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\title[Supersolvable posets and fiber-type arrangements]{Supersolvable posets and fiber-type arrangements}

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\author{Christin Bibby\thanks{\href{mailto:bibby@math.lsu.edu}{bibby@math.lsu.edu}. C.B. was partially supported by NSF DMS-2204299.}\addressmark{1} \and Emanuele Delucchi\thanks{\href{mailto:emanuele.delucchi@supsi.ch}{emanuele.delucchi@supsi.ch}}\addressmark{2}}

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\address{\addressmark{1}Louisiana State University, Baton Rouge, USA \\ \addressmark{2}University of Applied Arts and Sciences of Southern Switzerland}

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\abstract{
We develop a theory of modularity and supersolvability for chain-finite geometric posets, extending that of Stanley for finite lattices and building a new connection between combinatorics and topology. 
From a combinatorial point of view, our theory features results about factorizations of the characteristic polynomials, dovetails with established notions on geometric semilattices, and behaves well under quotients by translative group actions. 
We also establish a topological counterpart in the context of toric and abelian arrangements, 
akin to Terao's fibration theorem connecting bundles of hyperplane arrangements to supersolvability of their intersection lattice. From this, we obtain a combinatorially determined class of $K(\pi,1)$ toric arrangements. 
Moreover, we characterize combinatorially when our toric arrangement bundles are pulled back from Fadell--Neuwirth's bundles of configuration spaces, and establish an analogue of Falk--Randell's formula relating the Poincar\'e polynomial to the lower central series of the fundamental group.

}

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\keywords{supersolvable lattice, hyperplane arrangement, configuration space}

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\begin{document}

\maketitle
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\section{Introduction}


The theory of {\bf supersolvable lattices} is a cornerstone of enumerative, algebraic and topological combinatorics. Its foundations were laid in work by Stanley \cite{stanley,stanley-modular}, motivated by the study of subgroups in supersolvable groups and building on the classical notion of {\bf modular elements} in lattices.  



Supersolvable lattices arise in a variety of contexts, e.g., the study of factorizations of characteristic polynomials \cite{Hallam-Sagan} and of shellable posets \cite{BjornerSCM}, as well as in convex geometry \cite{Adaricheva} and representation theory \cite{CMB}. More generally, modularity is a key concept in lattice theory, see \cite[II.\S 7]{Birkhoff}. 
Several of these connections interact in the context of matroid theory, where modular flats exhibit a rich structure \cite{brylawski}. 

 
A seminal result by Terao \cite{teraofibthm} shows the equivalence between supersolvability of the lattice of flats of a matroid and an inductive fibration property of the complement manifold of any {\bf arrangement of hyperplanes} that realizes the given matroid over $\mathbb C$. This opens up a powerful bridge between combinatorics and topology. 


More precisely, Terao's result states that the intersection lattice of a complex hyperplane arrangement $\A$ is supersolvable if and only if the arrangement $\A$ is "fiber type", i.e., 
there is a tower of arrangements $\emptyset=\A_0\subsetneq\A_1\subsetneq \ldots \subsetneq \A_d=\A $ such that the natural projection of complements $M(\A_i)\to M(\A_{i-1})$ is a linear fibration.
%
Falk and Randell's study of fiber-type arrangements \cite{FR} unveiled a wealth of combinatorial-topological structure echoeing classical features of configuration spaces. This includes for instance a combinatorial formula for the lower central series of the fundamental group of the arrangement's complement \cite[Theorem 4.1]{FR} and the proof that the fibrations arising in fiber-type arrangements are pullbacks from the classical Fadell-Neuwirth bundle for the configuration space of points in the plane \cite{cohen}. 

 A major point of interest of fiber-type arrangements is related to the long-standing $K(\pi,1)$-problem for hyperplane arrangements, asking for a combinatorial characterization of asphericity of the arrangement's complement. Indeed, fiber-type hyperplane arrangements have aspherical complements, and they are characterized by a combinatorial condition: supersolvability of the intersection lattice.


\bigskip

Here\footnote{This is an extended abstract of \cite{BD}.} we devise a general theory of modular elements and supersolvability for posets beyond lattices -- so-called "geometric posets" (see \Cref{def:geom}). 
On the combinatorial side we derive some fundamental results about factorization of characteristic polynomials (\Cref{thm:charpoly}) 
and quotients by poset automorphisms (\Cref{thm:quotient}). 



When the geometric poset is a semilattice, our definition of supersolvability agrees with that given by Falk and Terao \cite{falk-terao} in studying intersection posets of affine hyperplane arrangements. Moreover, 
we prove that a geometric semilattice is supersolvable if and only if its canonical  extension to a geometric lattice is supersolvable \cite[Theorem 4.2.4]{BD}. 
This leads to a first topological consequence of our work: 
an affine analogue of Terao's fibration theorem \cite[Theorem 4.3.3]{BD}. 


Just as for classical lattice supersolvability, our theory has a strong topological counterpart in terms of {\bf toric arrangements} (see \Cref{def_arrgts}).  Indeed, the notion of "geometric poset" \Cref{def:geom} seems to provide the right level of generality for studying intersection data of an arrangement of subtori in a complex torus.


