Modular flats of oriented matroids and poset quasi-fibrations

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.


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(π, 1)-spaces where π 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 π.They play an important role in algebraic topology e.g. as their cohomology coincides with the group cohomology of π.Moreover, if a K(π, 1)-space is homotopy equivalent to a finite CW-complex then the group π is torsion free.In the following, we call complex arrangements whose complements are aspherical simply K(π, 1) or aspherical arrangements.
One of the most prominent positive results regarding K(π, 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(π, 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(π, 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(π, 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.

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, ≤) be a poset.Its dual or opposite poset P ∨ has the same ground set as P but its order relation ≤ ∨ is reversed: x ≤ ∨ y : ⇐⇒ y ≤ x.
To emphasize the order relation of a particular poset P we occasionally use the notation ≤ P .
We denote by ⋖ the cover relations of P .That is x ⋖ y if and only if x < y, and for all z ∈ P : x < z ≤ y implies z = y.
A subset I ⊆ P is called an order ideal if y ∈ I and x ∈ P with x ≤ y implies x ∈ I. Dually, F ⊆ P is called an order filter if F ∨ is an order ideal in P ∨ .We write P ≤y = {x ∈ P | x ≤ y} for the principal order ideal generated by y ∈ P and similarly P ≥x = {y ∈ P | x ≤ y} for the principal order filter generated by x ∈ P .
A map f : P → Q between two posets is order preserving or a poset map if for all x ≤ P y (x, y ∈ P ) we have f A linear extension ⊣ of P is a total order on the ground set P such that for all x, y ∈ P : x ≤ y implies x ⊣ y.Linear extensions of finite posets always exits.Given an order ideal I ⊆ P , a linear extension ⊣ I on I can always be extended to a linear extension ⊣ on P with x ⊣ y if and only if x ⊣ I y for all x, y ∈ 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 Σ be a regular cell complex and σ, τ ∈ Σ.If τ ⊆ σ then τ is called a face of σ and we simply write τ ≤ σ.In what follows, we thus regard a regular cell complex Σ itself as a poset.Cells of dimension zero are referred to as vertices.
Conversely, to every poset P is associated its order complex ∆(P ).It is the simplicial complex with n-simplices given by all chains of length n in P , i.e. ∆(P ) n := {{x 0 , . . ., x n } | x i ∈ P with x 0 < x 1 < . . .< x n }.
For a simplicial complex ∆ we denote its topological realization by |∆|.The composition of forming the order complex and taking its realization yields the simplicial realization functor |∆(−)| : Pos → Top from the category Pos of posets and order preserving maps to the category Top of topological spaces and continuous maps.For a regular cell complex Σ the image of its face poset under this functor is naturally homeomorphic to Σ: We call an order preserving map f : P → Q a homotopy equivalence provided |∆(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 Σ is called pure if all maximal cells have the same dimension.
A subset Γ ⊆ Σ is called a (closed) subcomplex if with a cell σ ∈ Γ all of its faces also belong to Γ, i.e.Γ is an order ideal of Σ.For a subset of cells S ⊆ Σ we denote by Σ(S) := {τ ∈ Σ | τ ≤ σ for a σ ∈ S} the subcomplex of all faces of cells in S. In poset terminology, Σ(S) = σ∈S Σ ≤σ is the order ideal generated by S. For a cell σ ∈ Σ we denote by δσ the subcomplex consisting of all proper faces of σ.
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 σ ∈ {+, −, 0} E .We denote by z(σ) := {e ∈ E | σ e = 0} its zero set.The opposite vector −σ is defined by The composition σ • τ of two sign vectors σ, τ ∈ {+, −, 0} E is defined by and their separating set S(σ, τ ) is defined as for its restriction to A.
The rank of M is defined as the length of a maximal chain in L.
The maximal elements in L are called topes; they are denoted by Note that this is only one of several equivalent ways to define oriented matroids, cf.[BLVS + 99, Ch. 3].
An element e ∈ E is called a loop of the oriented matroid M = (E, L) if σ e = 0 for all σ ∈ L. Two elements e, f ∈ E which are not loops are parallel if σ e = 0 implies σ f = 0 for all σ ∈ 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 A be an arrangement of hyperplanes in a real vector space We recall the definition of an affine oriented matroid.
Definition 2.3.Let M = (E, L) be an oriented matroid and g ∈ E.
Define M aff,g := (E, L + , g) by Then M aff,g is called an affine oriented matroid or the decone of 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 L \ {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, L \ {0} is a PL-sphere, so L ∨ represents a regular cell decomposition of a ball, cf.[BLVS + 99, Prop.4.7.26].
There is another poset associated to an oriented matroid M = (E, L) which is essential for our considerations.
Definition 2.5.The geometric lattice L(M) of M is defined as with partial order given by inclusion.Elements of L(M) are called flats.
Moreover, for X, Y ∈ L = L(M) we write L X := L ≤X for the lattice of the localization at X, L Y := L ≥Y for the lattice of the contraction to Y and L Y X := [X, Y ] := L ≤X ∩ L ≥Y for an interval.
The lattice L = L(M) is graded by rank, i.e. the rank of X ∈ L is the length of a maximal chain in L ≤X .
Note that the map z : L → L is a cover and rank preserving, order reversing surjection [BLVS + 99, Prop.4.1.13].The associated order preserving map L ∨ → L is also denoted by z.
(1) For a subset A ⊆ E the restriction M| A = (A, L| A ) of M to A is defined by Remark 2.7.For the covectors of the contraction M/X we also use the notation We often identify L X with z −1 (L X ) by the obvious isomorphism z −1 (L X ) ≃ L X .Therefor, we write we frequently write L Y X := L X /Y .Remark 2.8.Let X ∈ L and let ρ X : L → L X be the localization map.Then for any α ∈ L with z(α) = X we have a section ι α : L X → L to ρ X which is defined by: An easy calculation shows that both the localization map ρ X and the section ι α preserve the composition of covectors, i.e. we have ρ We recall the notion of an extension of an oriented matroid.
Definition 2.9.Let M = (E, L) be an oriented matroid.Then another oriented matroid If M is an extension of M, then we have a natural injective map 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 < . . .< X r of flats in L such that X i is modular for all 0 ≤ i ≤ r.
In this case, we call 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 M to be of rank 3. Then M is supersolvable if and only if there is a flat X ∈ L of rank 2 such that for all flats Y ∈ L of rank 2 we have X ∩ Y = ∅.
Modular flats have the following property (which is actually equivalent to modularity).
2.4.Topes and convex subsets.Recall that we denote the topes of an oriented matroid M = (E, L), i.e. the maximal elements of L by T .As remarked earlier, without loss of generality, we assume that M is simple. of Q is defined as the intersection of all halfspaces containing Q: Remark 2.16.Note that the definition of L ∨ [Q] ⊆ L ∨ yields a subcomplex since L ∨ has the intersection property, i.e. a face is uniquely determined by its vertices.Furthermore, an easy calculation shows that ( We record the following alternative characterization of convex subsets.
Two easy but useful facts are provided by the following lemmas.
Lemma 2.18.Let Q ⊆ T be a convex subset of topes.Then for a tope B ∈ Q there is a linear extension ⊣ of T (M, B) such that with respect to ⊣ all the topes in Q come first.
Proof.The convexity of Q and Lemma 2.17 imply that Q forms an order ideal in T (M, B) for every B ∈ Q, cf.[BLVS + 99, pp.172ff.].This, in turn, implies the statement.
We consider the localization map ρ X : L ∨ → L ∨ X .For the subset of topes (vertices of L ∨ ) in the poset fiber (ρ X ↓ σ) we write T (ρ X ↓ σ).
Definition 2.20.Let Σ be a pure d-dimensional regular cell complex.
A linear ordering σ 1 , σ 2 , . . ., σ t of its maximal cells is called a shelling if either d = 0, or d ≥ 1 and the following conditions are satisfied: (i) δσ j ∩ ( j−1 i=1 δσ i ) is pure of dimension d − 1 for 2 ≤ j ≤ t, (ii) δσ j has a shelling in which the (d − 1)-cells of δσ j ∩ ( j−1 i=1 δσ i ) come first for 2 ≤ j ≤ t, (iii) δσ 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 L of an oriented matroid.More precisely, we have the following theorem.
With the preceding results we can delineate the following property of subcomplexes generated by subsets of the topes T = T (M) of an oriented matroid M.
Corollary 2.23.Let ∅ = Q ⊆ T be a convex subset of topes.Then both the subcomplexes L(Q) and L(T \ Q) are shellable balls.
Proof.Let B ∈ Q.By Lemma 2.18 there is a linear extension of T (M, B) such that all the topes in Q come first.Conversely, with respect to the opposite linear extension of T (M, −B) all the topes in T \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 M(A) associated to a real hyperplane arrangement A. Gel'fand and Rybnikov [GR90] observed, that the same construction applies more generally to oriented matroids.Definition 2.24.Let L be the poset of covectors of an oriented matroid M. Then the Salvetti poset S = S(M) is defined as The affine Salvetti poset S aff,g of the affine oriented matroid M aff,g is defined as S aff,g := {(σ, T ) ∈ S | σ 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 M = M(A) is the oriented matroid associated to a hyperplane arrangement A in a real vector space V , then the Salvetti complex is homotopy equivalent to the complement of the complexified arrangement: Let X ∈ L(M).The projection map ρ X : L → L X respects the composition of covectors by Remark 2.8.Hence, ρ X induces an order preserving map ρ X : S = S(M) → S(M X ) =: S X , (σ, T ) → (ρ X (σ), ρ X (T )) between the Salvetti complexes.Moreover, for each α ∈ L with z(α) = X we get a section ι α : L X → L to ρ X by Remark 2.8 which respects the composition of covectors.Hence, this also yields a section to ρ X and provides a simple argument for the following lemma.
Lemma 2.26.If S is a K(π, 1) complex then so is S X = S(M X ) for all X ∈ L(M).
Proof.Application of the homotopy group functors directly yields that the map π i ( ρ X ) : π i (S) → π i (S X ) has a section π i ( ι α ) 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 S consisting of all the faces of a maximal cell can be identified with the dual covector complex.

S S X
S ≤(0,T ) (S X ) ≤(0,ρ X (T )) Localization maps restricted to subcomplexes of maximal cells are localization maps of the dual covector complexes.

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].For an acyclic matching M on P its critical elements C(M) ⊆ P are defined as C(M) := {a ∈ P | a / ∈ e for all e ∈ M}.In the same way we define acyclic matchings for a regular cell complex Σ (which is identified with its poset of faces).
Note that each acyclic matching M on a poset P gives an acyclic matching M ∨ on the dual poset P ∨ with C(M) = C(M ∨ ).
Restriction maps of face posets and corresponding geometric lattices.
Theorem 2.30.Let Σ be a regular cell complex and let Γ ⊆ Σ be a subcomplex.Assume that M is an acyclic matching on Σ with C(M) = Γ.Then Γ is a strong deformation retract of Σ.In particular, the inclusion Γ ֒→ Σ 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].
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 → Q be a poset map.Assume that for all q ∈ Q we have an acyclic matching M(q) on f −1 (q).Then M = q∈Q M(q) is an acyclic matching on P .

Combinatorics of modular flats of oriented matroids
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(M) be the geometric lattice of a simple oriented matroid and L its covector poset.Consider the localization map ρ X : L ∨ → L ∨ X for the dual covector complexes for an X ∈ L. We state a lemma which connects the notion of a modular flat of corank one to special subsets of topes.
We give an example to illustrate Lemma 3.1.
Example 3.2.Consider the hyperplane arrangement A in R 3 consisting of the hyperplanes and M = M(A) the associated oriented matroid with topes T .Then the flat X = {H 1 , H 2 , H 3 } ∈ L(M) is modular.A stereographic projection of A (or rather A ∩ 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 Proof.Suppose σ| X∨Y = τ | X∨Y , i.e. there is an e ∈ X ∨ Y such that σ e = τ e = 0.So e ∈ S(σ, τ ) ∩ X ∨ Y .By the fourth covector axiom (see Definition 2.1) there is a γ ∈ L with γ e = 0 and ), the oriented matroid axiom (4) in Definition 2.1 implies the statement.
(ρ X ↓ σ) . The dual covector complex L ∨ = L ∨ (A) of the arrangement A, and its contraction (L H ) ∨ , which is mapped isomorphically to the localization L ∨ X at a modular flat X.
We clearly see that the localization map ρ X , restricted to (L H ) ∨ is an isomorphism in accordance with Lemma 3.5.

A stratification of subcomplexes of the Salvetti complex
Again, let X ∈ L be a modular flat of corank 1.We now describe a special stratification of the poset fiber ( ρ X ↓ (0, B ′ )) for B ′ ∈ 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 Then it also follows from Lemma 3.1 that the order filter where S(T i−1 , T i ) consists of a single element in E \ X.Now, we define locally closed subcomplexes N i ⊆ ( ρ X ↓ (0, B ′ )) by Note that ( ρ X ↓ (0, B ′ )) = 0≤i≤k N i .Furthermore, we have the following useful lemma.
Lemma 4.1.For all 0 ≤ i ≤ k we have 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].

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.Invoking a homotopy colimit construction [WZ Ž99, 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.

Acyclic matchings for subcomplexes and the main result
Let M = (E, L) be an oriented matroid and L := L(M) its geometric lattice.
Firstly, we derive certain acyclic matchings on the dual covector complex L ∨ .Recall from Definition 2.15 the subcomplex L ∨ [Q] for a convex subset Q ⊆ T of topes.Proposition 6.1.Let ∅ = Q ⊆ T be a convex subset and L ∨ [Q] ⊆ L ∨ its associated dual subcomplex.Then there is an acyclic matching M on Proof.By Corollary 2.23 the subcomplex L(T \ Q) is a shellable ball.Hence, by Proposition 2.31 there is an acyclic matching M ′ on L(T \Q) whose critical cells consist of only one vertex of v ∈ L(T \ Q).Since L(T \ Q) is an order ideal in L, the matching M ′ can be extended to a matching M on L by a trivial matching on the order filter L \ L(T \ Q) (e.g.use Theorem 2.32).Hence, this new matching M has critical cells: C(M) = L\L(T \Q)⊔{v}.By Remark 2.16 we have (L\L(T \Q)) ∨ = L ∨ [Q].We can extend the matching M ∨ on L ∨ by one further matched pair (v, 0), since 0 ∈ L ∨ is the unique maximal element of L ∨ and v ⋖ L ∨ 0. This gives the claimed matching on L ∨ .
Let X ∈ L be some flat and ρ X : L ∨ → L ∨ X the associated localization map.From the preceding proposition we immediately get the following.Corollary 6.2.Let σ ∈ L ∨ X .Then there is an acyclic matching on L ∨ with C(M) = (ρ X ↓ σ).
Proof.Firstly, let T 0 < T 1 < . . .< T k be a linear order on the set of topes T (ρ X ↓ B ′ ) as described by Lemma 3.1.We utilize the stratification of ( ρ X ↓ (0, B ′ )) = 0≤i≤k N i from Section 4. Note that each N i ⊆ ( ρ X ↓ (0, B ′ )) is an order filter in S({(0, T 0 ), . . ., (0, T i )}).Thus, we have a natural order preserving map 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 ρ X is a poset quasi-fibration.
To see the last statement, pick a minimal element (B ′ , B ′ ) ∈ S X .We easily see that (ρ X ↓ B ′ ) = N aff,g for an oriented matroid N of rank two on E \ X ⊔ {g} by Lemma 3.1.Then ( ρ X ↓ (B ′ , B ′ )) = S aff,g (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 ∈ L(M) is modular of corank 1 and S X is aspherical, then S is aspherical as well.

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 ∈ P .Then we write π 1 (P, x) = π 1 (|∆(P )|, x) for the fundamental group of P .Similarly, if f : P → Q is an order preserving map, then we write π for the induced map between fundamental groups.
Firstly, we have the following lemma.
Lemma 7.1.Assume that f : P → Q is a poset quasi-fibration which has a section s : Q → P .Let x ∈ P and a = f (x) and assume further that |∆(Q)| and |∆(f ↓ a)| are K(π, 1)-spaces.Then the fundamental group π 1 (P, x) is a semi-direct product: Proof.By Theorem 5.1, the long exact sequence of the poset quasifibration degenerates to the short exact sequence: The map π 1 (s) gives a section to π 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 M = (E, L) be an oriented matroid and assume that X ∈ L(M) is modular and of corank 1. Assume further that S X is aspherical.Then for x ∈ S and a = ρ X (x) ∈ S X we have where F is a free group with |E \ X| generators.
Proof.By Theorem 6.4 the map ρ X : S → S X is a poset quasi fibration with ( ρ X ↓ a) homotopy equivalent to the affine Salvetti complex of an oriented matroid of rank 2 on |E \ X| + 1 elements.Hence ( ρ X ↓ a) ≃ S 1 ∨ . . .∨ S 1 is a wedge of |E \ X| circles.Thus, π 1 (( ρ X ↓ a), a) = F is a free group with |E \ X| generators.Furthermore, ρ 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.

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 M = (E, L) be an oriented matroid of rank 3 and let X 1 , X 2 ∈ L(M) be two distinct flats of rank 2 with X 1 ∩ X 2 = ∅.Then there is a one element extension M on E ⊔ {g} of M such that X 1 , X 2 are flats of rank 2 and g ∈ X 1 ∩ X 2 .Moreover, the extension can be chosen such that g / ∈ Y for any flat Y ∈ L \ {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 M be an oriented matroid of rank 3. Then there is an extension M of M which is supersolvable.
Proof.Let X ∈ L(M) be some flat of rank 2. If there is no Y ∈ L(M) of rank 2 with X ∩ Y = ∅, then by Lemma 2.12 the oriented matroid M is already supersolvable and there is nothing to show.Otherwise, let Y ∈ L(M) be a flat of rank 2 with X ∩ Y = ∅.By Lemma 8.1 we can extend M by a single element g to M such that X ∩ Y = {g}.Then we have Hence, after a finite number of single element extensions via Lemma 8.1 we arrive at an extension M ′ of 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 ≥ 4. E.g. the Edmonds-Fukuda-Mandel oriented matroid 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 ≥ 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.
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 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, M can be extended to a supersolvable oriented matroid 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 M is evidently not realizable.

2. 3 .
Modular flats and supersolvable geometric lattices.For this section, let L = L(M) be the geometric lattice of a simple oriented matroid M = (E, L).The notion of a modular flat in a geometric lattice was introduced by Stanley [Sta71].
Definition 2.14.Let B ∈ T .We define a partial order on T by R ≤ T : ⇐⇒ S(B, R) ⊆ S(B, T ).The resulting ranked poset T (M, B) with rank function rk(T ) := |S(B, T )| is called the tope poset with respect to B. Definition 2.15.Let T be the topes of an oriented matroid M = (E, L). (i) Let T, R ∈ T be two topes.Then the distance dist(T, R) of T and R is defined as dist(T, R) := |S(T, R)|.(ii) Let e ∈ E. Denote by T + e := {T ∈ T | T e = +} ⊆ T the positive halfspace with respect to e.The negative halfspace T − e ⊆ T is defined analogously.Set H = H(M) = {T ǫ e | e ∈ E, ǫ ∈ {+, −}}, and an H ∈ H is called a halfspace of M. (iii) Let Q ⊆ T be a subset of topes.Then the convex hull conv(Q)

Definition 2. 29 .
Let P = (P, ≤) be a poset.We associate a directed graph G(Σ) = (V, E) with vertex set V = P the ground set of P and directed edges E := {(a, b) | a ⋖ b} its cover relations, i.e. the Hasse diagram of the poset.A subset M ⊆ E is called a matching on P if every element of P appears in at most one (a, b) ∈ M. Let G(P, M) = (V, E ′ ) be the new directed graph with E ′ := E \ M ∪{(b, a) | (a, b) ∈ M}, i.e. the directed graph constructed from G(P ) by inverting all edges in M. A matching M on Σ is called acyclic if G(P, M) has no directed cycles.

π
i (|∆(j)|) π i (|∆(f )|) where j : (f ↓ a) ֒→ P is the natural inclusion.Theorem 5.1 motivates the following notion.Definition 5.2.Let f : P → Q be a poset map such that for all a ≤ b (a, b ∈ Q) the inclusion (f ↓ a) ֒→ (f ↓ b) is a homotopy equivalence.Then f is called a poset quasi-fibration.

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