# Modular flats of oriented matroids and poset quasi-fibrations

Paul Mücksch  Paul Mücksch, Kyushu University, Department of Mathematics, 744,
Motooka, Nishi-ku, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

We study the combinatorics of modular flats of oriented matroids and the
topological consequences for their Salvetti complexes. We show that the
natural map to the localized Salvetti complex at a modular flat of corank one
is what we call a poset quasi-fibration – a notion derived from Quillen’s
fundamental Theorem B from algebraic $K$-theory. As a direct consequence, the
Salvetti complex of an oriented matroid whose geometric lattice is
supersolvable is a $K(\pi,1)$-space – a generalization of the classical result
for supersolvable hyperplane arrangements due to Falk, Randell and Terao.
Furthermore, the fundamental group of the Salvetti complex of a supersolvable
oriented matroid is an iterated semidirect product of finitely generated free
groups – analogous to the realizable case.

Our main tools are discrete Morse theory, the shellability of certain
subcomplexes of the covector complex of an oriented matroid, a nice
combinatorial decomposition of poset fibers of the localization map, and an
isomorphism of covector posets associated to modular elements.

We provide a simple construction of supersolvable oriented matroids. This
gives many non-realizable supersolvable oriented matroids and by our main
result aspherical CW-complexes.

###### Key words and phrases:

Oriented matroid, supersolvable lattice, Salvetti complex, poset quasi-
fibration, discrete Morse theory

###### 2010 Mathematics Subject Classification:

Primary: 52C40, 55P20; Secondary: 55R10, 57Q70

##  1\. Introduction

Complements of complex hyperplane arrangements provide a rich source of
interesting topological spaces. One of the most intricate and fascinating
problems is to decide when arrangement complements are Eilenberg-MacLane
$K(\pi,1)$-spaces where $\pi$ is the fundamental group of the space. By
definition, this is the case if their universal covering space is
contractible. Synonyms are aspherical space or classifying space for the group
$\pi$. They play an important role in algebraic topology e.g. as their
cohomology coincides with the group cohomology of $\pi$. Moreover, if a
$K(\pi,1)$-space is homotopy equivalent to a finite CW-complex then the group
$\pi$ is torsion free. In the following, we call complex arrangements whose
complements are aspherical simply $K(\pi,1)$ or aspherical arrangements.

One of the most prominent positive results regarding $K(\pi,1)$-arrangements
is Deligne’s theorem [Del72], stating that complexified real simplicial
arrangements are aspherical. Subsequently, Salvetti [Sal87] introduced a
regular cell complex modeling the homotopy type of the complement of a
complexified real arrangement. Gel’fand and Rybnikov [GR90] realized that the
construction of the Salvetti complex only depends on the oriented matroid
associated to the real arrangement (see Section 2.6). This in turn led to a
generalization of Deligne’s result to oriented matroids in general, i.e. the
Salvetti complex of an oriented matroid whose covector complex is a simplicial
cell decomposition of the sphere is indeed aspherical, independently
established by Cordovil [Cor94] and Salvetti [Sal93].

The combinatorial notions of a modular flat in a geometric lattice and
supersolvability were introduced by Stanley [Sta71, Sta72] (see Section 2.3).
Regarding the $K(\pi,1)$ problem for hyperplane arrangements, a further corner
stone in the theory is due to Falk and Randell [FR85] and Terao [Ter86], see
also [OT92, Ch. 5]. Thanks to their theorem, complex arrangements with
supersolvable geometric lattices are aspherical. Their proof is entirely
geometric and does not use any complexes modeling the homotopy type of the
complement. More precisely, they showed that the natural projection of the
ambient space of the arrangement to the quotient space by a modular
intersection of corank one, restricted to the complement space is a fiber
bundle map with fiber a punctured complex plane. The associated long exact
sequence of homotopy groups then yields their result. As a special case for
real arrangements one obtains that supersolvability of their intersection
lattices implies that their Salvetti complexes are concrete instances of
finite $K(\pi,1)$-complexes. A nice generalization of Falk, Randell and
Terao’s result to arrangements on connected abelian Lie groups is obtained in
the very recent work of Bibby and Delucchi [BD22].

The natural question appears whether the “real” version of Falk, Randell and
Terao’s theorem extends to all oriented matroids as does Deligne’s theorem.

It is exactly the aim of this paper to provide a positive answer to this
question and a novel combinatorial proof of the “supersolvability theorem” for
real arrangements. More precisely, we show that the natural projection map
from the Salvetti complex to its localization at a modular corank one flat in
the geometric lattice of an oriented matroid (see Section 2.6) is what we call
a _poset quasi-fibration_. The poset fibers of this localization map are
homotopy equivalent to the affine Salvetti complex of an oriented matroid of
rank two (Theorem 6.4). The notion of a poset quasi-fibration is directly
derived from Quillen’s fundamental Theorem B from algebraic $K$-theory [Qui73]
(see Section 5).

Since every poset quasi-fibration comes with a long exact homotopy sequence,
as an application we obtain the following theorem which indeed extends Falk,
Randell and Terao’s result to oriented matroids.

######  Theorem 1.1.

The Salvetti complex of a supersolvable oriented matroid is aspherical.

We also provide a simple argument that the Salvetti complex of each
localization of an oriented matroid with aspherical Salvetti complex is
aspherical as well (Lemma 2.26). In the case of hyperplane arrangements this
useful necessary condition for asphericity is due to Oka , cf. [Par93, Lem.
1.1]. Furthermore, we show that the fundamental group of the Salvetti complex
of a supersolvable oriented matroid is an iterated semidirect product of
finitely generated free groups (see Section 7) which is completely analogous
to the realizable case.

A main ingredient in our study is discrete Morse theory which was established
by Forman [For98] (see Section 2.7). We use its reformulation in terms of
acyclic matchings on face posets of cell complexes due to Chari [Cha00], which
is now standard for applications to combinatorial topology, see also Kozlov’s
book [Koz08]. The acyclic matchings which are relevant to us are derived from
the shellability of the covector complex of an oriented matroid (see Section
2.5), a central result in the combinatorial topology of oriented matroids.

In the recent breakthrough work by Paolini and Salvetti [PS21] which settled
the $K(\pi,1)$ conjecture for affine Artin groups, discrete Morse theory in
combination with certain shellability results also played an important role.

In a separate erratum [DM22] we comment on results stated by Delucchi in
[Del08] which are unfortunately not correct. Resolving these issues would lead
to a generalization of our results, similar to the more general fiber bundles
for hyperplane arrangements described in the work of Falk and Proudfoot
[FP02].

This article is organized as follows. In Section 2 we recall all the required
notions from combinatorial topology and oriented matroid theory. In Section 3
we further elaborate on the special properties of modular flats of oriented
matroids. Section 4 describes a useful stratification of the poset fibers of
the localization map between Salvetti complexes. Section 5 introduces the
special class of order preserving maps which play a central role in our study.
In Section 6 we derive certain acyclic matchings which are then applied to
prove our main result: Theorem 6.4. In the following Section 7 we describe the
structure of the fundamental group of the Salvetti complex of a supersolvable
oriented matroid. To conclude, in Section 8 we provide a simple construction
which yields many non-realizable supersolvable oriented matroids.

##  2\. Preliminaries

In this preliminary section we review all the objects and notions required for
our study.

###  2.1. Posets and regular CW-complexes

We introduce some notation and collect basic informations on the combinatorial
topology of posets and regular CW-complexes. Throughout this paper, all
partially ordered sets (or posets for short) and all complexes are assumed to
be finite. A concise reference for us is [BLVS+99, App. 4.7].

We start by recalling some basic poset terminology. Let $P=(P,\leq)$ be a
poset. Its _dual_ or _opposite_ poset $P^{\vee}$ has the same ground set as
$P$ but its order relation $\leq^{\vee}$ is reversed: $x\leq^{\vee}y:\iff
y\leq x$.

To emphasize the order relation of a particular poset $P$ we occasionally use
the notation $\leq_{P}$.

We denote by $\lessdot$ the _cover relations_ of $P$. That is $x\lessdot y$ if
and only if $x<y$, and for all $z\in P$: $x<z\leq y$ implies $z=y$.

A subset $I\subseteq P$ is called an _order ideal_ if $y\in I$ and $x\in P$
with $x\leq y$ implies $x\in I$. Dually, $F\subseteq P$ is called an _order
filter_ if $F^{\vee}$ is an order ideal in $P^{\vee}$. We write $P_{\leq
y}=\\{x\in P\mid x\leq y\\}$ for the _principal_ order ideal generated by
$y\in P$ and similarly $P_{\geq x}=\\{y\in P\mid x\leq y\\}$ for the principal
order filter generated by $x\in P$.

A map $f:P\to Q$ between two posets is _order preserving_ or a _poset map_ if
for all $x\leq_{P}y$ $(x,y\in P)$ we have $f(x)\leq_{Q}f(y)$. If $q\in Q$ then
we write $(f\downarrow q):=f^{-1}(Q_{\leq q})$ for the _poset fibers_ of $f$.

A _linear extension_ $\dashv$ of $P$ is a total order on the ground set $P$
such that for all $x,y\in P$: $x\leq y$ implies $x\dashv y$. Linear extensions
of finite posets always exits. Given an order ideal $I\subseteq P$, a linear
extension $\dashv_{I}$ on $I$ can always be extended to a linear extension
$\dashv$ on $P$ with $x\dashv y$ if and only if $x\dashv_{I}y$ for all $x,y\in
I$. Moreover, a linear extension on $I$ can always be extended to $P$ in such
a way that all the elements of $I$ come first. An analogue statement holds for
order filters.

We assume familiarity with the notion of _CW-complexes_ , also called _cell
complexes_ , e.g. see [Hat02]. Important for our context are the _regular_
cell complexes since they are completely determined by their _face poset_ of
closed cells ordered by inclusion. Let $\Sigma$ be a regular cell complex and
$\sigma,\tau\in\Sigma$. If $\tau\subseteq\overline{\sigma}$ then $\tau$ is
called a _face_ of $\sigma$ and we simply write $\tau\leq\sigma$. In what
follows, we thus regard a regular cell complex $\Sigma$ itself as a poset.
Cells of dimension zero are referred to as _vertices_.

Conversely, to every poset $P$ is associated its _order complex_ $\Delta(P)$.
It is the simplicial complex with $n$-simplices given by all chains of length
$n$ in $P$, i.e.