The study of toric arrangements is a recent field of research that
has given rise to combinatorial structures such as arithmetic Tutte polynomials, arithmetic matroids, matroid schemes and group actions on semimatroids \cite{moci,dD,bibby22}, to name a few. In particular, the poset of intersections of a toric arrangement has been studied from different points of view \cite{Lawrence,ERS}. 
%
Our notion of supersolvability applies to intersection posets of toric arrangements, and has several implications for the topology of the arrangement complement, see \S\ref{sec:toric}.




\section{Supersolvable posets}



We recall basic ideas about posets and supersolvable geometric lattices.
We then define M- and TM-ideals (\Cref{def:modideal}) and the corresponding notions of supersolvability (\Cref{def:ss}).

\subsection{Generalities about posets} Let $\P$ be a partially ordered set (or ``poset'') with partial order relation $\leq$. For $x,y\in \P$ write $x< y$ when $x\leq y$ and $x\neq y$, and $x \lessdot y$ when $x<y$ and $x\leq z<y$ implies $x=z$. Given any $x\in \P$ let $\P_{<x}:=\{y\in \P\st y<x\}$, partially ordered by the restriction of $<$. The posets $\P_{\leq x}$, $\P_{>x}$ and $\P_{\geq x}$ are defined analogously.  The \textit{interval} between two elements $x,y\in \P$ is the set $[x,y]:=\P_{\geq x}\cap \P_{\leq y}$. We refer to \cite{stanley} for standard poset terminology and notation. Departing slightly from standard notation, for any two elements $x,y\in\P$, we define $x\vee y$ to be the \textit{set} of minimal
upper bounds and $x\wedge y$ to be the \text{set} of maximal lower bounds. That
is:
\[ x\vee y := \min\{z\in\P\st z\geq x \text{ and } z\geq y\},
\quad
 x\wedge y := \max\{z\in\P\st z\leq x \text{ and } z\leq y\}. \]
More generally, denote by $\bigvee T$ and $\bigwedge T$ the sets of minimal upper
bounds and maximal lower bounds of a set $T\subseteq\P$.

A {\em complement} of an element $x$ in a chain-finite poset $\P$ is any $z\in \P$ such that $x\vee z\subseteq \max \P$ and $x\wedge z\subseteq \min\P$. Given a subset $X\subseteq \P$ we say that $z\in\P$ is a complement to $X$ if $z$ is a complement of every $x\in X$.\footnote{Notice that this definition generalizes the usual one for lattices.}



\subsection{Locally geometric posets}

\label{def:geomlattice}
Recall that a chain-finite lattice $\L$ is called \defhno{geometric} if and only if, for all $x,y\in \L$:
\begin{center}
	$x\lessdot y$ if and only if there is an atom $a\in A(\L)$ with $a\not\leq x$, $y=x\vee a$.
\end{center}

\begin{definition}[Locally geometric and geometric posets]\label{def:geom}
A graded, bounded below poset  $\P$ is \defh{locally geometric} if, for every $x\in\P$, the subposet 
$\P_{\leq x}$ is a geometric lattice. 
A locally geometric poset $\P$ is \defh{geometric} if for all $x,y\in\P$:
\begin{itemize}
\item[\mylabel{A}{$(\ddagger\ddagger)$}] if $\rk(x)<\rk(y)$ and $I\subseteq A(\P)$ is such that $\bigvee I \ni y$ and $\vert I \vert = \rk(y)$, then there is $a\in I$ such that $a\not \leq x$ and $a\vee x\neq\emptyset$.
\end{itemize}
\end{definition}





\begin{remark}
We do not require $\P$ itself to even be a (semi)lattice. If $\P$ is a lattice, then it is locally geometric if and only if
it is geometric. 
%
A geometric (semi)lattice in the sense of \cite{WW} is precisely a (semi)lattice satisfying condition \ref{A}.
Further note that if $\P$ is locally geometric, then so are $\P_{\leq x}$ and $\P_{\geq x}$ for
any $x\in\P$. 
\end{remark}


\begin{example}
A classical example of a geometric lattice is a Boolean lattice $B_n$, the set of all subsets of $[n]=\{1,2,\dots,n\}$ ordered by inclusion. A simplicial poset, in which every closed interval is isomorphic to a Boolean lattice, is then a locally geometric poset. 
One such example is depicted in \Cref{fig:g}: this is a geometric poset that is not a lattice nor a semilattice.

\begin{figure}[ht]
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture}[scale=.8]
\solidnodes
\node (0) at (0,0) {};
\node (1) at (-1,1.5) {};
\node (2) at (1,1.5) {};
\node (a) at (-1,3) {};
\node (b) at (1,3) {};
\draw[-] (2)--(0)--(1)--(a)--(2)--(b)--(1);
\end{tikzpicture}
  \caption{A geometric poset.}
  \label{fig:g}
\end{subfigure}%
\quad
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture}[scale=.8]
\solidnodes
\node (0) at (0,0) {};
\node (1) at (-1.5,1.5) {};
\node (2) at (-.3,1.5) {};
\node (3) at (.3,1.5) {};
\node (4) at (1.5,1.5) {};
\node (a) at (-1.1,3) {};
\node (b) at (1.1,3) {};
\draw[-] (2)--(0)--(1)--(a)--(2);
\draw[-] (3)--(0)--(4)--(b)--(3);
\end{tikzpicture}
\caption{A locally geometric but non-geometric poset}
\label{fig:ex:lg}  
\end{subfigure}
\quad
\begin{subfigure}[t]{.3\textwidth}
  \centering
  \begin{tikzpicture}[scale=.8]
\solidnodes
\node (0) at (0,0) {};
\node (1) at (-1,1.5) {};
\node (2) at (1,1.5) {};
\node (a) at (-1,3) {};
\node (b) at (1,3) {};
\draw[-] (a)--(1)--(0)--(2) --(b);
\end{tikzpicture}
\caption{A ranked, but not locally geometric, poset.
}
\label{fig:ex:nlg}
\end{subfigure}
\caption{}
\end{figure}
\end{example}

\begin{example}
	The poset of \Cref{fig:ex:lg} is locally geometric but not geometric. 
\end{example}


