_H=\mathbf {a}$ ) or $f_H(\mathbf {x})=\mathbf {x}^{\mathbf {-a}}-\lambda ^{-1}\eta ^{-q}$ (when $\mathbf {m}_H=-\mathbf {a}$ ). If $s(i,j)=\big |\{H\in \mathscr {B} \mid \mathbf {m}_H=\mathbf {a}\}\big |$ , then $m(i,j)-s(i,j)=\big |\{H\in \mathscr {B} \mid \mathbf {m}_H=-\mathbf {a}\}\big |$ , and

$$\begin{align*}{\boldsymbol{\nu}}(i,j):= \frac{1}{m(i,j)}\big[s(i,j)\mathbf{m}_i+(m(i,j)-s(i,j))\mathbf{m}_j\big]=\mathbf{m}_i+(m(i,j)-s(i,j))\mathbf{a} =\mathbf{m}_j-s(i,j)\mathbf{a}\in \mathbb{Z}^d \end{align*}$$

is an integer vector. Write ${\boldsymbol {\nu }}(i,j)=\big (\nu _1(i,j),\dots ,\nu _d(i,j)\big )$ .

Theorem 6.4. Let $\mathscr {A}$ be an essential toric arrangement in $(\mathbb {C}^\times )^{d+1}$ , with $\mathscr {P}(\mathscr {A}_Y)$ a corank-one TM-ideal of $\mathscr {P}(\mathscr {A})$ , $\mathscr {A}\smallsetminus \mathscr {A}_Y=\{H_1,\dots ,H_l\}$ , and $\mathscr {B}=\mathscr {A}/Y$ . On the basis $\{X_k \mid 1\le k \le d\} \cup \{X_H \mid H \in \mathscr {B}\}$ of $H_1(M(\mathscr {B}))$ , the homological root homomorphism $\mathbf {b}_*\colon H_1(M(\mathscr {B}))\to H_1( {\mathrm {Conf}}_n(\mathbb {C}))$ , where $n=l+1$ , is given by

$$\begin{align*}\mathbf{b}_*(X_{k})=\sum_{j=1}^{n-1}\Big[m_{j,k}A_{1,j+1} + \sum_{i=1}^{j-1} \nu_k(i,j) A_{i+1,j+1}\Big] \quad \text{and} \quad \mathbf{b}_*(X_{H})=\sum A_{i+1,j+1}, \end{align*}$$

the latter sum running over all $i<j$ for which H is a connected component of $\pi (H_i \cap H_j)$ .

Proof. We again use the Kronecker pairing. For $X\in H_1(M(\mathscr {B}))$ , the coefficient of $A_{r,s}$ in $\mathbf {b}_*(X)$ is equal to $\langle \omega _{r,s},\mathbf {b}_*(X)\rangle =\langle \mathbf {b}^*(\omega _{r,s}),X\rangle $ .

We determine the map $\mathbf {b}^*\colon H^1( {\mathrm {Conf}}_n(\mathbb {C}))\to H^1(M(\mathscr {B}))$ in cohomology. If $r=1$ and $s=j+1$ , then

(6.5) $$ \begin{align} \mathbf{b}^*(\omega_{1,j+1})={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(b_{j+1}(\mathbf{x})-b_1(\mathbf{x}))= {\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mu_j\mathbf{x}^{\mathbf{m}_j})= \sum_{k=1}^d m_{j,k}\zeta_k. \end{align} $$

If $r=i+1$ and $s=j+1$ for $1\le i<j\le l$ , then $b_{j+1}(\mathbf {x})-b_{i+1}(\mathbf {x})= \mu _j\mathbf {x}^{\mathbf {m}_j}-\mu _i\mathbf {x}^{\mathbf {m}_i}$ . In the notation established in Remark 6.3, we have