$\Delta(P)_{n}:=\\{\\{x_{0},\ldots,x_{n}\\}\mid x_{i}\in P\text{ with
}x_{0}<x_{1}<\ldots<x_{n}\\}.$

For a simplicial complex $\Delta$ we denote its topological realization by
$|\Delta|$. The composition of forming the order complex and taking its
realization yields the _simplicial realization_ functor

$|\Delta(-)|:\operatorname{\mathtt{Pos}}\to\operatorname{\mathtt{Top}}$

from the category $\operatorname{\mathtt{Pos}}$ of posets and order preserving
maps to the category $\operatorname{\mathtt{Top}}$ of topological spaces and
continuous maps. For a regular cell complex $\Sigma$ the image of its face
poset under this functor is naturally homeomorphic to $\Sigma$:
$|\Delta(\Sigma)|\cong\Sigma$.

We call an order preserving map $f:P\to Q$ a homotopy equivalence provided
$|\Delta(f)|$ is a homotopy equivalence.

Maps between finite regular cell complexes are given as order preserving maps
between their face posets. A poset map between the face posets of regular cell
complexes which is a homotopy equivalence in the above sense is indeed a
homotopy equivalence of their defining topological spaces.

A regular cell complex $\Sigma$ is called _pure_ if all maximal cells have the
same dimension.

A subset $\Gamma\subseteq\Sigma$ is called a (closed) subcomplex if with a
cell $\sigma\in\Gamma$ all of its faces also belong to $\Gamma$, i.e. $\Gamma$
is an order ideal of $\Sigma$. For a subset of cells $S\subseteq\Sigma$ we
denote by $\Sigma(S):=\\{\tau\in\Sigma\mid\tau\leq\sigma$ for a $\sigma\in
S\\}$ the subcomplex of all faces of cells in $S$. In poset terminology,
$\Sigma(S)=\bigcup_{\sigma\in S}\Sigma_{\leq\sigma}$ is the order ideal
generated by $S$. For a cell $\sigma\in\Sigma$ we denote by $\delta\sigma$ the
subcomplex consisting of all proper faces of $\sigma$.

###  2.2. Oriented Matroids

We recall some basics from oriented matroid theory. An oriented matroid can be
regarded as a combinatorial abstraction of a real hyperplane arrangement, i.e.
a finite set of hyperplanes in a finite dimensional real vector space. The
standard reference is the book by Björner et al. [BLVS+99]. In our study, we
adopt the point of view that oriented matroids constitute a special class of
regular cell complexes, see Remark 2.4 below.

First, let us recall some notation for sign vectors. Let $E$ be some finite
set and consider a sign vector $\sigma\in\\{+,-,0\\}^{E}$. We denote by
$z(\sigma):=\\{e\in E\mid\sigma_{e}=0\\}$ its _zero set_. The _opposite_
vector $-\sigma$ is defined by

$(-\sigma)_{e}:=\begin{cases}-&\text{ if }\sigma_{e}=+,\\\ +&\text{ if
}\sigma_{e}=-,\\\ 0&\text{ if }\sigma_{e}=0.\end{cases}$

The _composition_ $\sigma\circ\tau$ of two sign vectors
$\sigma,\tau\in\\{+,-,0\\}^{E}$ is defined by

$(\sigma\circ\tau)_{e}:=\begin{cases}\sigma_{e}&\text{ if }\sigma_{e}\neq
0,\\\ \tau_{e}&\text{ else},\end{cases}$

and their _separating set_ $S(\sigma,\tau)$ is defined as

$S(\sigma,\tau):=\\{e\in E\mid\sigma_{e}=-\tau_{e}\neq 0\\}.$

If $A\subseteq E$ and $\sigma\in\\{+,-,0\\}^{E}$ we write
$\sigma|_{A}:=(\sigma_{e}\mid e\in A)\in\\{+,-,0\\}^{A}$ for its restriction
to $A$.

######  Definition 2.1.

An _oriented matroid_ $\mathcal{M}=(E,\mathcal{L})$ is given by a finite
ground set $E$ and a subset of sign vectors
$\mathcal{L}\subseteq\\{+,-,0\\}^{E}$ called _covectors_ of $\mathcal{M}$,
subject to satisfying the following axioms:

  1. (1)

$\mathbf{0}:=(0,0,\ldots,0)\in\mathcal{L}$,

  2. (2)

if $\sigma\in\mathcal{L}$, then $-\sigma\in\mathcal{L}$,

  3. (3)

if $\sigma,\tau\in\mathcal{L}$, then $\sigma\circ\tau\in\mathcal{L}$,

  4. (4)

if $\sigma,\tau\in\mathcal{L}$ and $e\in S(\sigma,\tau)$ there exists an
$\eta\in\mathcal{L}$ such that $\eta_{e}=0$ and
$\eta_{f}=(\sigma\circ\tau)_{f}=(\tau\circ\sigma)_{f}$ for all $f\in
E\setminus S(\sigma,\tau)$.

Let $\leq$ be the partial order on the set $\\{+,-,0\\}$ defined by $0<+$,
$0<-$, with $+$ and $-$ incomparable. This induces the product partial order
on $\\{+,-,0\\}^{E}$ in which sign vectors are compared component wise. The
_face poset_ or covector poset $(\mathcal{L},\leq)$ of $\mathcal{M}$ is
obtained by considering the induced partial order on the subset
$\mathcal{L}\subseteq\\{+,-,0\\}^{E}$.

The _rank_ of $\mathcal{M}$ is defined as the length of a maximal chain in
$\mathcal{L}$.

The maximal elements in $\mathcal{L}$ are called _topes_ ; they are denoted by
$\mathcal{T}=\mathcal{T}(\mathcal{M})$.

Note that this is only one of several equivalent ways to define oriented
matroids, cf. [BLVS+99, Ch. 3].

An element $e\in E$ is called a _loop_ of the oriented matroid
$\mathcal{M}=(E,\mathcal{L})$ if $\sigma_{e}=0$ for all
$\sigma\in\mathcal{L}$. Two elements $e,f\in E$ which are not loops are
_parallel_ if $\sigma_{e}=0$ implies $\sigma_{f}=0$ for all
$\sigma\in\mathcal{L}$.

In the rest of the sequel, without loss of generality, we always assume an
oriented matroid to be simple, i.e. it contains no loops or parallel elements.
This is justified by the fact that the covector poset of an oriented matroid
and its simplification are isomorphic, cf. [BLVS+99, Lem. 4.1.11].

We have the following principal examples of oriented matroids.

######  Example 2.2.

Let $\mathcal{A}$ be an arrangement of hyperplanes in a real vector space
$V\simeq\mathbb{R}^{\ell}$. For $H\in\mathcal{A}$ choose defining linear forms
$\alpha_{H}\in V^{*}$ with $\ker(\alpha_{H})=H$. Then
$\mathcal{M}(\mathcal{A})=(\mathcal{A},\mathcal{L}(\mathcal{A}))$ with

$\mathcal{L}(\mathcal{A}):=\\{(\operatorname{sgn}(\alpha_{H}(v))\mid
H\in\mathcal{A})\mid v\in V\\}\subseteq\\{+,-,0\\}^{\mathcal{A}}$

is an oriented matroid.

We recall the definition of an affine oriented matroid.

######  Definition 2.3.

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid and $g\in E$. Define
$\mathcal{M}_{\operatorname{aff},g}:=(E,\mathcal{L}^{+},g)$ by

$\mathcal{L}^{+}:=\\{\sigma\in\mathcal{L}\mid\sigma_{g}=+\\}.$

Then $\mathcal{M}_{\operatorname{aff},g}$ is called an _affine_ oriented
matroid or the _decone_ of $\mathcal{M}$ with respect to $g$.

######  Remark 2.4.

For our purposes, oriented matroids are best thought of as special instances
of regular cell complexes. This is justified by the topological representation
theorem due to Folkman and Lawrence [FL78], see also [BLVS+99, Thm. 5.2.1].
Its core implication that the reduced covector poset
$\mathcal{L}\setminus\\{\operatorname{\mathbf{0}}\\}$ is the face poset of a
regular cell decomposition of a sphere induced by an arrangement of tamely
embedded codimension one subspheres suffices for our study, see also Theorem
2.21. Moreover, $\mathcal{L}\setminus\\{\operatorname{\mathbf{0}}\\}$ is a PL-
sphere, so $\mathcal{L}^{\vee}$ represents a regular cell decomposition of a
ball, cf. [BLVS+99, Prop. 4.7.26].

There is another poset associated to an oriented matroid
$\mathcal{M}=(E,\mathcal{L})$ which is essential for our considerations.

######  Definition 2.5.

The _geometric lattice_ $L(\mathcal{M})$ of $\mathcal{M}$ is defined as

$L(\mathcal{M}):=\\{z(\sigma)\mid\sigma\in\mathcal{L}\\}\subseteq 2^{E},$

with partial order given by inclusion. Elements of $L(\mathcal{M})$ are called
_flats_. For two elements $X,Y\in L(\mathcal{M})$, their join is $X\vee
Y=\inf\\{Z\in L(\mathcal{M})\mid X\cup Y\subseteq Z\\}$, their meet is
$X\wedge Y=X\cap Y$.

Moreover, for $X,Y\in L=L(\mathcal{M})$ we write $L_{X}:=L_{\leq X}$ for the
lattice of the _localization_ at $X$, $L^{Y}:=L_{\geq Y}$ for the lattice of
the _contraction_ to $Y$ and $L_{X}^{Y}:=[X,Y]:=L_{\leq X}\cap L_{\geq Y}$ for
an interval.

The lattice $L=L(\mathcal{M})$ is graded by _rank_ , i.e. the rank of $X\in L$
is the length of a maximal chain in $L_{\leq X}$.

Note that the map $z:\mathcal{L}\to L$ is a cover and rank preserving, order
reversing surjection [BLVS+99, Prop. 4.1.13]. The associated order preserving
map $\mathcal{L}^{\vee}\to L$ is also denoted by $z$.

######  Definition 2.6 (Restriction, contraction, and localization).

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid.

  1. (1)

For a subset $A\subseteq E$ the _restriction_
$\mathcal{M}|_{A}=(A,\mathcal{L}|_{A})$ of $\mathcal{M}$ to $A$ is defined by

$\mathcal{L}|_{A}:=\\{\sigma|_{A}\mid\sigma\in\mathcal{L}\\}.$

  2. (2)

For $X\subseteq E$ the _contraction_ $\mathcal{M}/X=(E\setminus
X,\mathcal{L}/X)$ is defined by

$\mathcal{L}/X:=\\{\sigma|_{E\setminus X}\mid\sigma\in\mathcal{L}\text{ with
}X\subseteq z(\sigma)\\}.$

  3. (3)

For a flat $X\in L(\mathcal{M})$ the _localization_ of $\mathcal{M}$ at $X$ is
$\mathcal{M}_{X}:=\mathcal{M}|_{X}$. The natural projection of $\mathcal{L}$
to $\mathcal{L}_{X}:=\mathcal{L}|_{X}$ is denoted by
$\rho_{X}:\mathcal{L}\to\mathcal{L}_{X},\sigma\mapsto\sigma|_{X}$.