\subsection{Supersolvable geometric lattices}

There are several equivalent definitions for a modular element in a geometric lattice (see eg. \cite[Theorem 3.3]{brylawski}). The one we state below is  the most useful for our setting and is due to Stanley \cite{stanley-modular}. 
We also extend Stanley's definition of supersolvable lattices \cite[Corollary 2.3]{stanley} to the context of \emph{chain-finite} lattices.

Let $\L$ be a chain-finite lattice. Then $\L$ has a unique minimal element $\zero$ and a unique maximal element $\one$. Let $x\in \L$. The \textit{complements} of $x$ in $\L$ are the elements $y\in\L$ such that $x\wedge y=\zero$ and $x\vee y=\one$. 
%
%



\begin{definition}\label{def:sslattice}\label{def:modular}
An element $x$ in a geometric lattice $\L$ is \defh{modular} if 
the complements of $x$ form an antichain.
A geometric lattice $\L$ is \defhno{supersolvable} if there is a chain $\zero=y_0<y_1<\cdots<y_n=\one$ where each $y_i$ is a modular element with $\rk(y_i)=i$.
\end{definition}




\subsection{Ideals in locally geometric posets}

Let $\P$ be a locally geometric poset. An \emph{order ideal} in $\P$ is a
downward-closed subset. An order ideal is \emph{pure} if all maximal elements
have the same rank. An order ideal $\Q $ is \emph{join-closed} if $T\subseteq \Q $
implies $\bigvee T\subseteq \Q $.

\begin{definition}[M-ideals and TM-ideals]\label{def:modideal}\label{def:TM}
An \defh{M-ideal} of a locally geometric poset $\P$ is a pure,
join-closed order ideal $\Q \subseteq\P$ such that:
\begin{deflist}
\item\label[definition]{def:mod:1}
if $y\in \Q $ and $a\in A(\P)$ such that $a\vee y=\emptyset$ then $a\in \Q $, and
\item\label[definition]{def:mod:2}
for every $x\in\max(\P)$, there is some $y\in\max(\Q )$ such that $y$ is a modular
element in the geometric lattice $\P_{\leq x}$.
\end{deflist}
An M-ideal $\Q$ in a locally geometric poset $\P$ is a \defh{TM-ideal} if $\vert a\vee y \vert = 1$ for all $y\in \Q$ and all $a\in A(\P) \setminus A(\Q)$.
\end{definition}


\begin{remark}\label{cor:niceGL}\label{lem:latticemodid}
Our definition of an M-ideal extends \Cref{def:sslattice}:
An order ideal $\Q $ in a geometric lattice $\L$ is an M-ideal if and only if
$\Q =\L_{\leq y}$ for some modular element $y$.
\end{remark}




\begin{example}\label{ex:ss:mod}
Consider the poset $\P$ in \Cref{figex4}. The subposet $\{\zero, 2, 3\}$ is a TM-ideal; the subposet $\{\zero,4\}$ is an M-ideal that is not a TM-ideal.
Note that in every locally geometric poset $\P$, both $\P$ and $\{\zero\}$ are M-ideals.
\end{example}