$$\begin{align*}\mu_j\mathbf{x}^{\mathbf{m}_j}-\mu_i\mathbf{x}^{\mathbf{m}_i} =\mu_j \mathbf{x}^{\mathbf{m}_i} (\mathbf{x}^{\mathbf{m}_j-\mathbf{m}_i} - \mu_i\mu_j^{-1})= \mu_j \mathbf{x}^{\mathbf{m}_i} ((\mathbf{x}^{\mathbf{a}})^{m(i,j)} - \mu_i\mu_j^{-1})= \mu_j \mathbf{x}^{\mathbf{m}_i} \prod_{q=1}^{m(i,j)} (\mathbf{x}^{\mathbf{a}}-\lambda\eta^q), \end{align*}$$

and the vanishing of each factor $\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q$ gives a hypersurface H of $\mathscr {B}$ , also defined by the vanishing of $f_H(\mathbf {x})=\mathbf {x}^{\mathbf {m}_H}-\mu _H$ . In the $s(i,j)$ instances where $\mathbf {m}_H=\mathbf {a}$ , we have $\mathrm {dlog}(\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q)=\mathrm {dlog}\,f_H$ , while in the $m(i,j)-s(i,j)$ instances where $\mathbf {m}_H=-\mathbf {a}$ , we have $\mathrm {dlog}(\mathbf {x}^{\mathbf {a}}-\lambda \eta ^q)=\mathrm {dlog}\,f_H+\mathrm {dlog}(\mathbf {x}^{\mathbf {a}})$ . These observations yield

$$\begin{align*}\begin{aligned} \mathbf{b}^*(\omega_{i+1,j+1})&={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(b_{j+1}(\mathbf{x})-b_{i+1}(\mathbf{x}))={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mu_j\mathbf{x}^{\mathbf{m}_j}-\mu_i\mathbf{x}^{\mathbf{m}_i})\\ &={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\mathbf{m}_i}) +\sum_{q=1}^{m(i,j)}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\mathbf{a}}-\lambda\eta^q)\\ &={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\mathbf{m}_i}) +\sum_{\mathbf{m}_H=\mathbf{a}}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}\,f_H +\sum_{\mathbf{m}_H=-\mathbf{a}}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}(\mathrm{dlog}\,f_H+\mathrm{dlog}(\mathbf{x}^{\mathbf{a}}))\\ &={\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\mathbf{m}_i}) +{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}(m(i,j)-s(i,j)){\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\mathbf{a}})+ \sum_{\genfrac{}{}{0pt}{}{H\in\mathscr{B}}{H\subset \pi(H_i\cap H_j)}}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}\,f_H. \end{aligned} \end{align*}$$

Since ${\boldsymbol {\nu }(i,j)}=\mathbf {m}_i+(m(i,j)-s(i,j))\mathbf {a}$ , we have $\mathrm {dlog} ( \mathbf {x}^{\boldsymbol {\nu }(i,j)})= \mathrm {dlog}(\mathbf {x}^{\mathbf {m}_i})+ (m(i,j)-s(i,j))\mathrm {dlog}(\mathbf {x}^{\mathbf {a}})$ . Consequently,

(6.6) $$ \begin{align} \mathbf{b}^*(\omega_{i+1,j+1})= {\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}(\mathbf{x}^{\boldsymbol{\nu}(i,j)})+ \sum_{\genfrac{}{}{0pt}{}{H\in\mathscr{B}}{H\subset \pi(H_i\cap H_j)}}{\textstyle\frac{1}{2\pi\boldsymbol{\iota}}}\mathrm{dlog}\,f_H =\sum_{k=1}^d \nu_k(i,j)\zeta_k + \sum_{\genfrac{}{}{0pt}{}{H\in\mathscr{B}}{H\subset \pi(H_i\cap H_j)}} \omega_H. \end{align} $$

It is now a straightforward exercise using Proposition 6.1, (6.5), and (6.6) to determine the homological root homomorphism $\mathbf {b}_*\colon H_1(M(\mathscr {B}))\to H_1( {\mathrm {Conf}}_n(\mathbb {C}))$ . For $X\in H_1(M(\mathscr {B}))$ , write $\mathbf {b}_*(X)=\sum _{1\le r<s\le n} c_{r,s}A_{r,s}$ , where $c_{r,s}=\langle \omega _{r,s},\mathbf {b}_*(X)\rangle =\langle \mathbf {b}^*(\omega _{r,s}),X\rangle $ .