######  Remark 2.7.

For the covectors of the contraction $\mathcal{M}/X$ we also use the notation
$\mathcal{L}^{X}:=\mathcal{L}/X$ where $L^{X}=\\{Y\in L(\mathcal{M})\mid Y\geq
X\\}$. We often identify $\mathcal{L}^{X}$ with $z^{-1}(L^{X})$ by the obvious
isomorphism $z^{-1}(L^{X})\simeq\mathcal{L}^{X}$. Therefor, we write
$\mathcal{L}^{X}\subseteq\mathcal{L}$ or
$\mathcal{L}^{X}\hookrightarrow\mathcal{L}$.

If $L_{X}^{Y}=[Y,X]\subseteq L(\mathcal{M})$ we frequently write
$\mathcal{L}_{X}^{Y}:=\mathcal{L}_{X}/Y$.

######  Remark 2.8.

Let $X\in L$ and let $\rho_{X}:\mathcal{L}\to\mathcal{L}_{X}$ be the
localization map. Then for any $\alpha\in\mathcal{L}$ with $z(\alpha)=X$ we
have a section $\iota_{\alpha}:\mathcal{L}_{X}\to\mathcal{L}$ to $\rho_{X}$
which is defined by:

$\iota_{\alpha}(\sigma)_{e}:=\begin{cases}\sigma_{e}&\text{if }e\in X,\\\
\alpha_{e}&\text{else}.\end{cases}$

An easy calculation shows that both the localization map $\rho_{X}$ and the
section $\iota_{\alpha}$ preserve the composition of covectors, i.e. we have
$\rho_{X}(\sigma\circ\tau)=\rho_{X}(\sigma)\circ\rho_{X}(\tau)$ for all
$\sigma,\tau\in\mathcal{L}$, and
$\iota_{\alpha}(\sigma^{\prime}\circ\tau^{\prime})=\iota_{\alpha}(\sigma^{\prime})\circ\iota_{\alpha}(\tau^{\prime})$
for all $\sigma^{\prime},\tau^{\prime}\in\mathcal{L}_{X}$.

We recall the notion of an extension of an oriented matroid.

######  Definition 2.9.

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid. Then another
oriented matroid $\widetilde{\mathcal{M}}=(E^{\prime},\mathcal{L}^{\prime})$
with $E\subseteq E^{\prime}$ is called an _extension_ of $\mathcal{M}$ if
$\widetilde{\mathcal{M}}|_{E}=\mathcal{M}$.

If $\widetilde{\mathcal{M}}$ is an extension of $\mathcal{M}$, then we have a
natural injective map $L(\mathcal{M})\hookrightarrow
L(\widetilde{\mathcal{M}}),X\mapsto\widetilde{X}$, where
$\widetilde{X}=\inf\\{X^{\prime}\in L(\widetilde{\mathcal{M}})\mid X\subseteq
X^{\prime}\\}$.

###  2.3. Modular flats and supersolvable geometric lattices

For this section, let $L=L(\mathcal{M})$ be the geometric lattice of a simple
oriented matroid $\mathcal{M}=(E,\mathcal{L})$. The notion of a modular flat
in a geometric lattice was introduced by Stanley [Sta71].

######  Definition 2.10.

Let $X\in L$. Then $X$ is _modular_ if for all $Y,Z\in L$ with $Z\leq Y$ we
have

$Z\vee(X\wedge Y)=(Z\vee X)\wedge Y.$

The following notion is also due to Stanley [Sta72].

######  Definition 2.11.

The geometric lattice $L$ is called _supersolvable_ if there is a maximal
chain $X_{0}<X_{1}<\ldots<X_{r}$ of flats in $L$ such that $X_{i}$ is modular
for all $0\leq i\leq r$.

In this case, we call $\mathcal{M}$ itself supersolvable.

Supersolvable oriented matroids of rank $3$ are easily describes by the
following lemma, the proof of which is left to the reader.

######  Lemma 2.12.

Assume $\mathcal{M}$ to be of rank $3$. Then $\mathcal{M}$ is supersolvable if
and only if there is a flat $X\in L$ of rank $2$ such that for all flats $Y\in
L$ of rank $2$ we have $X\cap Y\neq\emptyset$.

Modular flats have the following property (which is actually equivalent to
modularity).

######  Lemma 2.13 ([Bry75, Thm. 3.3]).

Let $X\in L$ be a modular flat. Then for $Y\in L$ the map $p_{X}:[Y,X\vee
Y]\to[X\wedge Y,X],Z\mapsto Z\wedge X$ is a poset isomorphism with inverse
$s_{Y}:[X\wedge Y,X]\to[Y,X\vee Y],W\mapsto W\vee Y$.

###  2.4. Topes and convex subsets

Recall that we denote the topes of an oriented matroid
$\mathcal{M}=(E,\mathcal{L})$, i.e. the maximal elements of $\mathcal{L}$ by
$\mathcal{T}$. As remarked earlier, without loss of generality, we assume that
$\mathcal{M}$ is simple.

######  Definition 2.14.

Let $B\in\mathcal{T}$. We define a partial order on $\mathcal{T}$ by

$R\leq T:\iff S(B,R)\subseteq S(B,T).$

The resulting ranked poset $\mathcal{T}(\mathcal{M},B)$ with rank function
$\operatorname{rk}(T):=|S(B,T)|$ is called the _tope poset_ with respect to
$B$.

######  Definition 2.15.

Let $\mathcal{T}$ be the topes of an oriented matroid
$\mathcal{M}=(E,\mathcal{L})$.

  1. (i)

Let $T,R\in\mathcal{T}$ be two topes. Then the _distance_
$\operatorname{dist}(T,R)$ of $T$ and $R$ is defined as
$\operatorname{dist}(T,R):=|S(T,R)|$.

  2. (ii)

Let $e\in E$. Denote by $\mathcal{T}_{e}^{+}:=\\{T\in\mathcal{T}\mid
T_{e}=+\\}\subseteq\mathcal{T}$ the _positive halfspace_ with respect to $e$.
The _negative halfspace_ $\mathcal{T}_{e}^{-}\subseteq\mathcal{T}$ is defined
analogously. Set
$\mathfrak{H}=\mathfrak{H}(\mathcal{M})=\\{T_{e}^{\epsilon}\mid e\in
E,\epsilon\in\\{+,-\\}\\}$, and an $\mathcal{H}\in\mathfrak{H}$ is called a
_halfspace_ of $\mathcal{M}$.

  3. (iii)

Let $\mathcal{Q}\subseteq\mathcal{T}$ be a subset of topes. Then the _convex
hull_ $\operatorname{conv}(\mathcal{Q})$ of $\mathcal{Q}$ is defined as the
intersection of all halfspaces containing $\mathcal{Q}$:

$\operatorname{conv}(\mathcal{Q})=\bigcap\limits_{\mathcal{Q}\subseteq\mathcal{H},\mathcal{H}\in\mathfrak{H}}\mathcal{H}.$

  4. (iv)

A subset of topes $\mathcal{Q}\subseteq\mathcal{T}$ is called _convex_ if
$\operatorname{conv}(\mathcal{Q})=\mathcal{Q}$.

  5. (v)

For a subset $\mathcal{Q}\subseteq\mathcal{T}$ we set
$\mathcal{L}(\mathcal{Q}):=\\{\sigma\in\mathcal{L}\mid\sigma\leq T$ for some
$T\in\mathcal{Q}\\}$ and
$\mathcal{L}^{\vee}[\mathcal{Q}]:=\\{\sigma\in\mathcal{L}^{\vee}\mid\mathcal{T}(\sigma)\subseteq\mathcal{Q}\\}$,
where
$\mathcal{T}(\sigma)=\\{T\in\mathcal{T}\mid\sigma\leq_{\mathcal{L}}T\\}$.

######  Remark 2.16.

Note that the definition of
$\mathcal{L}^{\vee}[\mathcal{Q}]\subseteq\mathcal{L}^{\vee}$ yields a
subcomplex since $\mathcal{L}^{\vee}$ has the _intersection property_ , i.e. a
face is uniquely determined by its vertices. Furthermore, an easy calculation
shows that
$(\mathcal{L}\setminus\mathcal{L}(\mathcal{T}\setminus\mathcal{Q}))^{\vee}=\mathcal{L}^{\vee}[\mathcal{Q}]$.

We record the following alternative characterization of convex subsets.

######  Lemma 2.17.

[BLVS+99, Prop. 4.2.6] Let $\mathcal{Q}\subseteq\mathcal{T}$. Then
$\mathcal{Q}$ is convex if and only if $T,R\in\mathcal{Q}$ and
$Q\in\mathcal{T}$ with
$\operatorname{dist}(T,Q)+\operatorname{dist}(Q,R)=\operatorname{dist}(T,R)$
implies $Q\in\mathcal{Q}$.

Two easy but useful facts are provided by the following lemmas.

######  Lemma 2.18.

Let $\mathcal{Q}\subseteq\mathcal{T}$ be a convex subset of topes. Then for a
tope $B\in\mathcal{Q}$ there is a linear extension $\operatorname{\dashv}$ of
$\mathcal{T}(\mathcal{M},B)$ such that with respect to $\operatorname{\dashv}$
all the topes in $\mathcal{Q}$ come first.

###### Proof.

The convexity of $\mathcal{Q}$ and Lemma 2.17 imply that $\mathcal{Q}$ forms
an order ideal in $\mathcal{T}(\mathcal{M},B)$ for every $B\in\mathcal{Q}$,
cf. [BLVS+99, pp. 172ff.]. This, in turn, implies the statement. ∎

We consider the localization map
$\rho_{X}:\mathcal{L}^{\vee}\to\mathcal{L}_{X}^{\vee}$. For the subset of
topes (vertices of $\mathcal{L}^{\vee}$) in the poset fiber
$(\rho_{X}\downarrow\sigma)$ we write $\mathcal{T}(\rho_{X}\downarrow\sigma)$.

######  Lemma 2.19.

Let $\sigma$ in $\mathcal{L}_{X}^{\vee}$. Then
$\mathcal{T}(\rho_{X}\downarrow\sigma)$ is a convex subset of $\mathcal{T}$.