\begin{figure}[ht]
\begin{subfigure}[t]{.3\textwidth}
  \centering
  \begin{tikzpicture} [scale=.8]
\node (T) at (0,0) {\scriptsize $\zero$};
\node (1) at (-1,1.5) {\scriptsize $1$};
\node (2) at (0,1.5) {\scriptsize $2$};
\node (3) at (1,1.5) {\scriptsize $3$};
\node (11) at (-1,3) {\scriptsize $a$};
\node (-11) at (1,3) {\scriptsize $b$};
\draw[-] (T)--(1);
\draw[-] (T)--(2);
\draw[-] (T)--(3);
\draw[-] (1)--(11);
\draw[-] (1)--(-11);
\draw[-] (2)--(11);
\draw[-] (3)--(11);
\draw[-] (3)--(-11);
\solidnodes
\end{tikzpicture}
\caption{
A supersolvable, but not strictly supersolvable, poset.
}
\label{fig:ex:ss}
\end{subfigure}
\quad
\begin{subfigure}[t]{.3\textwidth}
\centering
  \begin{tikzpicture} [scale=.8]
\node (T) at (0.5,0) {\scriptsize $\zero$};
\node (1) at (-1,1.5) {\scriptsize $1$};
\node (2) at (0,1.5) {\scriptsize $2$};
\node (3) at (1,1.5) {\scriptsize $3$};
\node (4) at (2,1.5) {\scriptsize $4$};
\node (11) at (-.5,3) {\scriptsize $a$};
\node (-11) at (1.5,3) {\scriptsize $b$};
\draw[-] (T)--(1);
\draw[-] (T)--(2);
\draw[-] (T)--(3);
\draw[-] (T)--(4);
\draw[-] (1)--(11);
\draw[-] (1)--(-11);
\draw[-] (2)--(11);
\draw[-] (4)--(11);
\draw[-] (3)--(-11);
\draw[-] (4)--(-11);
\solidnodes
\end{tikzpicture}
\caption{A strictly supersolvable poset.}
\label{figex4}
\end{subfigure}
\quad
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture} [scale=.8]
\node (0) at (0,0) {\scriptsize $\zero$};
\node (1) at (-1.5,1.5) {\scriptsize $\{1\}$};
\node (2) at (0,1.5) {\scriptsize $\{2\}$};
\node (3) at (1.5,1.5) {\scriptsize $\{3\}$};
\node (a) at (-1.5,3) {\scriptsize $\{1,2\}$};
\node (b) at (0,3) {\scriptsize $\{1,3\}$};
\node (d) at (1.5,3) {\scriptsize $\{2,3\}$};
\foreach \x in {1,2,3} \draw[-] (0) -- (\x);
\foreach \x in {a,b} \draw[-] (1.north) -- (\x.south);
\foreach \x in {a,d} \draw[-] (2.north) -- (\x.south); 
\foreach \x in {b,d} \draw[-] (3.north) -- (\x.south); 
\end{tikzpicture}
\caption{
A ``locally'' supersolvable, but not 
supersolvable, poset.} 
\label{fig:ex:locss}  
\end{subfigure}
\caption{
}
\label{fig:ex:sss}
\end{figure}


\subsection{Supersolvability in geometric posets}
We are now prepared to present our definition of a supersolvable locally geometric poset, which extends the definition of a supersolvable geometric lattice (cf.\ \Cref{def:sslattice}).
\begin{definition}\label{def:ss}
A locally geometric poset $\P$ is \defh{supersolvable} if there is a chain
\[
\zero = \Q_0 \subset \Q_1 \subset \cdots \subset \Q_n = \P\]
 where each $\Q_i$ is an M-ideal of $\Q_{i+1}$ with $\rk(\Q_i)=i$. If moreover every $\Q_i$ is a TM-ideal of $\Q_{i+1}$ we call $\P$ \defh{strictly supersolvable}.
\end{definition}

\begin{example}
Any rank-one locally geometric poset is strictly supersolvable. 
\label{ex:ss:ss}
The poset $\P$ from \Cref{ex:ss:mod} 
is strictly supersolvable via the chain $\zero\subset\{\zero,2,3\}\subset\P$. 
\end{example}

\begin{example}\label{ex:MnotTM}
The poset $\P$ from
\Cref{fig:ex:ss} is not strictly
supersolvable: its only proper M-ideals are $\P_{\leq 1}$ and $\P_{\leq
3}$, and the fact that $|1\vee3|=2$ means neither is a TM-ideal.
\end{example}





If a locally geometric poset is supersolvable, then every closed interval $\P_{\leq x}$ is a supersolvable
geometric lattice. However, this ``local'' supersolvability is not enough for $\P$
itself to be supersolvable, as demonstrated in the following example.


\begin{example}\label{ex:locss}
Consider the poset $\P$ depicted in \Cref{fig:ex:locss}. 
Every closed interval in $\P$ is supersolvable (since every Boolean lattice is), however it is not itself supersolvable.
Indeed, the only proper order ideals which are pure and join-closed are
principal, that is, $\P_{\leq x}$ for some rank-one element $x$. However, such
an order ideal cannot satisfy \Cref{def:TM}.(2) %{def:mod:2}
since no single element is
covered by all maximal elements.
\end{example}


\begin{remark}
A geometric lattice $\L$ satisfies \Cref{def:ss} if and only if it satisfies the supersolvability criterion of \Cref{def:sslattice}. In a geometric semilattice $\L$, M-ideals and TM-ideals are equivalent, thus $\L$ is supersolvable if and only if it is strictly supersolvable.
\end{remark}

For geometric posets, M-ideals can be characterized using partitions of atoms \cite[Theorem 4.1.2.]{BD}, providing the following characterization of supersolvability, reminiscent of \cite[Remark 2.6]{falk-terao} for geometric semilattices.


\begin{proposition}(\cite[Corollary 4.1.3]{BD})\label{prop:distinct}
Let $\P$ be a geometric poset. Then $\P$ is supersolvable if and only if there is a chain $\{\zero\}=\Q_0\subset \Q_1 \subset \ldots \subset \Q_n=\P$ of pure, join-closed order ideals of $\P$ with $\rk(\Q_i)=i$ and so that for every $i=1,\ldots,n$ and any two distinct
$a_1,a_2 \in A(\Q_{i})\setminus A(\Q_{i-1})$ and every $x\in a_1\vee a_2$ there is $a_3\in A(\Q_{i-1})$ with $x > a_3$.
\end{proposition}




\begin{example}%[Dowling]
Dowling posets \cite{BG} form a class of locally geometric posets that generalize partition lattices and Dowling lattices, which are
known to be supersolvable geometric lattices \cite{dowling,stanley}.
We can show \cite[Proposition 2.6.1.]{BD} that, for any positive integer $n$, finite group $G$, and finite $G$-set $S$, the
Dowling poset $D_n(G,S)$ is strictly supersolvable.
\end{example}


\subsection{Group actions}

Let $G$ be a group. An \textit{action} of $G$ on a poset $\P$ is any group homomorphism $G\to\Aut(\P)$ from $G$ to the group of automorphisms of $\P$. Given a group element $g\in G$ it is customary to denote the associated automorphism by $g:\P\to\P$. For $x\in \P$ we will write $gx$ for $g(x)$. Following \cite{dD}, we focus on the following special type of action.

\begin{definition}
\label{def:translative}
Let $\P$ be a poset with an action of a group $G$. We call the action \defh{translative} if $x\vee gx\neq\emptyset$ implies $x=gx$ for every $x\in \P$ and every $g\in G$.
\end{definition}

Write $Gx=\{gx\st g\in G\}$ for the orbit of an $x\in\P$ under $G$. 
 On the set of orbits  $\P/G:=\{Gx\st x\in \P\}$ we consider the relation given by $Gx\leq Gy$ if there is $g\in G$ with $x\leq gy$.  If the action is translative, this is a partial order relation on $\P/G$.



\begin{thm}\label{thm:quotient}
	Let $\P$ be a locally geometric poset with a translative action of a
group $G$ and let $\Q$ be a $G$-invariant subposet of $\P$. If $\Q$ is an
M-ideal in $\P$, then $\Q/G$ is an M-ideal in $\P/G$. Moreover, if $\P$
satisfies \ref{A}, the converse also holds.
\end{thm}


\subsection{Characteristic polynomial}
The {\em characteristic polynomial} of any bounded-below poset $\P$ with a rank
function $\rk$ is 
\[ \chi_{\P}(t):=\sum_{x\in \P} \mu_{\P}(x)t^{\rk(\P)-\rk(x)}, \]
where $\mu_{\P}$ is the M\"obius function of $\P$.
A feature of supersolvable geometric lattices is that their characteristic
polynomial decomposes into linear factors over $\mathbb{Z}$. We show that this
is true also for \textit{strictly} supersolvable posets. 

\begin{thm}\label{thm:charpoly}
Let $\Q $ be a TM-ideal of a locally geometric poset $\P$ with
$\rk(\Q )=\rk(\P)-1$, and let $a=|A(\P)\setminus A(\Q )|$. Then
\[\chi_\P(t) = \chi_\Q (t)\cdot(t-a).\]
In particular, if $\P$ is strictly supersolvable via the chain of TM-ideals $\zero=\Q_0\subset \Q_1\subset\cdots\subset Q_n=\P$, and $a_i=|A(\Q_i)\setminus A(\Q_{i-1})|$ for each $i$, then 
\[\chi_\P(t) = \prod_{i=1}^n (t-a_i).\]
\end{thm}


\begin{remark}
The assumption that $\Q $ is a TM-ideal in \Cref{thm:charpoly} is
necessary, as demonstrated in the following examples. Accordingly, a poset being
supersolvable is not enough for its characteristic polynomial to factor
completely over $\mathbb Z$.
\end{remark}

\begin{example}\label{ex:charpoly}
Consider the poset $\P$ depicted in \Cref{fig:ex:sss} (see also \Cref{ex:sss}).
Its characteristic polynomial is
\[\chi_{\P}(t) = t^2-4t+4 = (t-2)(t-2).\]
This agrees with the fact that the TM-ideal $\Q=\{\zero,2,3\}$ in \Cref{figex4}
has $\chi_\Q(t)=t-2$ and $|A(\P)\setminus A(\Q)|=2$.
\end{example}

\begin{example}
Consider again the poset $\P$ in \Cref{fig:ex:ss}. It is supersolvable, with $\{\zero,1\}$ and $\{\zero,3\}$
both M-ideals. However, it is not strictly supersolvable and its
characteristic polynomial $\chi_{\P}(t) = t^2-3t+3$ does not factor over the
integers. 
\end{example}



\section{Toric Arrangement Bundles}\label{sec:toric}

\subsection{Toric Arrangements}
Fix a finitely generated free abelian group $\Gamma\cong\Z^d$ and let $T=\Hom(\Gamma,\C^\times)\cong (\C^\times)^d$ be the complex torus.

\begin{definition}\label{def_arrgts}
A \defh{toric arrangement} is a collection $\{H_\alpha \colon \alpha\in \A\}$ for some finite set $\A\subseteq\Gamma$, where 
\[H_\alpha := \{t\in T\st t(\alpha)=0\}.\]
We will often refer to an arrangement $\{H_\alpha \colon \alpha\in\A\}$ simply by $\A$ when there is no confusion.
%
The \defh{complement} of $\A$ is denoted by
\[ M(\A) := 
T\setminus \bigcup_{\alpha\in\A} H_\alpha.\]
We only consider toric arrangements that are \defh{essential}, i.e., where
$\A$ generates a full subgroup of $\Gamma$.
\end{definition}


\begin{example}\label{ex:A}
Let $\Gamma=\mathbb Z^2$.
The arrangement $\A=\{\alpha_1=(1,0),\alpha_2=(0,1),\alpha_3=(1,2)\}$
yields three subtori in $\C^\times\times\C^\times$, cut out by equations $x=1$, $y=1$, and $xy^2=1$. 
The real part, in $S^1\times S^1$, is depicted in \Cref{fig:ex:A}.
\begin{figure}[ht]
\begin{subfigure}[t]{.6\textwidth}
\centering
\begin{tikzpicture}[scale=2]
\draw[thick,-] (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw[thick,red,-] (0,0)--(1,0);
\draw[thick,green,-] (0,0)--(0,1);
\draw[thick,blue,-] (0,0)--(1,.5);
\draw[thick,blue,-] (0,.5)--(1,1);
\node at (0,-.25) {};
\solidnodes
\node at (0,0) {};
\node at (0,.5) {};
\end{tikzpicture}
\caption{The arrangement $\A$ from Example \ref{ex:A} is depicted in $S^1\times S^1$, 
with $H_1$ in \textcolor{green}{green}, $H_2$ in \textcolor{red}{red}, and $H_3$
in \textcolor{blue}{blue}. }
\label{fig:ex:A}
\end{subfigure}
\hfill
\begin{subfigure}[t]{.3\textwidth}
\centering
\begin{tikzpicture}[scale=.8]
\node (T) at (0,0) {\scriptsize $T$};
\node (1) at (-1,1.5) {\scriptsize $H_1$};
\node (2) at (0,1.5) {\scriptsize $H_2$};
\node (3) at (1,1.5) {\scriptsize $H_3$};
\node (11) at (-1,3) {\scriptsize $(1,1)$};
\node (-11) at (1,3) {\scriptsize $(1,-1)$};
\draw[-] (T)--(1);
\draw[-] (T)--(2);
\draw[-] (T)--(3);
\draw[-] (1)--(11);
\draw[-] (1)--(-11);
\draw[-] (2)--(11);
\draw[-] (3)--(11);
\draw[-] (3)--(-11);
\solidnodes
\end{tikzpicture}
\caption{The poset of layers $\P(\A)$.}
\label{fig:ex:P(A)}
\end{subfigure}
\caption{}
\end{figure}

\end{example}

\subsection{Poset of layers}
The intersection data of a toric arrangement may be encoded in a geometric poset.

\begin{definition}
The \defh{poset of layers} of an arrangement $\A$ is the set $\P(\A)$
whose elements are the nonempty connected components of intersections
$\bigcap_{\alpha\in S} H_\alpha$ where $S\subseteq\A$, partially ordered by
reverse inclusion.
\end{definition}


By convention, $T$ is the unique minimum element of $\P(\A)$. 
The atoms of $\P(\A)$ are precisely the connected components
of the $H_\alpha$, where $\alpha\in\A$.


\begin{remark}
The poset of layers for a toric arrangement is indeed a geometric poset.
The lift of all $H_\alpha$,  $\alpha\in\A$, to the universal cover of $T$ is an arrangement $\A^{\upharpoonright}$ of affine subspaces in $\C^d$.
Its poset of layers $\P(\A^{\upharpoonright})$ is a geometric semilattice and the action on $\A^{\upharpoonright}$ of the group of deck transformations induces a translative action of $\Z^d$ on $\P(\A^{\upharpoonright})$. 
Then $\P(\A)$ is isomorphic to the quotient $\P(\A^\upharpoonright)/\Z^d$ (see \cite[Lemma 9.8]{dD}), and thus it is geometric (via \Cref{thm:quotient}).
\end{remark}

\begin{example}
Let $\A$ be the toric arrangement from \Cref{ex:A}. The Hasse diagram for its poset of layers $\P(\A)$ is depicted in \Cref{fig:ex:P(A)}.
Notice that this poset was seen in \Cref{fig:ex:ss} and is supersolvable, with M-ideal given by $\P_{\leq H_1}$ or $\P_{\leq H_3}$ (see \Cref{ex:MnotTM}).
\end{example}

\begin{example}\label{ex:sss}
Let $\Gamma=\Z^2$ and $\A=\{\alpha_1=(1,0),\alpha_2=(0,2),\alpha_3=(1,2)\}$.
\Cref{fig:fib:sss} depicts the corresponding $H_1$, $H_2$, and $H_3$ in $S^1\times S^1$ and \Cref{figex4} depicts the Hasse diagram for
its poset of layers.
As seen in \Cref{ex:ss:mod} this poset is strictly supersolvable; the maximal elements of its proper TM-ideal are the two connected components of $H_2$.
\end{example}



\begin{figure}[ht]
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture}[scale=1.8]
\draw[->] (.5,-.3)--(.5,-.8);
\node at (1.5,.5) {$\supseteq$};
%% top
\draw[thick,-] (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw[thick,red,-] (0,0)--(1,0);
\draw[thick,green,-] (0,0)--(0,1);
\draw[thick,blue,-] (0,0)--(1,.5);
\draw[thick,blue,-] (0,.5)--(1,1);
\node at (0,-.25) {};
\solidnodes
\node at (0,0) {};
\node at (0,.5) {};
%% bottom
\draw[thick,-] (0,-1) -- (1,-1);
\node[green,fill=green] at (0,-1) {};
%% fiber
\draw[thick,-] (2,0)--(2,1);
\node[red,fill=red] at (2,0) {};
\node[blue,fill=blue] at (2,.3) {};
\node[blue,fill=blue] at (2,.8) {};
\end{tikzpicture}
\caption{A fibration
whose fibers are homeomorphic to $S^1$ with three punctures.}
\label{fig:fib:good}
\end{subfigure}
\hfill
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture}[scale=1.8]
%% misc
\foreach \x in {0.5,2,1.5} \draw[->] (\x,-.3)--(\x,-.8);
\node at (1.25,.5) {$\supseteq$};
\node at (2,-1) {\scriptsize $-1$};
\node at (1.5,-1) {\scriptsize $t\neq -1$};
%% top
\draw[thick,-] (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw[thick,red,-] (0,0)--(0,1);
\draw[thick,green,-] (0,0)--(1,0);
\draw[thick,blue,-] (0,0)--(.5,1);
\draw[thick,blue,-] (.5,0)--(1,1);
\node at (0,-.25) {};
\solidnodes
\node at (0,0) {};
\node at (.5,0) {};
%% bottom
\draw[thick,-] (0,-1) -- (1,-1);
\node[red,fill=red] at (0,-1) {};
%% fibers
\draw[thick,-] (2,0)--(2,1);
\draw[thick,-] (1.5,0)--(1.5,1);
\node[cyan,fill=cyan] at (2,0) {};
\node[green,fill=green] at (1.5,0) {};
\node[blue,fill=blue] at (1.5,.3) {};
\end{tikzpicture}
\caption{This is not a fibration, as indicated by the two non-homeomorphic fibers.}
\label{fig:fib:bad}
\end{subfigure}
\hfill
\begin{subfigure}[t]{.3\textwidth}
  \centering
\begin{tikzpicture}[scale=1.8]
%% misc
\foreach \x in {0.5} \draw[->] (\x,-.3)--(\x,-.8);
\node at (1.5,.5) {$\supseteq$};
%% top
\draw[thick,-] (0,0)--(1,0)--(1,1)--(0,1)--(0,0);
\draw[thick,red,-] (0,0)--(0,1);
\draw[thick,red,-] (0.5,0)--(0.5,1);
\draw[thick,green,-] (0,0)--(1,0);
\draw[thick,blue,-] (0,0)--(.5,1);
\draw[thick,blue,-] (.5,0)--(1,1);
\node at (0,-.25) {};
\solidnodes
\node at (0,0) {};
\node at (.5,0) {};
%% bottom
\draw[thick,-] (0,-1) -- (1,-1);
\node[red,fill=red] at (0,-1) {};
\node[red,fill=red] at (0.5,-1) {};
%% fibers
\draw[thick,-] (2,0)--(2,1);
\node[green,fill=green] at (2,0) {};
\node[blue,fill=blue] at (2,.3) {};
\end{tikzpicture}
\caption{A fibration whose fibers are homeomorphic to $S^1$ with two punctures.}
\label{fig:fib:sss}
\end{subfigure}
\caption{Each figure represents a restriction of the projection $S^1\times S^1 \to S^1$.}
\end{figure}



\subsection{Characterization of fibrations}

A subgroup $Y$ of $T$ will be called \defh{admissible} if there is a rank-one direct summand $\Gamma'\subseteq \Gamma$ such that $Y$ is the image of the injection $\varepsilon^*:\Hom(\Gamma',\C^\times)\to\Hom(\Gamma,\C^\times)$ induced by the projection $\varepsilon:\Gamma\to\Gamma'$. 
When $Y$ is admissible, the corresponding projection \[p:T\to
T/Y\cong\Hom(\Gamma/\Gamma',\C^\times)\] is a
section of the map induced by the quotient $q:\Gamma\to\Gamma/\Gamma'.$
This allows us to define toric arrangements 
\[\A_Y:=\{\alpha\in\A\st H_\alpha\supseteq Y\} \qquad \qquad \qA:=q(\A_Y) \subseteq \Gamma/\Gamma',\]
in $T$ and in $T/Y$, respectively.
Then the projection $p:T\to T/Y$ restricts to a map on arrangement complements
$\pr:M(\A)\to M(\qA)$ and induces an isomorphism of posets $\P(\A_Y)\cong\P(\qA).$

\begin{definition} 
An arrangement $\A$ is {\bf fiber-type} if there is a chain of subgroups $Y_1\subseteq \ldots Y_{d-1}\subseteq Y_d=\mathbb (\C^\times)^d$ such that for every projection $p_i:Y_i\to Y_i/Y_{i-1}$ the induced map $\pr_i$ is a fibration on arrangement complements whose fiber is homeomorphic to $\mathbb C^\times$ minus a finite set of points.
\end{definition}

\begin{remark}\label{rem:aay}
The poset of layers $\P(\A_Y)$ may be viewed as a subposet of $\P(\A)$. 
Its atoms are the atoms of $\A$ that either contain $Y$ or are disjoint from it. 
For any $\alpha\not\in\A_Y$, every connected component of $H_\alpha$ will intersect $Y$ nontrivially. 
Moreover, if $Y\in \P(\A)$, then the maximal elements of $\P(\A_Y)$ are cosets of $Y$.
\end{remark}




We prove in \cite[Theorem 3.3.1.]{BD} that $\P(\A_Y)$ is an M-ideal of $\P(\A)$ if and only if there is an integer $\ell$ such that the fibers of the projection $M(\A)\to M(\qA)$ are all homeomorphic to $\C^\times$ with $\ell$ points removed.
The number of punctures can be counted by examining how the hypersurfaces not in $\P(\A_Y)$ intersect $Y$ or its translates.
When $\P(\A_Y)$ is an M-ideal, the map is moreover locally trivial. 
Iterating this then yields:


\begin{thm}[Fibration Theorem {\cite[Theorem A]{BD}}]\label{thm:A}
An essential toric arrangement is fiber-type if and only if its poset of layers is supersolvable. 
\end{thm}

\begin{example}\label{ex:fib}
Consider the arrangement from \Cref{ex:A} (see also \Cref{fig:ex:ss}). 
Letting $Y=H_1$,  
the projection $M(\A)\to M(\qA)$ is depicted in
Figure \ref{fig:fib:good}.
As the picture suggests, this map is a fibration with fiber homeomorphic to $T$
with three points removed. 
On the other hand, letting $Y=H_2$ the projection $M(\A)\to M(\qA)$ is not a fibration. This is evident in Figure \ref{fig:fib:bad}, which depicts two non-homeomorphic fibers. 

This agrees with our observation in \Cref{ex:ss:mod}
that in the poset
of layers $\P=\P(\A)$, the order ideal $\P_{\leq H_1}$ is an M-ideal while
$\P_{\leq H_2}$ is not. 


\end{example}

From \Cref{thm:A}, Falk and Randell's arguments in \cite{FR} can be adapted to prove the following results.

\begin{thm}[Asphericity, {\cite[Corollary B]{BD}}]
If the poset of layers of a toric arrangement is supersolvable, then the arrangement complement is a $K(\pi,1)$ space.
If the poset is strictly supersolvable, then the fundamental group is an iterated semidirect product of free groups.
\end{thm}

\begin{thm}[{Lower Central Series Formula, \cite[Theorem D]{BD}}]
Let $\A$ be a strictly supersolvable toric arrangement with complement $M(\A)$, let $\A_0=\emptyset\subsetneq \A_1\subsetneq\cdots\subsetneq \A_n$ be the associated tower of arrangements and set $a_i:=\vert \A_{i} \setminus \A_{i-1}\vert$ for $i=1,\ldots,n-1$. For $j\geq 1$, let $\varphi_j$ be the rank of the $j$th successive quotient in the lower central series of the fundamental group $\pi_1(M(\A))$. Then 
\begin{equation}\label{eq:lcs}
\prod_{j=1}^\infty (1-t^j)^{\varphi_j} = \prod_{i=1}^n (1-(a_i+1)t).
\end{equation}
\end{thm}

The right-hand side of \eqref{eq:lcs} encodes the Betti numbers, and is a specialization of the characteristic polynomial for the associated poset of layers (see \Cref{thm:charpoly}).

\subsection{Pullback of Fadell--Neuwirth bundles}

Suppose that $\pr\colon M(\A)\to M(\qA)$ is a toric arrangement bundle arising from a TM-ideal $\Q=\P(\A_Y)$,
and fix an order $H_1, H_2, \dots, H_\ell$ of the atoms in $\P(\A)$ that are not in $\Q$. 
The definition of a TM-ideal implies that for any $x\in M(\qA)$, and any $1\leq i \leq \ell$, there is a unique point in $H_i\cap p^{-1}(x)$. Through the identification $p^{-1}(x)\cong \C^\times \subseteq \C$, this defines a continuous map $g_i\colon M(\A)\to\C$. 
Via \Cref{prop:distinct}, the points $g_1(x),\dots,g_\ell(x)$ must be distinct and nonzero, thus determining a point in the ordered configuration space
$\Conf_{\ell+1}(\C) = \{(z_0,\dots,z_\ell)\in\C^{\ell+1} \st z_i\neq z_j \text{ when } i\neq j\}$.

In fact, the bundle $\pr$ is pulled back from Fadell--Neuwirth's bundle of configuration spaces (as in \cite{FN}) through this map $g$. Consequently, properties of Fadell--Neuwirth's bundles (eg. existence of a section, trivial homological monodromy) may thus be pulled back to obtain properties of toric arrangement bundles.

\begin{thm}[{\cite[Theorem 5.3.1]{BD}}]
The map $g:M(\A)\to \Conf_{\ell+1}(\C)$ 
given by $g(x) = (0,g_1(x),\dots,g_\ell(x))$ is continuous and  yields the following pullback diagram. 

\begin{center}
\begin{tikzcd}
M(\A) \arrow{r}{h}\arrow{d}{\pr}& \Conf_{\ell+2}(\C)\arrow{d}{\pi} \\
M(\qA) \arrow{r}{g} & \Conf_{\ell+1}(\C) 
\end{tikzcd}
\end{center}
\end{thm}


\begin{example}
Let $\A$ be the arrangement of \Cref{ex:sss}. The fiber bundle depicted in \Cref{fig:fib:sss} 
is pulled back through the map $g:M(\qA)\to \Conf_3(\C)$,  $g(x) = (0,1,x^2)$.
\end{example}



\acknowledgements{%This is an extended abstract of \cite{BD}. 
For the work reported in this extended abstract, we acknowledge the hospitality of the Mathematisches Forschingsinstitut Oberwolfach.}

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