First, consider the generator $X=X_H$ corresponding to $H\in \mathscr {B}$ . If $r=1$ and $s=j+1$ , then $c_{1,j+1}=\sum _{k=1}^d m_{j,k} \langle \zeta _k,X_H\rangle =0$ , using (6.5). If $r=i+1$ and $s=j+1$ , using (6.6), we have

$$\begin{align*}c_{i+1,j+1}= \sum_{k=1}^d m_{i,k}\langle\zeta_k,X_H\rangle + \sum_{\genfrac{}{}{0pt}{}{K\in\mathscr{B}}{K\subset \pi(H_i\cap H_j)}} \langle\omega_K,X_H \rangle= \sum_{\genfrac{}{}{0pt}{}{K\in\mathscr{B}}{K\subset \pi(H_i\cap H_j)}} \langle\omega_K,X_H\rangle =\delta_{K,H}. \end{align*}$$

Thus, $\mathbf {b}_*(X_H)=\sum A_{i+1,j+1}$ , where the sum is over $i<j$ for which H is a connected component of $\pi (H_i \cap H_j)$ , as asserted.

Next, consider the generator $X=X_k$ corresponding to the coordinate hyperplane $\{x_k=0\}$ . If $r=1$ and $s=j+1$ , then $c_{1,j+1}=\sum _{q=1}^d m_{j,q} \langle \zeta _q,X_k\rangle =m_{j,k}$ , using (6.5). If $r=i+1$ and $s=j+1$ , using (6.6), we have

$$\begin{align*}c_{i+1,j+1}= \sum_{q=1}^d \nu_q(i,j)\langle\zeta_q,X_k\rangle + \sum_{\genfrac{}{}{0pt}{}{H\in\mathscr{B}}{H\subset \pi(H_i\cap H_j)}} \langle\omega_H ,X_k \rangle=\nu_k(i,j). \end{align*}$$

This completes the proof.

Example 6.5. Let $\mathscr {C}$ be the rank-two circuit in $(\mathbb {C}^\times )^2$ considered in Section 5, defined by $x-\mu ^j=0$ , $1\le j \le n$ , $x^{m_1}y=1$ , and $x^{m_2}y=1$ , where $m=m_2-m_1>0$ , $n=km$ , and $\mu =\exp (2\pi \boldsymbol {\iota } /n)$ . With $H_1$ given by $y=x^{-m_1}$ and $H_2$ by $y=x^{-m_2}$ , the point $x=\mu ^j$ is a component of $\pi (H_1\cap H_2)$ when $j=qk$ , so that $\mu ^j$ is an m-th root of unity. As in §5.2, denote the generators of the first homology of $M(\mathscr {B})=\mathbb {C}\smallsetminus \{0,1,\mu ,\dots ,\mu ^{n-1}\}$ by $X_j$ , $0\le j \le n$ , where $X_0$ is the class of a loop about $0$ , and $X_j$ is the class of a loop about $\mu ^j$ for $j \ge 1$ . By Theorem 6.4, the homological root homomorphism associated to the root map $\mathbf {b}(x)=(0,x^{-m_1},x^{-m_2})$ is given by

$$\begin{align*}\mathbf{b}_*(X_0)=-m_1 A_{1,2}-m_2A_{1,3}-m_2A_{2,3}, \qquad \mathbf{b}_*(X_j)=\begin{cases}A_{2,3}&\text{if }j=qk, \\ 0 &\text{otherwise}. \end{cases} \end{align*}$$

Noting that

$$\begin{align*}\mathbf{b}_*(X_0)=(m_2-m_1) A_{1,2}-m_2(A_{1,2}+A_{1,3}+A_{2,3})=m A_{1,2}-m_2(A_{1,2}+A_{1,3}+A_{2,3}), \end{align*}$$

it is readily checked that the resulting LCS Lie algebra $\mathfrak {h}(\mathscr {C})$ and cohomology ring $H^*(M(\mathscr {C}))$ are isomorphic to $\mathfrak {h}(\mathscr {C}_{n,m})$ and $H^*(M(\mathscr {C}_{n,m}))$ , obtained from the homological root homomorphism for the “standard form” rank-two circuit $\mathscr {C}_{n,m}$ recorded in §5.2 (and easily recoverable from Theorem 6.4).