###### Proof.

We have $\mathcal{T}(\rho_{X}\downarrow\sigma)=\bigcap\limits_{e\in X\setminus
z(\sigma)}\mathcal{T}_{e}^{\sigma_{e}}$. ∎

###  2.5. Shellability and subcomplexes

We briefly elaborate on shellability of regular cell complexes, cf. [BLVS+99,
App. 4.7].

######  Definition 2.20.

Let $\Sigma$ be a pure $d$-dimensional regular cell complex. A linear ordering
$\sigma_{1},\sigma_{2},\ldots,\sigma_{t}$ of its maximal cells is called a
_shelling_ if either $d=0$, or $d\geq 1$ and the following conditions are
satisfied:

  1. (i)

$\delta\sigma_{j}\cap(\bigcup_{i=1}^{j-1}\delta\sigma_{i})$ is pure of
dimension $d-1$ for $2\leq j\leq t$,

  2. (ii)

$\delta\sigma_{j}$ has a shelling in which the $(d-1)$-cells of
$\delta\sigma_{j}\cap(\bigcup_{i=1}^{j-1}\delta\sigma_{i})$ come first for
$2\leq j\leq t$,

  3. (iii)

$\delta\sigma_{1}$ has a shelling.

A complex which admits a shelling is called _shellable_.

For our investigation, the principal example of a shellable regular cell
complex is the covector complex $\mathcal{L}$ of an oriented matroid. More
precisely, we have the following theorem.

######  Theorem 2.21 ([BLVS+99, Thm. 4.3.3]).

The reduced covector poset
$\mathcal{L}\setminus\\{\operatorname{\mathbf{0}}\\}$ of an oriented matroid
of rank $r$ is isomorphic to the face poset of a shellable regular cell
decomposition of the $(r-1)$-sphere. Furthermore, every linear extension of
$\mathcal{T}(\mathcal{M},B)$ is a shelling of
$\mathcal{L}\setminus\\{\operatorname{\mathbf{0}}\\}$.

We list the following useful fact, cf. [BLVS+99, Prop. 4.7.26].

######  Proposition 2.22.

Let $\Sigma$ be a regular cell decomposition of the $d$-sphere which has a
shelling $\sigma_{1},\ldots,\sigma_{t}$. Then for $k<t$ the subcomplex
$\Sigma(\\{\sigma_{1},\ldots,\sigma_{k}\\})$ is a shellable $d$-ball with
shelling $\sigma_{1},\ldots,\sigma_{k}$.

With the preceding results we can delineate the following property of
subcomplexes generated by subsets of the topes
$\mathcal{T}=\mathcal{T}(\mathcal{M})$ of an oriented matroid $\mathcal{M}$.

######  Corollary 2.23.

Let $\emptyset\neq\mathcal{Q}\subseteq\mathcal{T}$ be a convex subset of
topes. Then both the subcomplexes $\mathcal{L}(\mathcal{Q})$ and
$\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$ are shellable balls.

###### Proof.

Let $B\in\mathcal{Q}$. By Lemma 2.18 there is a linear extension of
$\mathcal{T}(\mathcal{M},B)$ such that all the topes in $\mathcal{Q}$ come
first. Conversely, with respect to the opposite linear extension of
$\mathcal{T}(\mathcal{M},-B)$ all the topes in
$\mathcal{T}\setminus\mathcal{Q}$ come first. This, in turn, yields the
statement with Theorem 2.21 and Proposition 2.22. ∎

###  2.6. The Salvetti complex of an oriented matroid

We start by defining a certain poset which was first constructed by Salvetti
[Sal87] for an oriented matroid $\mathcal{M}(\mathcal{A})$ associated to a
real hyperplane arrangement $\mathcal{A}$. Gel’fand and Rybnikov [GR90]
observed, that the same construction applies more generally to oriented
matroids.

######  Definition 2.24.

Let $\mathcal{L}$ be the poset of covectors of an oriented matroid
$\mathcal{M}$. Then the _Salvetti poset_
$\mathcal{S}=\mathcal{S}(\mathcal{M})$ is defined as

$\mathcal{S}:=\\{(\sigma,T)\mid T\in\mathcal{T}\text{ and
}\sigma\in\mathcal{L}_{\leq T}\\}\subseteq\mathcal{L}\times\mathcal{T},$

with partial order

$(\sigma,T)\leq_{\mathcal{S}}(\tau,R):\iff\sigma\geq_{\mathcal{L}}\tau\text{
and }\sigma\circ R=T.$

The _affine_ Salvetti poset $\mathcal{S}_{\operatorname{aff},g}$ of the affine
oriented matroid $\mathcal{M}_{\operatorname{aff},g}$ is defined as
$\mathcal{S}_{\operatorname{aff},g}:=\\{(\sigma,T)\in\mathcal{S}\mid\sigma_{g}=+,T_{g}=+\\}$.

The following theorem due to Salvetti [Sal87] is a fundamental result in the
homotopy theory of complements of complexified real arrangements.

######  Theorem 2.25 ([Sal87]).

The Salvetti poset is the face poset of a regular cell complex. If
$\mathcal{M}=\mathcal{M}(\mathcal{A})$ is the oriented matroid associated to a
hyperplane arrangement $\mathcal{A}$ in a real vector space $V$, then the
Salvetti complex is homotopy equivalent to the complement of the complexified
arrangement:

$\mathcal{S}(\mathcal{M})\cong
V\otimes\mathbb{C}\setminus\left(\bigcup\limits_{H\in\mathcal{A}}H\otimes\mathbb{C}\right).$

Let $X\in L(\mathcal{M})$. The projection map
$\rho_{X}:\mathcal{L}\to\mathcal{L}_{X}$ respects the composition of covectors
by Remark 2.8. Hence, $\rho_{X}$ induces an order preserving map

$\widetilde{\rho}_{X}:\mathcal{S}=\mathcal{S}(\mathcal{M})\to\mathcal{S}(\mathcal{M}_{X})=:\mathcal{S}_{X},(\sigma,T)\mapsto(\rho_{X}(\sigma),\rho_{X}(T))$

between the Salvetti complexes. Moreover, for each $\alpha\in\mathcal{L}$ with
$z(\alpha)=X$ we get a section $\iota_{\alpha}:\mathcal{L}_{X}\to\mathcal{L}$
to $\rho_{X}$ by Remark 2.8 which respects the composition of covectors.
Hence, this also yields a section

$\widetilde{\iota}_{\alpha}:\mathcal{S}_{X}\to\mathcal{S},(\sigma^{\prime},T^{\prime})\mapsto(\iota_{\alpha}(\sigma^{\prime}),\iota_{\alpha}(T^{\prime}))$

to $\widetilde{\rho}_{X}$ and provides a simple argument for the following
lemma.

######  Lemma 2.26.

If $\mathcal{S}$ is a $K(\pi,1)$ complex then so is
$\mathcal{S}_{X}=\mathcal{S}(\mathcal{M}_{X})$ for all $X\in L(\mathcal{M})$.

###### Proof.

Application of the homotopy group functors directly yields that the map
$\pi_{i}(\widetilde{\rho}_{X}):\pi_{i}(\mathcal{S})\to\pi_{i}(\mathcal{S}_{X})$
has a section $\pi_{i}(\widetilde{\iota}_{\alpha})$ and thus is surjective for
all $i$. ∎

The realizable analogue of the previous result was first observed by Oka for
complements of complex hyperplane arrangements, cf. [Par93, Lem. 1.1].
Corollary 6.5 gives an inverse to Lemma 2.26 for a modular flat $X$ of corank
one.

The subcomplex of $\mathcal{S}$ consisting of all the faces of a maximal cell
can be identified with the dual covector complex.

######  Lemma 2.27.

Let $(\operatorname{\mathbf{0}},T)\in\mathcal{S}$ be a maximal element of
$\mathcal{S}$. Then
$\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T)}\simeq\mathcal{L}^{\vee}$.

###### Proof.

The maps
$\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T)}\to\mathcal{L}^{\vee},(F,R)\mapsto
F$ and
$\mathcal{L}^{\vee}\to\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T)},F\mapsto(F,F\circ
T)$ are mutually inverse and order preserving. ∎

${\mathcal{S}}$${\mathcal{S}_{X}}$${\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T)}}$${(\mathcal{S}_{X})_{\leq(\operatorname{\mathbf{0}},\rho_{X}(T))}}$${\mathcal{L}^{\vee}}$${\mathcal{L}_{X}^{\vee}}$$\scriptstyle{\widetilde{\rho}_{X}}$$\scriptstyle{\widetilde{\rho}_{X}|}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\rho_{X}}$
Figure 1. Localization maps restricted to subcomplexes of maximal cells are
localization maps of the dual covector complexes.

######  Corollary 2.28.

The diagram shown in Figure 1 commutes.

###  2.7. Discrete Morse theory

We review some basic concepts from Forman’s discrete Morse theory [For98] from
the viewpoint of acyclic matchings on posets respectively regular cell
complexes adopted by Chari [Cha00]. For a comprehensive exposition of the
theory we refer to Kozlov’s book [Koz08].

######  Definition 2.29.

Let $P=(P,\leq)$ be a poset. We associate a directed graph
$G(\Sigma)=(\mathcal{V},\mathcal{E})$ with vertex set $\mathcal{V}=P$ the
ground set of $P$ and directed edges $\mathcal{E}:=\\{(a,b)\mid a\lessdot
b\\}$ its cover relations, i.e. the Hasse diagram of the poset.

A subset $\operatorname{\underline{M}}\subseteq\mathcal{E}$ is called a
_matching_ on $P$ if every element of $P$ appears in at most one
$(a,b)\in\operatorname{\underline{M}}$.

Let $G(P,\operatorname{\underline{M}})=(\mathcal{V},\mathcal{E}^{\prime})$ be
the new directed graph with
$\mathcal{E}^{\prime}:=\mathcal{E}\setminus\operatorname{\underline{M}}\cup\\{(b,a)\mid(a,b)\in\operatorname{\underline{M}}\\}$,
i.e. the directed graph constructed from $G(P)$ by inverting all edges in
$\operatorname{\underline{M}}$. A matching $\operatorname{\underline{M}}$ on
$\Sigma$ is called _acyclic_ if $G(P,\operatorname{\underline{M}})$ has no
directed cycles.

For an acyclic matching $\operatorname{\underline{M}}$ on $P$ its _critical
elements_ $C(\operatorname{\underline{M}})\subseteq P$ are defined as

$C(\operatorname{\underline{M}}):=\\{a\in P\mid a\notin e\text{ for all
}e\in\operatorname{\underline{M}}\\}.$

In the same way we define acyclic matchings for a regular cell complex
$\Sigma$ (which is identified with its poset of faces).

Note that each acyclic matching $\operatorname{\underline{M}}$ on a poset $P$
gives an acyclic matching $\operatorname{\underline{M}}^{\vee}$ on the dual
poset $P^{\vee}$ with
$C(\operatorname{\underline{M}})=C(\operatorname{\underline{M}}^{\vee})$.

We record the following special case of the main theorem of discrete Morse
theory, cf. [For98], [Cha00], [Koz08, Ch. 11].

######  Theorem 2.30.

Let $\Sigma$ be a regular cell complex and let $\Gamma\subseteq\Sigma$ be a
subcomplex. Assume that $\operatorname{\underline{M}}$ is an acyclic matching
on $\Sigma$ with $C(\operatorname{\underline{M}})=\Gamma$. Then $\Gamma$ is a
strong deformation retract of $\Sigma$. In particular, the inclusion
$\Gamma\hookrightarrow\Sigma$ is a homotopy equivalence.

The next proposition, due to Chari [Cha00], connects shellability with
discrete Morse theory. Another, more explicit description of acyclic matchings
for shellable regular cell complexes is due to Delucchi [Del08].

######  Proposition 2.31 ([Cha00, Prop. 4.1]).

Let $\Sigma$ be a regular cell decomposition of the $d$-ball with shelling
$\sigma_{1},\ldots,\sigma_{t}$. Further, let $v\in\sigma_{1}$ be a vertex.
Then there is an acyclic matching $\operatorname{\underline{M}}$ on $\Sigma$
with $C(\operatorname{\underline{M}})=\\{v\\}$, i.e. $\Sigma$ collapses onto
$v$.

We conclude by recalling the following useful tool to construct acyclic
matchings on posets, and hence on cell complexes.

######  Theorem 2.32 (Patchwork theorem [Koz08, Thm. 11.10]).

Let $f:P\to Q$ be a poset map. Assume that for all $q\in Q$ we have an acyclic
matching $\operatorname{\underline{M}}(q)$ on $f^{-1}(q)$. Then
$\operatorname{\underline{M}}=\bigcup\limits_{q\in
Q}\operatorname{\underline{M}}(q)$ is an acyclic matching on $P$.

##  3\. Combinatorics of modular flats of oriented matroids

${\mathcal{L}^{\vee}}$${\mathcal{L}_{X}^{\vee}}$${\sigma}$${\sigma|_{X}}$${L}$${L_{X}}$${z(\sigma)}$${z(\sigma)\wedge
X=z(\sigma|_{X})}$$\scriptstyle{\rho_{X}}$$\scriptstyle{z}$$\scriptstyle{z}$$\scriptstyle{\rho_{X}}$$\scriptstyle{z}$$\scriptstyle{z}$$\scriptstyle{p_{X}}$$\scriptstyle{p_{X}}$
Figure 2. Restriction maps of face posets and corresponding geometric
lattices.

We study the properties of modular flats of oriented matroids in more detail.
The basic idea is to use the commutative square of poset maps depicted in
Figure 2 to derive refined statements about covector posets from the
corresponding statements about geometric lattices.

Let $L=L(\mathcal{M})$ be the geometric lattice of a simple oriented matroid
and $\mathcal{L}$ its covector poset. Consider the localization map
$\rho_{X}:\mathcal{L}^{\vee}\to\mathcal{L}_{X}^{\vee}$ for the dual covector
complexes for an $X\in L$. We state a lemma which connects the notion of a
modular flat of corank one to special subsets of topes.

######  Lemma 3.1.

Let $X\in L$ be modular of corank $1$ and $B^{\prime}\in\mathcal{T}_{X}$. Then
there are exactly two topes $B_{1},B_{2}\in(\rho_{X}\downarrow B^{\prime})$
such that the restricted poset
$\mathcal{T}(\mathcal{M},B_{i})|_{\mathcal{T}(\rho_{X}\downarrow B^{\prime})}$
is linearly ordered.

In particular, for each
$T\in\mathcal{T}(\mathcal{M},B_{i})|_{\mathcal{T}(\rho_{X}\downarrow
B^{\prime})},T\neq B_{i}$ there is a unique
$T^{\prime}\in\mathcal{T}(\rho\downarrow B^{\prime})$ with $T^{\prime}\lessdot
T$ and $S(T^{\prime},T)=\\{e\\}$ such that $e\in S(R,T)$ for each $R<T$,
$R\in\mathcal{T}(\mathcal{M},B_{i})$.

###### Proof.

Since $X$ has corank $1$, there are exactly two covectors in $z^{-1}(X)$, say
$\alpha$ and $\beta=-\alpha$. Let $\iota_{\alpha},\iota_{\beta}$ be the
associated sections to $\rho_{X}$ from Remark 2.8. Set
$B_{1}:=\iota_{\alpha}(B^{\prime})$ and $B_{2}:=\iota_{\beta}(B^{\prime})$.
Then the modularity of $X$ forces every cell in $(\rho_{X}\downarrow
B^{\prime})$ to have dimension $\leq 1$. Otherwise, there would be a cell
$\tau\in(\rho_{X}\downarrow B^{\prime})$ with $\rho_{X}(\tau)=B^{\prime}$ and
$Y=z(\tau)\in L$ a flat of rank $2$ with $Y\wedge
X=z(\rho_{X}(\tau))=z(B^{\prime})=\emptyset$, contradicting the modularity of
$X$. It follows that $(\rho_{X}\downarrow B^{\prime})$ is a string with end
vertices $B_{1},B_{2}$. Thus, the two topes $B_{1},B_{2}$ have the desired
property which yields both statements. ∎

$\alpha$$\beta$$T_{2}$$T_{1}$$T_{0}$$\rho_{X}$$B^{\prime}$$H_{4}$$H_{2}$$H_{3}$$H_{5}$$H_{1}$
Figure 3. A linear order $\mathcal{T}(\rho_{X}\downarrow
B^{\prime})=\\{T_{0},T_{1},T_{2}\\}$ as described in Lemma 3.1.

We give an example to illustrate Lemma 3.1.

######  Example 3.2.

Consider the hyperplane arrangement $\mathcal{A}$ in $\mathbb{R}^{3}$
consisting of the hyperplanes

$\displaystyle\mathcal{A}=\\{$ $\displaystyle
H_{1}=\ker(x),H_{2}=\ker(y),H_{3}=\ker(x+y),$ $\displaystyle
H_{4}=\ker(z),H_{5}=\ker(x+z)\\}$

and $\mathcal{M}=\mathcal{M}(\mathcal{A})$ the associated oriented matroid
with topes $\mathcal{T}$. Then the flat $X=\\{H_{1},H_{2},H_{3}\\}\in
L(\mathcal{M})$ is modular. A stereographic projection of $\mathcal{A}$ (or
rather $\mathcal{A}\cap S^{2}$) is shown on the left hand side of Figure 3. On
the right hand side of Figure 3 we see its localization at $X$.

Then the topes in the fiber of $B^{\prime}\in\mathcal{T}_{X}$ are
$\mathcal{T}(\rho_{X}\downarrow B^{\prime})=\\{T_{0},T_{1},T_{2}\\}$ with
$T_{0}=\iota_{\alpha}(B^{\prime})$ and $T_{2}=\iota_{\beta}(B^{\prime})$.
Indeed, $\mathcal{T}(\mathcal{M},T_{0})|_{\mathcal{T}(\rho_{X}\downarrow
B^{\prime})}$ is linearly ordered as $T_{0}<T_{1}<T_{2}$. Furthermore,
$S(T_{1},T_{2})=\\{H_{5}\\}\subseteq S(T_{0},T_{2})=\\{H_{4},H_{5}\\}$. So
$H_{5}\in\mathcal{A}$ satisfies the second statement of Lemma 3.1.

Now, we consider the primal covector complex $\mathcal{L}$.

######  Lemma 3.3.

Let $X\in L$ be a modular flat and $\sigma,\tau\in\mathcal{L}$ with
$z(\sigma)=z(\tau)=Y\in L$ and $\sigma|_{X}=\tau|_{X}$. Then $\sigma|_{X\vee
Y}=\tau|_{X\vee Y}$.

###### Proof.

Suppose $\sigma|_{X\vee Y}\neq\tau|_{X\vee Y}$, i.e. there is an $e\in X\vee
Y$ such that $\sigma_{e}\neq\tau_{e}\neq 0$. So $e\in S(\sigma,\tau)\cap X\vee
Y$. By the fourth covector axiom (see Definition 2.1) there is a
$\gamma\in\mathcal{L}$ with $\gamma_{e}=0$ and
$\gamma_{f}=(\sigma\circ\tau)_{f}=(\tau\circ\sigma)_{f}=\sigma_{f}=\tau_{f}$
for all $f\in X\vee Y\setminus S(\sigma,\tau)$. In particular
$\gamma|_{X}=\tau|_{X}=\sigma|_{X}$ and $z(\sigma)=Y\subseteq z(\gamma)$. Set
$Z:=z(\gamma)$. Then $Z\in[Y,X\vee Y]$ with $Z>Y$. But we also have $Z\wedge
X=Y\wedge X$ by construction. Thanks to Lemma 2.13, we get $Z=(Z\wedge X)\vee
Y=(Y\wedge X)\vee Y=Y$, contradicting $Z>Y$. ∎

${L_{X\vee Y}}$${L_{X}}$${\mathcal{L}_{X\vee
Y}}$${\mathcal{L}_{X}}$${\mathcal{L}^{Y}_{X\vee Y}}$${\mathcal{L}_{X}^{X\wedge
Y}}$${L^{Y}_{X\vee Y}}$${L_{X}^{X\wedge
Y}}$$\scriptstyle{p_{X}}$$\scriptstyle{z}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\iota_{\alpha},\iota_{\beta}}$$\scriptstyle{z}$$\scriptstyle{\overline{\rho}_{X}=\rho_{X}|_{\mathcal{L}^{Y}_{X\vee
Y}}}$$\scriptstyle{z}$$\scriptstyle{\overline{\iota}}$$\scriptstyle{z}$$\scriptstyle{p_{X}|\quad\simeq}$$\scriptstyle{s_{Y}}$
Figure 4. Proof of Lemma 3.5.

######  Proposition 3.4.

Let $X,Y\in L(\mathcal{M})=L$ and let $\sigma\in\mathcal{L}_{X}^{X\wedge
Y}\subseteq\mathcal{L}_{X}$. Then for a minimal element
$\widehat{\sigma}\in\rho_{X}^{-1}(\sigma)\cap\mathcal{L}^{Y}$ we have
$z(\widehat{\sigma})=z(\sigma)\vee_{L}Y$.

###### Proof.

If $Y\subseteq X$ then the statement is trivial. So assume $Y\nsubseteq X$.
Let $\alpha\in\mathcal{L}$ with $z(\alpha)=X$ and set
$\beta=-\alpha\in\mathcal{L}$. The maps
$\iota_{\alpha},\iota_{\beta}:\mathcal{L}_{X}\to\mathcal{L}$ from Remark 2.8
give sections to $\rho_{X}$. Note that for all $\sigma\in\mathcal{L}_{X}$ we
have
$S(\iota_{\alpha}(\sigma),\iota_{\beta}(\sigma))=S(\alpha,\beta)=E\setminus
X$. Since $Y\setminus X=Y\setminus X\wedge Y\subseteq S(\alpha,\beta)$, the
oriented matroid axiom (4) in Definition 2.1 implies the statement. ∎

With the help of the previous proposition and Lemma 3.3 we can give the
following refinement of Lemma 2.13 to covector posets of oriented matroids.

######  Lemma 3.5.

Let $X\in L(\mathcal{M})$ be modular and $Y\in L(\mathcal{M})$ another flat.
Then the map

$\displaystyle\overline{\rho}_{X}:=\rho_{X}|_{\mathcal{L}^{Y}_{X\vee
Y}}:\mathcal{L}_{X\vee Y}^{Y}$ $\displaystyle\to\mathcal{L}_{X}^{X\wedge Y},$
$\displaystyle\sigma$ $\displaystyle\mapsto\sigma|_{X}$

is an isomorphism of posets.

###### Proof.

We may assume without loss of generality that $X\vee Y=E$. Firstly, we confirm
injectivity. Let $\sigma_{1},\sigma_{2}\in\mathcal{L}^{Y}$ with
$\rho_{X}(\sigma_{1})=\sigma_{1}|_{X}=\rho_{X}(\sigma_{2})=\sigma_{2}|_{X}$.
By the modularity of $X$ and Lemma 2.13 we have that
$z(\sigma_{i})=Y\vee(z(\sigma_{i})\wedge X)=Y\vee z(\rho_{X}(\sigma_{i}))$ for
$i=1,2$. So $z(\sigma_{1})=z(\sigma_{2})$ and $\sigma_{1}=\sigma_{1}|_{X\vee
Y}=\sigma_{2}|_{X\vee Y}=\sigma_{2}$ by Lemma 3.3. Consequently,
$\overline{\rho}_{X}$ is injective.

Secondly, we construct a section $\overline{\iota}:\mathcal{L}_{X}^{X\wedge
Y}\to\mathcal{L}_{X\vee Y}^{Y}$ to $\overline{\rho}_{X}$ which implies the
statement. By Proposition 3.4 and the commutativity of the diagram in Figure
4, for $\sigma\in\mathcal{L}_{X}^{X\wedge Y}$ we have that

$z(\rho_{X}^{-1}((\mathcal{L}_{X})_{\geq\sigma})\cap\mathcal{L}^{Y})=L^{Y}_{\leq
s_{Y}(z(\sigma))}.$

So there is a
$\widehat{\sigma}\in\rho_{X}^{-1}((\mathcal{L}_{X})_{\geq\sigma})\cap\mathcal{L}^{Y}$
with $z(\widehat{\sigma})=s_{Y}(z(\sigma))=Y\vee z(\sigma)$ and further, by
Lemma 2.13, $z(\overline{\rho}_{X}(\widehat{\sigma}))=z(\sigma)$. This implies
$\overline{\rho}_{X}(\widehat{\sigma})=\sigma$. Moreover, by the injectivity
of $\overline{\rho}_{X}$, this minimal $\widehat{\sigma}$ is the unique
minimal element of
$\rho_{X}^{-1}((\mathcal{L}_{X})_{\geq\sigma})\cap\mathcal{L}^{Y}$. For
$\sigma\leq\tau\in\mathcal{L}_{X}^{X\wedge Y}$ we have
$\rho_{X}^{-1}((\mathcal{L}_{X})_{\geq\sigma})\cap\mathcal{L}^{Y}\supseteq\rho_{X}^{-1}((\mathcal{L}_{X})_{\geq\tau})\cap\mathcal{L}^{Y}$
and thus $\widehat{\sigma}\leq\widehat{\tau}$ for the unique minimal elements
in these order filter of $\mathcal{L}^{Y}$. This yields our desired order
preserving section $\overline{\iota}$ by setting
$\overline{\iota}(\sigma):=\widehat{\sigma}$. ∎

$(\rho_{X}\downarrow\sigma)$$\mathcal{L}^{\vee}\supseteq(\mathcal{L}^{H})^{\vee}$$\rho_{X}$$\sigma$$\mathcal{L}_{X}^{\vee}$
Figure 5. The dual covector complex
$\mathcal{L}^{\vee}=\mathcal{L}^{\vee}(\mathcal{A})$ of the arrangement
$\mathcal{A}$, and its contraction $(\mathcal{L}^{H})^{\vee}$, which is mapped
isomorphically to the localization $\mathcal{L}_{X}^{\vee}$ at a modular flat
$X$.

We visualize Lemma 3.5 with an example.

######  Example 3.6.

Let $\mathcal{A}$ be the hyperplane arrangement in $\mathbb{R}^{3}$ consisting
of the hyperplanes

$\displaystyle\mathcal{A}=\\{$ $\displaystyle
H_{1}=\ker(x),H_{2}=\ker(y),H_{3}=\ker(x+y),$ $\displaystyle
H_{4}=\ker(z),H_{5}=\ker(x+z)\\}$

and $\mathcal{M}=\mathcal{M}(\mathcal{A})$ the associated oriented matroid
with covectors $\mathcal{L}=\mathcal{L}(\mathcal{A})$. Then the flat
$X=\\{H_{1},H_{2},H_{3}\\}\in L(\mathcal{M})$ is modular. The dual covector
complex $\mathcal{L}^{\vee}$, the localization $\mathcal{L}_{X}^{\vee}$ and
the map $\rho_{X}:\mathcal{L}^{\vee}\to\mathcal{L}_{X}^{\vee}$ together with a
poset fiber $(\rho_{X}\downarrow\sigma)$ are shown in Figure 5. The dual of
the restricted covector complex $(\mathcal{L}^{H})^{\vee}$ to
$H=\\{H_{4}\\}\in L(\mathcal{M})$ is given by the dotted subcomplex
transversely to the cells in $z^{-1}(L^{H})\simeq(\mathcal{L}^{H})^{\vee}$. We
clearly see that the localization map $\rho_{X}$, restricted to
$(\mathcal{L}^{H})^{\vee}$ is an isomorphism in accordance with Lemma 3.5.

##  4\. A stratification of subcomplexes of the Salvetti complex

Again, let $X\in L$ be a modular flat of corank $1$. We now describe a special
stratification of the poset fiber
$(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$ for
$B^{\prime}\in\mathcal{T}_{X}$. This is analogous to a result by Delucchi
[Del08, Lem. 4.17,Lem. 4.18]. Unfortunately, Delucchi’s more general statement
is not correct, cf. [DM22].

Recall that by Lemma 3.1 we have a natural linear order on the subset of topes
$\mathcal{T}(\rho_{X}\downarrow B^{\prime})$, say
$\mathcal{T}(\rho_{X}\downarrow B^{\prime})=\\{T_{0}<T_{1}<...<T_{k}\\}$. Then
it also follows from Lemma 3.1 that the order filter

$J_{i}:=\\{X\in L\mid S(T_{j},T_{i})\cap X\neq\emptyset\text{ for all
}j<i\\}\subseteq L$

is principal, i.e.

$J_{i}=\begin{cases}L&\text{if }i=0,\\\ L^{S(T_{i-1},T_{i})}&\text{ if }1\leq
i\leq k,\end{cases}$

where $S(T_{i-1},T_{i})$ consists of a single element in $E\setminus X$.

Now, we define locally closed subcomplexes
$N_{i}\subseteq(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$
by

$N_{i}:=\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T_{i})}\setminus(\bigcup_{j<i}\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T_{j})}).$

Note that
$(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))=\bigsqcup_{0\leq
i\leq k}N_{i}$.

Furthermore, we have the following useful lemma.

######  Lemma 4.1.

For all $0\leq i\leq k$ we have
$N_{i}\cong(\mathcal{L}^{S(T_{i-1},T_{i})})^{\vee}$.

###### Proof.

This follows from the fact that $J_{i}$ is a principal order filter. See the
proof of [Del08, Lem. 4.18] or [LP21, Lem. 4.12]. ∎

##  5\. Poset quasi-fibrations

We introduce a new class of poset maps with a desirable topological property.

Firstly, we record the following theorem from Quillen’s foundational work
[Qui73] on algebraic $K$-Theory, originally expressed in terms of small
categories and functors. We state it in its special incarnation in terms of
posets and order preserving maps.

######  Theorem 5.1 (Theorem B for posets [Qui73, pp. 89]).

Let $f:P\to Q$ be an order preserving map between posets. If for all $a\leq b$
$(a,b\in Q)$ the inclusion $(f\downarrow a)=f^{-1}(Q_{\leq a})\hookrightarrow
f^{-1}(Q_{\leq b})=(f\downarrow b)$ is a homotopy equivalence, then for $x\in
P$ with $f(x)=a$ the homotopy fiber of $F(|\Delta(f)|,a)$ is homotopy
equivalent to $|\Delta(f\downarrow a)|$.

Consequently, we have a long exact homotopy sequence

${\ldots}$${\pi_{i+1}(|\Delta(Q)|,a)}$${\pi_{i}(|\Delta(f\downarrow
a)|,x)}$${\pi_{i}(|\Delta(P)|,x)}$${\pi_{i}(|\Delta(Q)|,a)}$${\ldots}$$\scriptstyle{\pi_{i}(|\Delta(j)|)}$$\scriptstyle{\pi_{i}(|\Delta(f)|)}$

where $j:(f\downarrow a)\hookrightarrow P$ is the natural inclusion.

Theorem 5.1 motivates the following notion.

######  Definition 5.2.

Let $f:P\to Q$ be a poset map such that for all $a\leq b$ $(a,b\in Q)$ the
inclusion $(f\downarrow a)\hookrightarrow(f\downarrow b)$ is a homotopy
equivalence. Then $f$ is called a _poset quasi-fibration_.