7.1 The case $n=2$

We begin in rank two, where we record the almost-direct product structure of the fundamental group of the complement of the type C toric arrangement in $(\mathbb {C}^\times )^2$ .

Consider the type C toric arrangements $\mathscr {C}_1$ in $\mathbb {C}^\times \subset \mathbb {C}$ and $\mathscr {C}_2$ in $(\mathbb {C}^\times )^2 \subset \mathbb {C}^2$ , given by the vanishing of the polynomials $x(x^2-1)$ and $x(x^2-1)y(y^2-1)(y-x)(xy-1)$ , respectively. The strictly supersolvable bundle $M(\mathscr {C}_2)\to M(\mathscr {C}_1)$ is equivalent to the pullback of the Fadell-Neuwirth bundle $ {\mathrm {Conf}}_6(\mathbb {C}) \to {\mathrm {Conf}}_5(\mathbb {C})$ along the map $\mathbf {b}\colon M(\mathscr {C}_1) \to {\mathrm {Conf}}_5(\mathbb {C})$ given by $\mathbf {b}(x)=(0,1,-1,x,x^{-1})$ .

In $M(\mathscr {C}_1)=\mathbb {C}\smallsetminus \{-1,0,1\}$ , define loops $\gamma _1(t)=1-\frac {1}{2}\exp (2 \pi \boldsymbol {\iota } t)$ , $\gamma _{-1}(t)=-1+\frac {1}{2}\exp (2 \pi \boldsymbol {\iota } t)$ , $0\le t \le 1$ , and paths $\gamma _0^+(t)=\frac {1}{2}\exp (2 \pi \boldsymbol {\iota } t)$ , $\gamma _0^-(t)=\frac {1}{2}\exp (2 \pi \boldsymbol {\iota } (t+\frac {1}{2}))$ , $0 \le t \le \frac {1}{2}$ , where $\boldsymbol {\iota }=\sqrt {-1}$ ; see Figure 6. The fundamental group $\pi _1(M(\mathscr {C}_1))$ , based at $*=x_0=\frac {1}{2}$ , is generated by the homotopy classes of the loops $\gamma _0^+\centerdot \gamma _{-1} \centerdot \bar {\gamma }_0^+$ , $\gamma _0^+\centerdot \gamma _0^-$ , and $\gamma _1$ , where $\bar \lambda (t)=\lambda (1-t)$ is the reverse path and $\lambda \centerdot \mu $ is concatenation. Denoting these classes by $\mathsf {n}_{1}$ (loop about $x=-1$ ), $\mathsf {z}_1$ (about $x=0$ ), and $\mathsf {p}_1$ (about $x=1$ ), $\pi _1(M(\mathscr {C}_1),x_0)$ is the free group $F_3=\langle \mathsf {z}_1,\mathsf {p}_1,\mathsf {n}_1\rangle $ on three generators.

Figure 6

Loops and paths in $M(\mathscr {C}_1)$ .

Figure 6 Long description

The three circles are centered at minus 1, 0 and 1, each with radius 1 over 2, so that (from left to right) the first is tangent to the second and the second is tangent to the third. The point 1 over 2, where two circles meet, is marked with an asterisk. The circles are oriented counterclockwise. The first and third circle represent loops around minus 1 and 1, and they are labelled gamma sub minus 1 and gamma sub 1, respectively. The middle circle represents two oriented paths, one from asterisk to the point minus 1 over 2, labelled gamma super plus sub 0, the other from the point minus 1 over 2 to asterisk, labelled gamma super minus sub 0.