Invoking a homotopy colimit construction [WZŽ99, Prop. 3.12 & Rem.], every
poset quasi-fibration can be combinatorially “turned into” a proper quasi-
fibration in the sense of Dold and Thom [DT58] (after simplicial realization)
with fibers homotopy equivalent to the poset fibers of the original order
preserving map.

##  6\. Acyclic matchings for subcomplexes and the main result

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid and
$L:=L(\mathcal{M})$ its geometric lattice.

Firstly, we derive certain acyclic matchings on the dual covector complex
$\mathcal{L}^{\vee}$. Recall from Definition 2.15 the subcomplex
$\mathcal{L}^{\vee}[\mathcal{Q}]$ for a convex subset
$\mathcal{Q}\subseteq\mathcal{T}$ of topes.

######  Proposition 6.1.

Let $\emptyset\neq\mathcal{Q}\subseteq\mathcal{T}$ be a convex subset and
$\mathcal{L}^{\vee}[\mathcal{Q}]\subseteq\mathcal{L}^{\vee}$ its associated
dual subcomplex. Then there is an acyclic matching
$\operatorname{\underline{M}}$ on $\mathcal{L}^{\vee}$ with
$C(\operatorname{\underline{M}})=\mathcal{L}^{\vee}[\mathcal{Q}]$.

###### Proof.

By Corollary 2.23 the subcomplex
$\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$ is a shellable ball. Hence, by
Proposition 2.31 there is an acyclic matching
$\operatorname{\underline{M}}^{\prime}$ on
$\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$ whose critical cells consist of
only one vertex of $v\in\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$. Since
$\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$ is an order ideal in
$\mathcal{L}$, the matching $\operatorname{\underline{M}}^{\prime}$ can be
extended to a matching $\operatorname{\underline{M}}$ on $\mathcal{L}$ by a
trivial matching on the order filter
$\mathcal{L}\setminus\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})$ (e.g. use
Theorem 2.32). Hence, this new matching $\operatorname{\underline{M}}$ has
critical cells:
$C(\operatorname{\underline{M}})=\mathcal{L}\setminus\mathcal{L}(\mathcal{T}\setminus\mathcal{Q})\sqcup\\{v\\}$.
By Remark 2.16 we have
$(\mathcal{L}\setminus\mathcal{L}(\mathcal{T}\setminus\mathcal{Q}))^{\vee}=\mathcal{L}^{\vee}[\mathcal{Q}]$.
We can extend the matching $\operatorname{\underline{M}}^{\vee}$ on
$\mathcal{L}^{\vee}$ by one further matched pair
$(v,\operatorname{\mathbf{0}})$, since
$\operatorname{\mathbf{0}}\in\mathcal{L}^{\vee}$ is the unique maximal element
of $\mathcal{L}^{\vee}$ and
$v\lessdot_{\mathcal{L}^{\vee}}\operatorname{\mathbf{0}}$. This gives the
claimed matching on $\mathcal{L}^{\vee}$. ∎

Let $X\in L$ be some flat and
$\rho_{X}:\mathcal{L}^{\vee}\to\mathcal{L}_{X}^{\vee}$ the associated
localization map. From the preceding proposition we immediately get the
following.

######  Corollary 6.2.

Let $\sigma\in\mathcal{L}_{X}^{\vee}$. Then there is an acyclic matching on
$\mathcal{L}^{\vee}$ with
$C(\operatorname{\underline{M}})=(\rho_{X}\downarrow\sigma)$.

###### Proof.

This readily follows from Proposition 6.1, the convexity of the subset of
topes $\mathcal{T}(\rho_{X}\downarrow\sigma)$ thanks to Lemma 2.19 and the
observation that we have
$\mathcal{L}^{\vee}[\mathcal{T}(\rho_{X}\downarrow\sigma)]=(\rho_{X}\downarrow\sigma)$.
∎

Now assume that $X$ is modular and of corank $1$. Recall the map
$\widetilde{\rho}_{X}:\mathcal{S}\to\mathcal{S}_{X},(\sigma,T)\mapsto(\rho_{X}(\sigma),\rho_{X}(T))$
between the Salvetti complexes. Next, we derive special acyclic matchings on
poset fibers of $\widetilde{\rho}_{X}$ of maximal elements of
$\mathcal{S}_{X}$ which yield a homotopy equivalence.

${\mathcal{S}}$${\mathcal{S}_{X}}$${N_{i}}$${(\mathcal{S}_{X})_{\leq(\operatorname{\mathbf{0}},B^{\prime})}}$${(\mathcal{L}^{S(T_{i-1},T_{i})})^{\vee}}$${\mathcal{L}_{X}^{\vee}}$${\mathcal{L}^{\vee}}$$\scriptstyle{\widetilde{\rho}_{X}}$$\scriptstyle{\widetilde{\rho}_{X}|_{N_{i}}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\rho_{X}|}$$\scriptstyle{\rho_{X}}$
Figure 6. Restriction of $\widetilde{\rho}_{X}$ to $N_{i}$ can be identified
with the restriction of $\rho_{X}$.

######  Proposition 6.3.

Let $a\in\mathcal{S}_{X}$ and let $B^{\prime}\in\mathcal{T}_{X}$ with
$a\leq(\operatorname{\mathbf{0}},B^{\prime})$. Then there is an acyclic
matching $\operatorname{\underline{M}}$ on
$(\widetilde{\rho}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$ with
$C(\operatorname{\underline{M}})=(\widetilde{\rho}\downarrow a)$. In
particular, $(\widetilde{\rho}\downarrow a)$ is a strong deformation retract
of $(\widetilde{\rho}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$ and
$(\widetilde{\rho}\downarrow
a)\hookrightarrow(\widetilde{\rho}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$
is a homotopy equivalence.

###### Proof.

Firstly, let $T_{0}<T_{1}<\ldots<T_{k}$ be a linear order on the set of topes
$\mathcal{T}(\rho_{X}\downarrow B^{\prime})$ as described by Lemma 3.1. We
utilize the stratification of
$(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))=\bigsqcup_{0\leq
i\leq k}N_{i}$ from Section 4. Note that each
$N_{i}\subseteq(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$
is an order filter in
$\mathcal{S}(\\{(\operatorname{\mathbf{0}},T_{0}),\ldots,(\operatorname{\mathbf{0}},T_{i})\\})$.
Thus, we have a natural order preserving map

$f:(\widetilde{\rho_{X}}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))\to\\{T_{0}<T_{1}<\ldots<T_{k}\\}$

with $f^{-1}(T_{i})=N_{i}$.

Now, by Lemma 4.1 and Lemma 2.27 we have
$N_{0}=\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T_{0})}\simeq\mathcal{L}^{\vee}$,
and for each $1\leq i\leq k$ we have
$\mathcal{S}_{\leq(\operatorname{\mathbf{0}},T_{i})}\supseteq
N_{i}\cong(\mathcal{L}^{S(T_{i-1},T_{i})})^{\vee}$. Hence, by Corollary 2.28
the map $\widetilde{\rho}_{X}$ restricted to $N_{0}$ is nothing else than the
map $\rho_{X}:\mathcal{L}^{\vee}\to\mathcal{L}_{X}^{\vee}$. Similarly, for
$1\leq i\leq k$ the map $\widetilde{\rho}_{X}$ restricted to $N_{i}$ is
$\widetilde{\rho}_{X}|_{N_{0}}=\rho_{X}|:{(\mathcal{L}^{S(T_{i-1},T_{i})})^{\vee}}\to\mathcal{L}_{X}^{\vee}$,
which is an isomorphism by Lemma 3.5, since $S(T_{i-1},T_{i})\cap
X=\emptyset$; see the commutative diagram shown in Figure 6. Assume that
$a=(\sigma,R^{\prime})$ for some $R^{\prime}\in\mathcal{T}_{X}$. Then:

  * •

$N_{0}\simeq(\rho_{X}\downarrow\operatorname{\mathbf{0}})\simeq\mathcal{L}^{\vee}$
$\supseteq$ $(\widetilde{\rho}_{X}\downarrow a)\cap
N_{0}\simeq(\rho_{X}\downarrow\sigma)$,

  * •

$N_{i}\simeq(\rho_{X}|\downarrow\operatorname{\mathbf{0}})\simeq(\mathcal{L}^{S(T_{i-1},T_{i})})^{\vee}$
$\supseteq$ $(\widetilde{\rho}_{X}\downarrow a)\cap
N_{i}\simeq(\rho_{X}|\downarrow\sigma)\simeq(\mathcal{L}_{X}^{\vee})_{\leq\sigma}$,
for $1\leq i\leq k$.

In both instances, by Corollary 6.2 we get a matching
$\operatorname{\underline{M}}(i)$ on $N_{i}$ with critical cells
$C(\operatorname{\underline{M}}(i))=(\widetilde{\rho}_{X}\downarrow a)\cap
N_{i}$. Patching these matchings together via Theorem 2.32 along the map $f$
yields our desired matching
$\operatorname{\underline{M}}=\bigcup_{i=0}^{k}\operatorname{\underline{M}}(i)$
on $(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))$
with $C(\operatorname{\underline{M}})=(\widetilde{\rho}_{X}\downarrow a)$ and
concludes the proof. ∎

From the previous proposition we can now readily derive our principal result.

######  Theorem 6.4.

Let $X\in L(\mathcal{M})$ be a modular flat of corank $1$. Then the map
$\widetilde{\rho}_{X}:\mathcal{S}\to\mathcal{S}_{X}$ is a poset quasi-
fibration. For $a\in\mathcal{S}_{X}$ the poset fiber
$(\widetilde{\rho}_{X}\downarrow a)$ is homotopy equivalent to the affine
Salvetti complex of an oriented matroid $\mathcal{N}$ of rank $2$.

###### Proof.

Let $a,b\in\mathcal{S}_{X}$ with $a\leq b$. We have to show that the inclusion
$(\widetilde{\rho_{X}}\downarrow
a)\hookrightarrow(\widetilde{\rho}_{X}\downarrow b)$ is a homotopy
equivalence. Picking some maximal element
$(\operatorname{\mathbf{0}},B^{\prime})$ of $\mathcal{S}_{X}$ with
$b\leq(\operatorname{\mathbf{0}},B^{\prime})$, we have a commutative triangle
of inclusion maps:

${(\widetilde{\rho}_{X}\downarrow(\operatorname{\mathbf{0}},B^{\prime}))}$${(\widetilde{\rho}_{X}\downarrow
a)}$${(\widetilde{\rho}_{X}\downarrow
b).}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{3}}$$\scriptstyle{i_{2}}$

By Proposition 6.3 the two inclusions $i_{1}$ and $i_{2}$ are homotopy
equivalences. Hence, so is the third inclusion $i_{3}$. This confirms that
$\widetilde{\rho}_{X}$ is a poset quasi-fibration.

To see the last statement, pick a minimal element
$(B^{\prime},B^{\prime})\in\mathcal{S}_{X}$. We easily see that
$(\rho_{X}\downarrow B^{\prime})=\mathcal{N}_{\operatorname{aff},g}$ for an
oriented matroid $\mathcal{N}$ of rank two on $E\setminus X\sqcup\\{g\\}$ by
Lemma 3.1. Then
$(\widetilde{\rho}_{X}\downarrow(B^{\prime},B^{\prime}))=\mathcal{S}_{\operatorname{aff},g}(\mathcal{N})$.
∎

The affine Salvetti complex of an oriented matroid of rank $2$ is a graph and
consequently aspherical. Hence, by the long exact sequence of homotopy groups
associated to a poset quasi-fibration due to Theorem 5.1 we immediately obtain
the following.

######  Corollary 6.5.

If $X\in L(\mathcal{M})$ is modular of corank $1$ and $\mathcal{S}_{X}$ is
aspherical, then $\mathcal{S}$ is aspherical as well.

######  Corollary 6.6 (Theorem 1.1).

The Salvetti complex of a supersolvable oriented matroid is aspherical.

##  7\. Fundamental groups

We analyze the structure of the fundamental group of the Salvetti complex of a
supersolvable oriented matroid. This is completely analogous to the realizable
case, cf. [FR85].

Let $P$ be a poset and $x\in P$. Then we write
$\pi_{1}(P,x)=\pi_{1}(|\Delta(P)|,x)$ for the fundamental group of $P$.
Similarly, if $f:P\to Q$ is an order preserving map, then we write
$\pi_{1}(f):=\pi_{1}(|\Delta(f)|):\pi_{1}(|\Delta(P)|,x)\to\pi_{1}(|\Delta(Q)|,f(x))$
for the induced map between fundamental groups.

Firstly, we have the following lemma.

######  Lemma 7.1.

Assume that $f:P\to Q$ is a poset quasi-fibration which has a section $s:Q\to
P$. Let $x\in P$ and $a=f(x)$ and assume further that $|\Delta(Q)|$ and
$|\Delta(f\downarrow a)|$ are $K(\pi,1)$-spaces. Then the fundamental group
$\pi_{1}(P,x)$ is a semi-direct product:

$\pi_{1}(P,x)\cong\pi_{1}((f\downarrow a),x)\rtimes\pi_{1}(Q,a).$

###### Proof.

By Theorem 5.1, the long exact sequence of the poset quasi-fibration
degenerates to the short exact sequence:

${1}$${\pi_{1}((f\downarrow
a),x)}$${\pi_{1}(P,x)}$${\pi_{1}(Q,a)}$${1.}$$\scriptstyle{\pi_{1}(f)}$

The map $\pi_{1}(s)$ gives a section to $\pi_{1}(f)$ and so the above short
exact sequence splits. ∎

With the preceding lemma and our main result from the last section we can say
the following.

######  Theorem 7.2.

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid and assume that $X\in
L(\mathcal{M})$ is modular and of corank $1$. Assume further that
$\mathcal{S}_{X}$ is aspherical. Then for $x\in\mathcal{S}$ and
$a=\widetilde{\rho}_{X}(x)\in\mathcal{S}_{X}$ we have

$\pi_{1}(\mathcal{S},x)\cong F\rtimes\pi_{1}(\mathcal{S}_{X},a),$

where $F$ is a free group with $|E\setminus X|$ generators.

###### Proof.

By Theorem 6.4 the map $\widetilde{\rho}_{X}:\mathcal{S}\to\mathcal{S}_{X}$ is
a poset quasi fibration with $(\widetilde{\rho}_{X}\downarrow a)$ homotopy
equivalent to the affine Salvetti complex of an oriented matroid of rank $2$
on $|E\setminus X|+1$ elements. Hence $(\widetilde{\rho}_{X}\downarrow
a)\simeq S^{1}\vee\ldots\vee S^{1}$ is a wedge of $|E\setminus X|$ circles.
Thus, $\pi_{1}((\widetilde{\rho}_{X}\downarrow a),a)=F$ is a free group with
$|E\setminus X|$ generators. Furthermore, $\widetilde{\rho_{X}}$ has a section
by Remark 2.8 and Lemma 7.1 yields the result. ∎

Consequently, the structure of the fundamental group of the Salvetti complex
of a supersolvable oriented matroid is as follows.

######  Corollary 7.3.

Let $\mathcal{M}$ be a supersolvable oriented matroid of rank $r$. Then
$\pi_{1}(\mathcal{S},x)\cong
F_{1}\rtimes(F_{2}\rtimes(\ldots(F_{r-1}\rtimes\mathbb{Z})\ldots))$ where the
$F_{i}$ are finitely generated free groups.

##  8\. Non-realizable supersolvable oriented matroids

We provide examples of non-realizable supersolvable oriented matroids. In
particular, we observe that every oriented matroid of rank $3$ can be extended
to a supersolvable oriented matroid.

Recall the definition of an extension of an oriented matroid (see Definition
2.9). Before we construct supersolvable extensions, we record a fundamental
result for arrangements of pseudolines. This is Levi’s Enlargement Lemma
[Lev26] which we state in its version for oriented matroids of rank $3$, see
[BLVS+99, Prop. 6.3.4].

######  Lemma 8.1 (Levi’s Enlargement Lemma).

Let $\mathcal{M}=(E,\mathcal{L})$ be an oriented matroid of rank $3$ and let
$X_{1},X_{2}\in L(\mathcal{M})$ be two distinct flats of rank $2$ with
$X_{1}\cap X_{2}=\emptyset$. Then there is a one element extension
$\widetilde{\mathcal{M}}$ on $E\sqcup\\{g\\}$ of $\mathcal{M}$ such that
$\widetilde{X_{1}},\widetilde{X_{2}}$ are flats of rank $2$ and
$g\in\widetilde{X_{1}}\cap\widetilde{X_{2}}$. Moreover, the extension can be
chosen such that $g\notin\widetilde{Y}$ for any flat $Y\in
L\setminus\\{X_{1},X_{2}\\}$ of rank $2$.

From the preceding lemma we can delineate the existence of supersolvable
extensions for oriented matroids of rank $3$.

######  Proposition 8.2.

Let $\mathcal{M}$ be an oriented matroid of rank $3$. Then there is an
extension $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ which is supersolvable.

###### Proof.

Let $X\in L(\mathcal{M})$ be some flat of rank $2$. If there is no $Y\in
L(\mathcal{M})$ of rank $2$ with $X\cap Y=\emptyset$, then by Lemma 2.12 the
oriented matroid $\mathcal{M}$ is already supersolvable and there is nothing
to show. Otherwise, let $Y\in L(\mathcal{M})$ be a flat of rank $2$ with
$X\cap Y=\emptyset$. By Lemma 8.1 we can extend $\mathcal{M}$ by a single
element $g$ to $\widetilde{\mathcal{M}}$ such that
$\widetilde{X}\cap\widetilde{Y}=\\{g\\}$. Then we have

$|\\{Y^{\prime}\in L(\widetilde{\mathcal{M}})\mid
Y^{\prime}\cap\widetilde{X}=\emptyset\\}|<|\\{Y\in L(\mathcal{M})\mid X\cap
Y=\emptyset\\}|.$

Hence, after a finite number of single element extensions via Lemma 8.1 we
arrive at an extension $\widetilde{\mathcal{M}}^{\prime}$ of $\mathcal{M}$
which is supersolvable thanks to Lemma 2.12. ∎

######  Remark 8.3.

Note that there is no analogue of Lemma 8.1 for oriented matroids of rank
$\geq 4$. E.g. the Edmonds-Fukuda-Mandel oriented matroid $\mathtt{EFM}(8)$ –
an orientation of the uniform matroid of rank $4$ on $8$ points – gives an
example that has three corank $1$ flats which cannot be connected by any
extension, cf. [BLVS+99, Prop. 10.4.5]. Consequently, it is not clear to us
that supersolvable extensions for higher rank oriented matroids exist in
general. In contrast, in the realizable case one easily sees that any
hyperplane arrangement is a subarrangement of a supersolvable arrangement.

The remark leaves the following question.

######  Question 8.4.

Do oriented matroids of rank $\geq 4$ always have supersolvable extensions?

Starting with a non-realizable oriented matroid, Proposition 8.2 yields a non-
realizable supersolvable oriented matroid. Consequently, we obtain many non-
realizable supersolvable oriented matroids.

######  Corollary 8.5.

There are infinitely many supersolvable oriented matroids which are not
realizable.

All of these non-realizable oriented matroids have aspherical Salvetti
complexes by Theorem 1.1.

Figure 7. A projective picture of an arrangement of pseudolines representing a
(reorienteation class of a) non-realizable oriented matroid and a
supersolvable extension.

We conclude with a simple example of a non-realizable supersolvable oriented
matroid of rank $3$. For a precise definition of an _arrangement of
pseudolines_ – a nice visual way to represent (reorientation classes of) rank
$3$ oriented matroids, we refer to [BLVS+99, Ch. 6].

######  Example 8.6.

Let $\mathcal{M}$ be the oriented matroid of rank $3$ associated to some
orientation of the pseudoline arrangement, illustrated as a projective picture
in Figure 7. By Pappus’ theorem, the oriented matroid is not realizable by an
arrangement of linear hyperplanes.

By Proposition 8.2, $\mathcal{M}$ can be extended to a supersolvable oriented
matroid $\widetilde{\mathcal{M}}$. A possible extension is illustrated by the
dotted lines in Figure 7. The intersection point marked by a solid circle
represents a modular flat $X$ of corank one. It is connected to every other
intersection point of pseudolines in the arrangement by some line passing
through $X$. The resulting supersolvable oriented matroid
$\widetilde{\mathcal{M}}$ is evidently not realizable.

## Acknowledgments

The author would like to thank Emanuele Delucchi and Masahiko Yoshinaga for
helpful discussions. He would also like to thank Gerhard Röhrle for useful
remarks on an earlier draft of the manuscript.

The author is grateful to the JSPS for financial support during a JSPS
fellowship for foreign researchers in Japan.

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