The above paths may be used to determine the map $\mathbf {b}_\sharp \colon F_3 \to P_5$ induced by $\mathbf {b}$ on fundamental groups, where $P_5=\pi _1( {\mathrm {Conf}}_5(\mathbb {C}),\mathbf {b}(x_0))$ is the five-string pure braid group. Ordering braid strands by increasing real part at the basepoint $\mathbf {b}(x_0)=(0,1,-1,\frac {1}{2},2)$ , one can check that

$$\begin{align*}\mathsf{z}_{1}\mapsto a_{2,3}\big(a_{1,5}a_{2,5}a_{3,5}a_{4,5}\big)^{-1}, \quad \mathsf{p}_{1}\mapsto a_{3,4}a_{3,5}a_{4,5}, \quad \mathsf{n}_{1}\mapsto\big(a_{1,3}a_{1,5}a_{3,5}\big)^{a_{2,5}a_{3,5}a_{4,5}}, \end{align*}$$

where the $a_{i,j}$ are the standard generators of the pure braid group and $u^v=v^{-1}uv$ . Conjugating by $\sigma _1^{-1}\sigma _3^{}\sigma _2^{}\in B_5$ yields an automorphism of $P_5$ , $u\mapsto u^{\sigma _1^{-1}\sigma _3^{}\sigma _2^{}}$ , which ensures that the strands of $P_5$ correspond to the order of the roots given by $\mathbf {b}(x)$ as in Section 6 (and slightly simplifies subsequent fundamental group calculations). Carrying this out yields

(7.1) $$ \begin{align} \mathbf{b}_\sharp(\mathsf{z}_{1})=a_{1,4}^{a_{2,4}a_{3,4}}a_{4,5}^{-1}a_{3,5}^{-1}a_{2,5}^{-1}a_{1,5}^{-1}, \quad \mathbf{b}_\sharp(\mathsf{p}_{1})=a_{2,4}a_{2,5}a_{4,5}, \quad \mathbf{b}_\sharp(\mathsf{n}_{1})=\big(a_{3,4}a_{3,5}a_{4,5}\big)^{a_{1,4}^{-1}a_{1,3}^{-1}a_{1,2}^{-1}}. \end{align} $$

The map $\mathbf {b}_\sharp \colon F_3 \to P_5$ , together with the Artin representation $\hat \alpha _5\colon P_5 \to {\mathrm {Aut}}(F_5)$ , determines the almost-direct product structure of the fundamental group $\pi _1(M(\mathscr {C}_2)) = F_5 \rtimes _\phi F_3$ , where $\phi =\hat \alpha _5 \circ \mathbf {b}_\sharp $ . Denote the generators of $F_5$ by $\mathsf {y}_1=\mathsf {z}_2$ , $\mathsf {y}_2=\mathsf {p}_2$ , $\mathsf {y}_3=\mathsf {n}_2$ , $\mathsf {y}_4=\mathsf {a}_{1,2}$ , and $\mathsf {y}_5=\mathsf {b}_{1,2}$ , when viewing them as elements of $\pi _1(M(\mathscr {C}_2))$ . Then, the group $\pi _1(M(\mathscr {C}_2))$ has generators $\mathsf {z}_1,\mathsf {p}_1,\mathsf {n}_1,\mathsf {z}_2,\mathsf {p}_2,\mathsf {n}_2,\mathsf {a}_{1,2},\mathsf {b}_{1,2}$ , and relations $u^{-1}v u = \phi (u)(v)$ , where $\phi (u)(v)=w(u,v) \cdot v\cdot w(u,v)^{-1}=[w(u,v),v]\cdot v$ for u and v generators of $F_3$ and $F_5$ , respectively. Letting $\mathsf {v}=\mathsf {z}_2^{\mathsf {p}_2\mathsf {n}_2\mathsf {a}_{1,2}}$ and $\mathsf {w}=\mathsf {z}_2^{}\mathsf {p}_2^{}\mathsf {n}_2^{}\mathsf {a}_{1,2}^{}\mathsf {b}_{1,2}^{}$ , calculations with the Artin representation (2.5) yield: