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Universita degli Studi di Pisa

FACOLTA DI MATEMATICA

Corso di Laurea magistrale in Matematica

Tesi di laurea magistrale

Hyperplane Arrangement and Discrete Morse Theory

Candidato:

Davide LofanoRelatore:

Mario Salvetti

Anno Accademico 2017–2018

Contents

Introduction v

1 Discrete Morse Theory 11.1 Collapsibility of posets . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Discrete Morse Theory for CW-complex . . . . . . . . . . . . . . . 31.3 Out-j Collapsibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Shellability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Discrete Morse theory in the infinite case . . . . . . . . . . . . . . . 11

2 Hyperplane arrangements 132.1 First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Poincare polynomial . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Oriented Matroids . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The complement of a hyperplane arrangement . . . . . . . . . . . . 162.2.1 The Salvetti complex . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Salvetti-Settepanella and the polar order . . . . . . . . . . . 182.3.2 Adiprasito and 2-arrangements . . . . . . . . . . . . . . . . 242.3.3 Delucchi and the central case . . . . . . . . . . . . . . . . . 28

3 Local Abelian Homology 353.1 Homology with local coefficients . . . . . . . . . . . . . . . . . . . . 353.2 Local homology of hyperplane arrangement . . . . . . . . . . . . . . 383.3 Local homology of the Braid arrangements . . . . . . . . . . . . . . 39

4 Minimality of infinite affine arrangement 534.1 Decomposition of the Salvetti complex . . . . . . . . . . . . . . . . 534.2 Construction of the acyclic matching . . . . . . . . . . . . . . . . . 554.3 Euclidean orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Local homology of line arrangement . . . . . . . . . . . . . . . . . . 71

iii

Contents

iv

Introduction

The aim of this thesis is to study the complement of a hyperplane arrangementusing the techniques of Discrete Morse theory.

A hyperplane arrangement is simply a set A = H1, H2, . . . of hyperplanes ina vector space. This object has been widely studied especially to find correlationsbetween its topological properties and the combinatorics of the intersections of thehyperplanes. One of the most studied topological objects is the complement, i.e.the vector space minus the hyperplanes and its homology and homotopy groups.One of the question that we are going to answer is if this complement is a minimalspace, meaning that it is homotopy equivalent to a CW−complex with as manyi−cells as the i−th Betti number. We will see that the answer is positive in varioussettings.

Having a minimal complex, if it is given explicitly, can also help in studyingvarious properties of the complement, we will focus in particular on abelian lo-cal homology. Local homology is an important tool for the study of hyperplanearrangements because it gives us informations on a special fibration on the com-plement, called Milnor fibration as well as informations about the characteristicvarieties. Even if there is plenty of research on the subject there is still a lotunknown about local homology even for the most famous arrangements, like theBraid ones.

In the first chapter we talk about Discrete Morse theory, first introduced byForman in [For98],[For02]. The aim is to reduce a CW−complex or in generalvarious type of topological and combinatorial object to smaller ones, called Morsecomplex with a series of elementary collapsments such that the properties of thecomplex are still the same. We will study explore the correlation of this theorywith shellability [Del08] and focus on different aspect, all of which will becameuseful in the following chapters.

In the second chapter, we first give a brief introduction to the theory of hyper-plane arrangement, presenting some of the most important known results, follow-ing [OT13], concerning in particular their combinatorial properties and homologygroups. In the special case of complexified real arrangement we introduce theSalvetti complex [Sal87], a CW−complex homotopy equivalent to the complemen-

v

Chapter 0. Introduction

tary of the arrangement. We then review three different articles that with similartechniques have reduced this complex to a minimal one (with as many cells as theBetti numbers).

The following chapter introduce the concept of local homology and its corre-lations with Discrete Morse theory, in particular how we could compute the localhomology of the Morse complex. We focus then our attention to a special kind ofhyperplane arrangement, called the Braid arrangement and we explicitly write aprogram in Sage to compute the boundary in local homology.

In the last chapter we try to give our contribute to the subject. It is a jointwork with Giovanni Paolini in which in a similar way to what has been done in[Del08] in the case of oriented matroids, we reduce the Salvetti complex in the caseof affine, locally finite hyperplane arrangment to a minimal Morse complex givinga special characterization to the critical cells.

vi

Chapter 1

Discrete Morse Theory

In this chapter we will introduce Discrete Morse Theory, an important tool tostudy the homology and cohomology of CW−complexes collapsing some cells.Our approach will be similar to the one presented in [Koz08] but sometimes wewill also reference to the originals Forman’s articles [For02],[For98]. We will alsogive briefly an introduction to shellability and how it compares to the DiscreteMorse Theory since it will be useful to understand [Del08].

1.1 Collapsibility of posets

We start here with the case of posets and then in the following section we will talkwith more details about the case of CW−complex. The idea behind the DiscreteMorse Theory is that of collapsibility, in particular of elementary collapse.

Definition 1.1.1. Given a generalized simplicial complex X, an elementary col-lapse is the removal of the interiors of two simplices σ and τ such that:

• dimσ = dim τ + 1,

• the only simplex containing σ is σ itself,

• the only simplices containing τ are σ and τ .

The idea is then to have a set of collapses and to encode them in a matching,consisting of collection of pairs. Of course not every matching arise from a series ofelementary collapsment, so what we want to do is to characterize said matchings.

Definition 1.1.2. 1. A partial matching of a poset (P,≺) is a subset M ⊆P × P such that

• (a, b) ∈M implies b a and @ c such that b c a, we call b = u(a),

1

Chapter 1. Discrete Morse Theory

• each a ∈ P belongs to at most one element in M.

2. Given the Hasse diagram of P we can orient all the edges so that they pointfrom the larger element to the smaller one and then we change the orientationof the edges in a partial matchingM. M is then called acyclic if this orientedgraph has no cycles.

The idea now is to remove all the matched elements in some appropriate orderwithout changing the homotopy of the underlying space. We call the unmatchedelements critical.

The next theorem is the crucial step in being able to prove so in the followingsection; the idea is to look for linear extensions of our poset that in a certain wayrespect the matching.

Theorem 1.1.3. A partial matching on P is acyclic if and only if there exists alinear extension L of P such that the elements a and u(a) follow consecutively inL for every pair (a, u(a)) in the matching.

Proof. We prove first the second arrow since it is easier. We have a partial match-ing M and a linear extension L. We want to show that given a short sequenceof the form a1 u(a1) a2 we have that a1 >L a2. From this follow immedi-ately that the matching is acyclic. But the proof is clear because we have thatu(a1) >L a2 since it is true in P and L is an extension. Moreover we also nowthat a1 is the biggest element smaller than u(a1) and that the extension is linearso a1 >L a2.

To prove the contrary now we have an acyclic matching M and we want todefine inductively the linear extension L. We denote with Q the set of elementsthat are already ordered in L and we start with Q = ∅. Let W be the set ofminimal elements in P\Q. At each step we have two possibilities:

• One of the elements c in W is critical.

We add c to L as the largest element and proceed with Q ∪ c.

• All elements in W are matched.

Let’s consider the subgraph of the Hasse diagram of P\Q induced by W ∪u(W ) with the edges oriented as in the definition above and call this graphG.

If there is an element a ∈ W such that there is only one edge towards oroutwards from u(a) (the one starting in a), then we can add a and u(a) ontop of L and proceed with Q ∪ a, u(a). If this is not true we want to saythat G has a cycle that contradicts our hypothesis. This follow from the factthat every vertex has an edge that point outwards.

2

1.2. Discrete Morse Theory for CW-complex

Theorem 1.1.4 (Patchwork lemma). Assume that φ : P → Q is an order-preserving map, and assume that we have acyclic matchings on subposets φ−1(q),for all q ∈ Q. Then the union of these matching is itself an acyclic matching onP .

Proof. The proof of this lemma is straightforward. Indeed the union is surely apartial matching so we only need to prove that it has no cycles.

Let’s suppose we have a cycle of the form

a1 b1 a2 . . . an = a1

Since the matching on each fiber is acyclic in at least one point we have to changefiber, let i be that point, meaning that φ(ai) = φ(bi) 6= φ(ai+1). But bi ai+1

implies that φ(bi) φ(ai+1) since the map is order preserving. This is true everytimes we change fiber so we can’t come back to the starting fiber if we have changeat least once. This implies that the matching is acyclic.

1.2 Discrete Morse Theory for CW-complex

We want now to focus our study to the special case of CW−complexes. We recallhere that we can associate to a CW−complex X the poset of its faces F(X) (witha minimal element 0 representing the empty face). Then we can construct anacyclic matching on F(X) and study what this tell us about the complex.

Definition 1.2.1 (cellular elementary collapse). Let Y be a subcomplex of aCW−complex X. We say that Y is obtained from X by a cellular elementarycollapse if X can be obtained from Y by attaching two cells: Bn−1

+ and Bn, whereBn−1

+ is one of the closed hemisphere on the boundary of Bn and Bn is attachedto Y by a map φ on the other hemisphere.

We are now ready to state and prove the main theorem in this chapter thatexplains in detail how, given an acyclic matching, we are able to “reduce” aCW−complex to a smaller one preserving homotopy and homology.

Theorem 1.2.2. Let X be a regular CW−complex, and let M be an acyclicmatching on F(X)\0. Let ci denote the number of critical i−dimensional cellsof X.

(i) If the critical cells form a subcomplex Xc of X, then there exists a sequenceof cellular collapses leading from X to Xc.

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(ii) In general, the space X is homotopy equivalent to Xc, where Xc is a CW−complex with ci cells in dimension i called the Morse complex of M.

(iii) There is a natural indexing of cells of Xc with the critical cell of X such thatfor any two cells σ and τ of Xc satisfying dimσ = dim τ + 1, the incidencenumber is given by

[τ : σ] =∑c

ω(c)

Where the sum is taken over all alternating paths c connecting σ with τ , i.e.sequences c = (σ, a1, u(a1), . . . , at, u(at), τ) such that σ a1, u(at) τ , andu(ai) ai+1, for I = 1, . . . , t− 1. The quantity ω(c) is defined by:

ω(c) = (−1)t[a1 : σ][τ : u(at)]t∏i=1

[ai : u(ai)]t−1∏i=1

[ai+1 : u(ai)]

where all the incidence numbers are taken in X.

Proof. (i) Since the critical cells form a subcomplex, watching closely at theproof of 1.1.3 we note that we can choose a linear extension L such that allthe critical cells come first. Hence, L gives a sequence of cellular collapsesfrom X to Xc if we read in decreasing order.

(ii) The proof here is by induction on the cardinality of F(X), L is again a linearextension given by 1.1.3. The base step when |F(X)| = 1 is clear so we cando the induction step. Let then σ ∈ X be the bigger cell with respect to L.There are two possibilities.

• σ is critical.Let X = X\Intσ and ϕ : δσ → X the attaching map of σ.

X has one less cell than X and the restriction of M is clearly stillan acyclic matching so for the inductive hypothesis we have that thereexists an homotopy equivalence h between X and a CW−complex Xc

with as many cells as the critical cells of X (that are the same as thecritical cells of X minus σ.

We are now able to attach σ to Xc toward the map h ϕ : δσ → Xc.The theorem now follows if we set Xc = Xc ∪hϕ σ.

• σ is not critical.There exist τ such that (τ, σ) ∈ M and since they are the highest ele-ment in L removing this pair from X is a cellular collapse, in particularX and X = X\(Intσ∪ Intτ) are homotopy equivalent. Again as beforethe matchingM\(τ, σ) is an acyclic matching on X and by induction

4

1.3. Out-j Collapsibility

X is homotopy equivalent to a CW−complex Xc with ci i−dimensionalcells since the critical cells of X are the same of X. But compositionof homotopy equivalences is still an homotopy equivalence so if we setXc = Xc we obtain the thesis.

(iii) Let σ be a critical cell of X of dimension, we want to study the attachingmap of σ after all the collapses.

Recall from the proof of the previous point and the construction of L in 1.1.3that the collapses can be performed in a order such that the dimension ofthe collapsing pair does not increase.

Let’s at first study what happens after one collapse of a pair (a, u(a)) witha of dimension n − 1. If a was not in the image of the attaching map ofσ then this collapse does not alter the attaching map. Otherwise a getsreplaced with δu(a)\Int(a). This process will continue in sequence until allthe collapses of pair of dimension (n, n− 1) are done.

Once all this pairs have been collapsed the cells in the image of the attachingmap of σ are the critical ones and those that are matched to the cells ofdimension n− 2. But the latters after their collapse leave no contribution tothe incidence number. Then the only thing that we are interested in is howoften and with which orientations the critical cells of dimension n − 1 willappear on δσ.

From our procedure above follows that the appearance of a critical cell τ isin one-to-one with the alternating paths between σ and τ , moreover whena gets replaced by δu(a)\Int(a) each cell b in the latter gets the incidencenumber −ε[a : u(a)][b : u(a)] where ε is the incidence number of a. Puttingtogether this two facts concludes the proof.

1.3 Out-j Collapsibility

This section is devoted to the study of a particular use of the Discrete Morse theorythat will be useful to review the article [Adi14] in the following chapter.

We begin with some definitions.

Definition 1.3.1. Let X be a regular CW−complex and Y a non-empty sub-complex. Given an acyclic matching on X we say that a face τ of Y is outwardlymatched with respect to the pair (X, Y ) if it is matched with a face that does notbelong to Y .

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Definition 1.3.2. Let X and Y as above. The pair (X, Y ) out-j collapses to thepair (X ′, Y ∩X ′), and we write

(X, Y ) out−j (X ′, Y ∩X ′)

if there exists a collapsing sequence that reduces X to X ′ and every outwardlymatched face with respect to the pair (X, Y ) has dimension j.

We say that the pair (X, Y ) is out-j collapsible if there is a vertex v of Y suchthat

(X, Y ) out−j (v, v).

Even if the notion of out-j collapses may seem at a first sight not very usefulwe want now to show that it is actually quite natural. In particular, the numberof outwardly matches faces is independent of the collapsing sequence.

Proposition 1.3.3. Let (X, Y ) be an out-j collapsible pair, with Y 6= ∅. Thenumber of outwardly matched j−faces of Y is independent of the out-j collapsingsequence chosen, and equal to (−1)j(χ(Y )−1) (where χ is the Euler characteristic).Moreover, the following facts are equivalent:

1. Y is contractible,

2. There exists a collapsing sequence that has no outwardly matched faces withrespect to (X, Y ),

3. Y is collapsible.

Proof. The second part is simply a consequence of the first one, indeed:

• If Y is contractible then χ(Y ) = 1 but then if we take any collapsing sequencewe have that the number of outwardly matched j−faces is equal to zero foreach j. This proves 1)⇒ 2).

• If the sequence has no outwardly matched faces, then we can restrict it to Yand obtain a collapsing sequence for Y . This proves 2)⇒ 3).

• It is always true that collapsible implies contractible, so 3)⇒ 1).

We need now to prove the first statement. Let’s then fix an out-j collapsingsequence. The proof is an easy use of double counting. To do it, we need to definesome sets:

• F = faces of Y and fi = #i−faces of F,

• O = outwardly matched faces of Y and oi = #i−faces in O,

6

1.4. Shellability

• Q = faces of Y matched with faces of Y and qi = #i−faces in Q.

Moreover, let v be the vertex onto which X collapses. We then have thatfi = oi + qi if i ≥ 1 and that f0 = o0 + q0 + 1. From this follow the chain ofequalities:

χ(Y ) =∑

(−1)ifi = 1 +∑

(−1)ioi +∑

(−1)iqi = 1 + (−1)joj

and this prove the proposition.

In the following we will also need this lemma that can be proved pretty easily.

Lemma 1.3.4. Let X be a regular CW−complex, Y a subcomplex and v any vertexof X. Assume that v /∈ Y or Lk(v, Y ) is nonempty, and that (Lk(v,X), Lk(v, Y ))is out-j collapsible. Then (X, Y ) out-j collapses to (X − v, Y − v) (where X − vis the subcomplex of X without all the faces that contained v). On the other handif v ∈ Y but Lk(v, Y ) = ∅, then (X, Y ) out-0 collapses to (X − v, Y − v) if andoly if Lk(v,X) is collapsible.

The last thing that we want to state in this section is a generalization of 1.2.2that include the outwardly matches introduce above, the prove is similar to theone written in 1.2.2. For more detail we send the reader to [Adi14].

Theorem 1.3.5. Let X be a regular CW−complex and Y a subcomplex. Let Mbe an acyclic matching on X that does not have any outwardly matched faces withrespect to the pair (X, Y ). Then X is up to homotopy equivalence obtained fromY by attaching one cell of dimension k for every critical k−cell of M not in Y .

1.4 Shellability

As a first thing we need to define what shellable means and see why it is useful.Then we will explore the connection with the Discrete Morse Theory.

We will first study the case of simplicial complex, again following [Koz08].

Definition 1.4.1. A simplicial complex X is called shellable if its maximal sim-plices can be arranged in linear order F1, F2, . . . , Ft in such a way that the sub-complex (

⋃k−1i=1 Fi) ∩ Fk is pure (meaning that all its maximal simplex have the

same dimension) and (dimFk − 1)− dimensional for all k = 2, . . . , t.

The idea behind shellability is that we want to use this order to build thecomplex gluing together the maximal simplices starting from the first one andgoing on.

7


The most important topological fact that we need to highlight is that for anyn−dimensional simplex, if we take the union of some of the (n− 1)−dimensionalsimplices in its boundary then or we obtain something homeomorphic to a sphere ifwe have taken the entire boundary or in all the other cases is contractible. Whilegluing our simplices then if we glue along the entire boundary we are adding agenerator in the right homology group, in the other cases we are simply addingsomething contractible. To make this clearer we need a definition and a theorem.

Definition 1.4.2. A maximal simplex σ is called spanning with respect to a givenshelling order if it is glued along its entire boundary.

Theorem 1.4.3. [Koz08, Theorem 12.3] Assume that X is a shellable simplicialcomplex with F1, F2, . . . , Ft being the corresponding shelling order of the maximalsimplices, and Σ being the set of spanning simplices. Then the following facts hold:

(i) The generalized simplicial complex obtained by the removal of the interiors ofthe spanning simplices, that is, the complex X = X\

⋃σ∈Σ Intσ, is collapsi-

ble.

(ii) The generalized simplicial complex X is homotopy equivalent to a wedge ofspheres that are indexed by the spanning simplices and have correspondingdimensions.

(iii) The cohomology groups of X with integer coefficients are free, and the set ofelementary cochains σ∗σ∈Σ can be taken as a basis.

In reality we are interested to work with CW−complexes and in this case thedefinition is a bit more complex but it is immediate to see that it reduces to theabove definition in case of simplicial complexes.

Definition 1.4.4. [BW97, Definition 13.1] A regular CW−complex X is calledshellable if its maximal cells can be arranged in linear order F1, F2, . . . , Ft in such away that if dimX ≥ 1 the following conditions are satisfied (δ(F ) is the subcomplexconsisting of all proper faces of F ):

• δ(F1) is shellable,

• (⋃k−1i=1 Fi) ∩ Fk is pure and (dimFk − 1)− dimensional for all k = 2, . . . , t,

• δ(Fk) admits a shelling in which the cells of (⋃k−1i=1 Fi)∩Fk come first, for all

k = 2, . . . , t.

Even for CW−complexes can be proved a theorem analogous to 1.4.3 as done in[BW97]. As a consequence of this we can immediately see the following propositionthat will be useful.

8

1.4. Shellability

Proposition 1.4.5. Let K be a regular CW−decomposition of a sphere. Then inevery shelling order of K the only spanning facet is the last one.

Proof. We prove it by contraposition. Let K be a CW−complex with a home-omorphism φ : K → Sd and a shelling order such that the last facet F of theorder is not the only spanning facet. If now we call K ′ the union of all the facetsexcept F we obtain that K ′ is not contractible since it contains at least one span-ning facet by Theorem 1.4.3. But K ′ is homeomorphic to Sd\φ(F\K ′) and this iscontractible since F\K ′ is. This gives us a contradiction.

We now want to see the correlation between shellability and Discrete MorseTheory. It is clear that there is one since both are about collapsibility of complexesbut we want to make clear how we can make an acyclic matching starting from ashelling order following [Del08].

The first thing we need is a new definition and a new way to see the shellability.

Notation. Given a poset (P,<) we will denote

• coat(p) = q < p | @x ∈ P : q < x < p ,

• P≤q = p ∈ P | p ≤ q,

• A totally ordered subset ω ⊂ P is called chain and its length is l(ω) = |ω|−1,

• l(P ) is the maximum length of a chain contained in P ,

• P is said bounded if it has a maximal and a minimal element.

Observation 1.4.6. The poset F of facets of a CW−complex is not bounded butwe can add to it a maximal element if necessary to make it bounded, we will callthis element 1 and the corresponding poset F .

Definition 1.4.7. A bounded poset (P,<) is said to admit a recursive coatomordering ≺ if l(P ) = 1 or if l(P ) > 1 and there is a total ordering on the setcoat(1) such that:

• for all p ∈ coat(1), the interval [0, p] admits a recursive coatom ordering ≺pin which the coatoms of the intervals [hat0, q] for q ≺ p come first.

• for all p ≺ q, if p, q > y then there is p′ ≺ q and z ∈ coat(q) such thatp′ > z ≥ y.

Theorem 1.4.8. [BW97, Theorem 13.2] If F is the poset of faces of a regularCW−complex X, then a total ordering of the maximal faces of X is a shellingorder if and only if it is a recursive coatom ordering of F .

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We want now to describe, using this order, an acyclic matching of the posetof cells of every shellable CW−complex. We will continue to follow [Del08] butspecialize to the case of poset of facets.

The first thing we need is a technical lemma of which we omit the proof.

Lemma 1.4.9. Let F be the poset of facets of a d−dimensional CW−complex Xand let Fk be the (d− k)−dimensional facets. Given a recursive coatom ordering≺ on F it is then possible to define a family of total orders (Fk,@k) with thefollowing properties: Given σ ∈ F i, and writing Qσ =

⋃σ′@ip

coat(σ′,

1. The order induced by @i+1 on Dσ = coat(σ)\Qσ can be extended to a re-cursive coatom ordering ≺σ of coat(p) in which the elements of Qσ comefirst.

2. For all σ′ @i σ, if σ′, σ > τ , then there exist σ′′ @i σ and ρ ∈ coat(σ) suchthat σ′′ > ρ ≥ τ .

Definition 1.4.10. Let F as before. πi : F i → F i+1 is defined by:

πi(σ) = max@i+1

τ ∈ F i+1 | σ > τ.

A shelling-type ordering of F is a linear extension C of F given by σ C τ if andonly if σ @i τ in case σ, τ are cells of the same dimension or σ vi πiπi+1 . . . πj−1(τ)in case σ ∈ F i, τ ∈ F j and i > j.

It is an easy check to see that this is a well defined linear order. Using this weare now finally able to explicitly write the matching.

Lemma 1.4.11. Every shelling type ordering C of F induces an acyclic matchingM on F .

Proof. We will write F i = σi1, . . . , σiki, where σij @i σij+1 for all j. We want now

to define the matching.We start withM = (σ0

1, π0(σ01)). For every j = 2, . . . , k0 we add σ0

j , π0(σ0j to

M if the second element is not already matched.We then further expand M is a similar way i. e. for i = 1, . . . , d − 1 and for

j = 1, . . . , ki if σij is not alreasy matched and πi(σij) 6= πi(σ

il) for all l < j then we

add (σij, πi(σij)) to M.

The shelling-type ordering assures us that what we have defined is acyclic.

Now that we have construct the matching we want to compare its critical cellswith the spanning facets and we obtain the best possible result.

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1.5. Discrete Morse theory in the infinite case

Theorem 1.4.12. Every shelling of a regular CW−complex X induces an acyclicmatching of the poset of faces of X. Moreover, the critical cells of this matchingcorrespond to the spanning faces of the given shelling.

We want now to briefly recall a couple of notions about polytope and polyhedrathat will come useful later.

Definition 1.4.13. A polyhedron is an intersection of finitely many closed halfs-paces in some Rd. A polytope is a bounded polyhedron.

Given a polyhedron P , denote by F(P ) the set of its faces (considering thepolyhedron P itself as a trivial face). The faces of codimension 1 are called facets.In addition, denote by F(∂P ) the boundary complex of P , i.e. the complex thatcontains only the proper faces of P .

Definition 1.4.14. We say that a facet G ∈ F(P ) is visible from a point p ∈ Rdif every line segment segment from p to a point of G does not intersect the interiorof P (cf. [Zie12, Theorem 8.12]). We say that a face F ∈ F(P ) is visible from p ifall the facets G ⊇ F of P are visible from p. In particular, notice that the entirepolyhedron P is always visible.

The notion of shellability is the same given above for a CW−complex, wherewe say that a polytope is shellable if it’s boundary complex admits a shelling order.

The following theorems tell us that a polytope is always shellable.

Lemma 1.4.15 ([Zie12, Lemma 8.10]). If F1, F2, . . . , Fs is a shelling order for theboundary of a polytope P , then so is the reverse order Fs, Fs−1, . . . , F1.

Theorem 1.4.16 ([BM72], [Zie12, Theorem 8.12]). Let P ⊆ Rd be a d-polytope,and let x ∈ Rd be a point outside P . If x lies in general position (that is, not inthe affine hull of a facet of P ), then the boundary complex of the polytope has ashelling in which the facets of P that are visible from x come first.

1.5 Discrete Morse theory in the infinite case

We want here to expand discrete Morse theory to not necessarily finite CW−complexes following [BW02].

The first thing needed is the definition of grading and of proper acyclic match-ing.

Definition 1.5.1 (Grading [BW02]). Let Q be a poset. A poset map ϕ : P → Qis called a Q-grading of P . The Q-grading ϕ is compact if ϕ−1(Q≤q) ⊆ P is finitefor all q ∈ Q. A matching M on P is homogeneous with respect to the Q-gradingϕ if ϕ(p) = ϕ(p′) for all (p, p′) ∈ M . An acyclic matching M is proper if it ishomogeneous with respect to some compact grading.

11


The following is a direct consequence of the definition of proper matching (cf.[BW02, Definition 3.2.5 and Remark 3.2.17]).

Lemma 1.5.2 ([BW02]). Let M be a proper acyclic matching on a poset P , andlet p ∈ P . Then there is a finite number of alternating paths starting from p, andeach of them has a finite length.

Proof. By definition, P is homogeneous with respect to a compact Q−grading ϕ.Let ϕ(p) = q. It is clear that the restriction on the matching M to ϕ−1(Q≤q) isstill acyclic, we call it Mq and the corresponding oriented graph GMq . This is asubgraph of GM and it is easy to see that every alternatin path starting in p isactually in this subgraph. Indeed every edge, or is inM and then does not changethe fiber with respect of ϕ, or is in E \M and then goes from a fiber to a smallerone.

The Lemma is now obvious, since GMq has no cycles and a finite number ofedges.

We are now ready to state the main theorem of discrete Morse theory in theinfinite case. We will see that it is really similar to our previous version 1.2.2 butwith one more hypothesis on the matching. This particular formulation followsfrom [BW02, Theorem 3.2.14 and Remark 3.2.17]

Theorem 1.5.3. Let X be a regular CW complex, and let P be its poset of cells.If M is a proper acyclic matching on P , then X is homotopy equivalent to a CWcomplex Xc (called the Morse complex of M) with cells in dimension-preservingbijection with the critical cells of X.

The construction of the Morse complex is explicit in terms of the CW complexX and the matching M and the proof is similar to 1.2.2 (see for example [BW02]),and in a similar way we are able to compute the incidence numbers of the Morsecomplex studying the alternating paths.

12

Chapter 2

Hyperplane arrangements

The aim of this chapter is to introduce the reader to the hyperplanes arrangementsand their structure. In the first part we will partially follow [OT13] for the generaldefinitions, several facts will be listed without proof which can however be foundin the previous cited reference.

Then we will focus on the use of Discrete Morse Theory to find perfect acyclicmatching and review article from Salvetti [SS07], Delucchi [Del08] and Adiprasito[Adi14].

2.1 First definitions

An arrangement of hyperplanes is a finite (sometimes even infinite) collection A =H1, · · · , Hr of codimension one affine subspace in a n−dimensional K−vectorspace V .

Definition 2.1.1. L = L(A) is the poset of nonempty intersections of elementsof A with order given by the reverse inclusion; meaning that

X ≤ Y ⇔ Y ⊆ X

.

Definition 2.1.2. • A is central if⋂A 6= ∅,

• A is linear if 0 ∈⋂A, is called affine otherwise,

• A is essential if rk(A) = n, where the rank of an arrangement is the rank(the codimension) of a maximal element of L(A).

It is an easy proof that the rank is well defined, i.e. that maximal elements ofL(A) have all the same rank.

13

Chapter 2. Hyperplane arrangements

Moreover we can define two operations on L(A) called the join and the meetas follow:

Definition 2.1.3. Given X, Y ∈ L(A) we define their meet by

X ∧ Y = ∩Z ∈ L | X ∪ Y ⊆ Z

If X ∩ Y 6= ∅ we define their join by

X ∨ Y = X ∩ Y

It is easy to prove that if A is a central arrangement than L(A) is a geometriclattice, in particular all joins exist.

It is now useful to have some definitions that let us in a certain way consideronly a part of an arrangement. Of course, if we take B ⊂ A we can consider B asan arrangement in the same V and we will call it a subarrangement, but the moreimportant constructions are the following:

Definition 2.1.4. For X ∈ L(A) we define the subarrangement AX as

AX = H ∈ A | X ⊂ H

and the restriction of A to X as an arrangement AX in the vector space X definedby

AX = X ∩H | H ∈ A\AX and X ∩H 6= ∅.

Another important construction is the coning, a method that allows to compareaffine and central arrangements.

First, for each hyperplane Hi there exists αi ∈ (K∗)n, ai ∈ K (not unique)such that Hi = v | αiv = ai. We can then define the arrangement’s polynomialQA(x) ∈ K[x1, · · · , xn] as

∏i(αi(x) − ai). We should note that this polynomial

defines completely the arrangement, meaning that we are able from QA to extractall the hyperplanes.

Then, given an affine arrangement A we define its cone, called cA as thearrangement with polynomial Q(cA) = x0Q

′ where Q′ is the homogenization ofQ(A) ∈ K[xo, · · · , xn].

The last construction that we want to define is the complexification. Let sup-pose that we have a field extension L over K and an arrangement AK in a K−vectorspace that we will call VK. This gives rise naturally to an arrangement over Lwhere the vector space will be V = VK ⊗K L and the collection of hyperplanesAL = H ⊗K L | H ∈ AK.

The most important example of this, and the only one that we will study isthat of complexified real arrangements, i.e. when K = R and L = C.

14

2.1. First definitions

2.1.1 Poincare polynomial

The Poincare polynomial is a very important tool in the study of arrangementsof hyperplanes but before being able to define it we need to study the Mobiusfunction.

Definition 2.1.5. Let A be an arrangement and L = L(A), the Mobius functionµ : L× L→ Z is defined as follows:

µ(X,X) = 1 if X ∈ L∑X≤Z≤Y

µ(X,Z) = 0 ifX, Y, Z ∈ L and X < Y

µ(X, Y ) = 0 otherwise.

From this function we are then able to define µ(X):

Definition 2.1.6. For X ∈ L µ(X) = µ(V,X).

At this point we are able to define the object we are interested in:

Definition 2.1.7. The Poincare polynomial of A is defined by:

π(A, t) =∑X∈L

µ(X)(−t)rk(X)

Example 2.1.8 (Braid arrangement and its Poincare polynomial). The braid ar-rangement is probably one of the most important family of reflection arrangementssince they arise naturally from the symmetric groups. But since we are not inter-ested in reflection arrangement at the moment we will simply define the n−th braidarrangement An as the set of hyperplanes Hij = ker(xi−xj) for 1 ≤ i < j ≤ n+ 1where the xi are the coordinates in Rn+1.

The Poincare polynomial of An is

π(An, t) = (1 + t)(1 + 2t) . . . (1 + nt)

a proof of which can be found in [OT13].

The reason we are so interested in this polynomial is because it is equal tothe Poincare polynomial of several graded algebras and, as we will see in the nextsections, it has a close relation with the cohomology of the complement of A.

15


2.1.2 Oriented Matroids

We want now to give a brief introduction to a very important argument, that is thedefinition of oriented matroids. These are combinatorial objects that in a certainway can be seen as a generalization of central hyperplane arrangement. We willhere only give the basic definitions, following [BLVS+99].

Definition 2.1.9. Given a ground set E, a collection V ∈ +,−, 0E is the setof vectors of an oriented matroid M if and only if the following properties aresatisfied:

1. (0, . . . , 0) ∈ V ,

2. if X ∈ V , then −X ∈ V ,

3. for all X, Y ∈ V , X Y ∈ V ,

4. for all X, Y ∈ V , given e, f ∈ E such that Xe = −Ye and not both Xf , Yfequal 0, there is Z ∈ V such that Ze = 0, Zf 6= 0, and if Zi 6= 0 then Ziequals Xi or Yi.

Let us now take a real linear arrangement of hyperplanes A = H1, . . . , Hn.Its combinatorial data is encoded by the associated oriented matroidMA constructas follows:

E is the set v1, . . . , vn where, for all i, vi is normal to Hi. An element X isin V if there exists λi ∈ R+ such that:

n∑i=1

Xiλivi = 0

Definition 2.1.10. An oriented matroid is called realizable if it is of the formMAfor some real linear arrangement A.

2.2 The complement of a hyperplane arrange-

ment

Given an arrangement A we will call with M(A) the complement V \⋃A and a

chamber will be a connected component of M(A). We will call the set of chamberas C(A).

The number of chambers if A is a real arrangement is determined by itsPoincare polynomial as follows.

16

2.2. The complement of a hyperplane arrangement

Theorem 2.2.1. Let A be a real arrangement, then

|C(A)| = π(A, 1)

We won’t prove this or the following theorems in this section since everythingcan be found with every detail in [OT13]. Here we will just give a sketch of whathas been done regarding the cohomology of M(A) and the results that will be usedin the following.

Starting from an hyperplane arrangement there are different graded algebrasthat one can build. One of these is the algebra of differential forms, called R(A).

Notation. Given an affine arrangement A in V a K−vector space we will denoteby:

• S the symmetric algebra of V ∗ and we will write S = K[V ],

• F the quotient field of S and we will write S = K(V ),

• Ω(V ) the graded exterior algebra of F ⊗ V ∗ with the usual differential d.

Definition 2.2.2. For every H ∈ A let αH ∈ S be a polynomial of degree 1 suchthat H = ker(αH) and let ωH = dαH/αH ∈ Ω1(V ). R(A) is the subalgebra ofΩ(V ) generated by 1 and ωH for H ∈ A.

Theorem 2.2.3. The Poincare polynomial of R(A) is

Poin(R(A), t) = π(A, t)

So we already have another characterization of the Poincare polynomial of anarrangement and a reason for which is called in this way, but this is not yet thecharacterization we are interested in.

Let’s call ηH = ωH/2πi and with [ηH ] the cohomology class of ηH in H1(M(A)).We then have the following theorem,

Theorem 2.2.4. The map η : R(A) → H∗(M(A) that sends ωH to [ηH ] is anisomorphism of graded algebras.

Since we have already stated that the Poincare polynomial of R(A) is the sameof π(A, t) we immediately obtain from the previous theorem that if A is a complexarrangement:

Poin(H∗(M(A)), t) = π(A, t).But

Poin(H∗(M(A)) =∑p≥0

bp(M)tp

where the bp(M) are the Betti number.If we put together what we have just said with 2.2.1 we immediately obtain

one of the most used theorem in the following.

17


Theorem 2.2.5. Let A be a real arrangement than the number of chambers of Ais equal to the sum of the Betti numbers of the complement of the complexificationof A.

2.2.1 The Salvetti complex

The Salvetti complex, first introduced in [Sal87], is a CW-complex homotopyequivalent to the complement M(A) in the case of complexified real arrangement.

This complex is particular helpful in studying this particular class of arrange-ment as we will see in the next section. For now let’s try to understand how it isbuilt.

A real hyperplane arrangement A gives rise to a subdivision of Rn into facets,where the chambers defined above are the codimension 0 facets. We will call theset of all the facets with F . This set has a natural partial order given by F ≺ Gif and only if the closure of F contains G. We will also call with B(F) the unionof the bounded facets, that is known to be a compact connected subset of V.

Then the k−cells in S are in one to one correspondence with the pair 〈C,F k〉where C is a chamber,k is the codimension of F ∈ F and C ≺ F k. Moreover acell 〈C,F k〉 is in the boundary of 〈D,Gj〉 if and only if:

• k < j and F k ≺ Gj,

• the chambers C and D are contained in the same chamber of AFk .

Notation. Given a chamber C and a facet F we will denote by C.F the uniquechamber containing F and lying in the same chamber as C in AF .

2.3 Minimality

In this section we will review three different article that have found with similarmethods perfect acyclic matching on different classes of arrangements.

2.3.1 Salvetti-Settepanella and the polar order

The first article, [SS07], is also the first to appear in 2007. In it the two authorshave used Discrete Morse theory on the Salvetti complex to obtain a minimalcomplex i.e. with exactly as many i−cells as the i−th Betti number for the com-plement of a complexified real arrangement. The idea is that of giving a totalorder C on F which they call polar ordering. Given the order the gradient fieldcan then be recursively defined as we will see; but let’s start from the beginning.

First, we need some notation. Let e1, . . . , en be an orthonormal basis of an−vector space V .

18

2.3. Minimality

Notation. • Vi =< e1, . . . , ei > for i = 0, . . . , n (V0 = 0),

• Wi =< ei, . . . , en > for i = 1, . . . , n,

• prW : V → W is the orthogonal projection onto a subspace,

• Given a point P ∈ V Pi := prWi(P ).

Definition 2.3.1. The polar coordinates of P will be given by the module ρ = ||P ||also called θ0 and the angles θi that OPi makes with ei for i = 1, . . . , n.

Definition 2.3.2. Given a sequence θ of angles (θ = θi, . . . , θn−1) we indicate byVi(θ) the set:

Vi(θ) = P | θi(P ) = θi, . . . , θn−1(P ) = θn−1that is an i−dimensional open half-subspace.

Given a codimensional−k facet F ∈ F we denote by

F (θ) = F ∩ Vi(θ)

When θ = ∅ then F (θ) = F .Moreover for δ ∈ (0, π/2) we define the open cone:

B(δ) = P | θi(P ) ∈ (0, δ) for i = 1, . . . , n− 1, ρ(P ) > 0

Let now A be an essential finite arrangement of hyperplanes in Rn and letMC(A) be the complement of the associated complexification.

The first important concept that we introduce is that of genericity of a systemwith respect to an arrangement. This definition is crucial for the following stepsand it is a bit different from the usual one and also from the ones that will be usedin the following subsection where we will review Adiprasito’s article.

Definition 2.3.3 (Generic system). We say that a system of polar coordinatesin Rn, defined by an origin O and a base e1, . . . , en, is generic with respect to thearrangement A if it satisfies the following conditions:

(a) the origin O is contained in a chamber C0 of A,

(b) there exist δ ∈ (0, π/2) such that

B(F) ⊂ B(δ)

(c) subspaces Vi(θ) which intersect clos(B) are generic with respect to A, in thesense that, for each codim−k subspace L ∈ L(A),

i ≥ k ⇒ Vi(θ) ∩ L ∩ clos(B) 6= ∅ and dim(|Vi(θ)| ∩ L) = i− k.

19


The first thing we want to do is to be sure that for every arrangement thereexists a generic system of polar coordinates and this follow from the followingtheorem, whose proof can be found in [SS07, p. 7-9].

Theorem 2.3.4. For each unbounded chamber C such that C ∩H∞ is relativelyopen, the set of points O ∈ C such that exists a polar coordinate system centeredin O and generic with respect to A forms an open subset of C.

Having fixed a generic system with center O and frame e1, . . . , en we are nowready to define a total order on F . Let δ be the one from the definition of genericity.

Let’s now fix a θ = θ1, . . . , θn−1 with θj ∈ [0, δ], j = i, . . . , n − 1 and acodimensional−k facet F with i ≥ k. Suppose that F (θ) is not empty and set

iF (θ) = minj ≥ 0 | Vj ∩ clos(F (θ) 6= ∅For the third condition of a generic system, taking L = |F (θ)| (the subspace

spanned by F (θ)) we immediately see that

iF (θ) ≥ codim(F (θ)). (2.3.1)

There are now two possibilities:

• iF (θ) ≥ i:

This means that Vi−1 ∩ clos(F (θ)) = ∅. Then there exists a 0−dimensionalfacet PF (θ) ∈ clos(F (θ)) such that:

θi−1(PF (θ)) = minθi−1(Q) | Q ∈ clos(F (θ))

the minimum exists because otherwise there is a facet in Vi(θ) completelycontained in Vi−1(θi−1(PF (θ)) since there are at least two points, but this isimpossible for genericity.

• iF (θ) < i:

In this case we put PF (θ) = O if F is the base chamber, or the unique pointsuch that:

θiF (θ)−1(PF (θ)) = minθiF (θ)−1(Q) | Q ∈ clos(F (θ)) ∩ ViF (θ)

where the minimum exists for an argument similar to the previous one.

We will denote with Θ(F (θ)) = (θ0(PF (θ)), . . . , θiF (θ)−1(PF (θ)), 0, . . . , 0).

Definition 2.3.5 (Polar Ordering). Given F,G ∈ F and θ as above and such thatF (θ), G(θ) 6= ∅, we set

F (θ) C G(θ)

if one of the following occur:

20

2.3. Minimality

(i) PF (θ) 6= PG(θ) and Θ(F (θ)) < Θ(G(θ)) according to the anti-lexicographicordering of the coordinates.

(ii) PF (θ) = PG(θ) (so Θ = Θ(F (θ)) = Θ(G(θ))) and either:

(a) dim(F (θ)) = 0 and F (θ) 6= G(θ),

(b) dim(F (θ)) > 0, dim(G(θ)) > 0, let i0 = iF (θ) = iG(θ)

• if i0 ≥ i then ∀ε > 0, ε δ, it must happen:

F (Θi−1 + ε, θi, . . . , θi0−1, 0, . . . 0) C G(Θi−1 + ε, θi, . . . , θi0−1, 0, . . . 0)

• if i0 < i then ∀ε > 0, ε δ, it must happen:

F (Θi0−1 + ε, 0, . . . , 0) C G(Θi0−1 + ε, 0, . . . , 0)

It is easy to see that the polar order gives a total ordering of the facets of Vi(θ)for any given θ, in particular it gives a total ordering on F .

We now need a theorem that compare the polar ordering with the ≺ partialordering that will be used in the following to prove that the polar gradient iseffectively an acyclic matching.

Theorem 2.3.6. Each codimensional−k facet F k ∈ F (k < n) such that F k∩Vk =∅ has the following property: among all codimensional−(k + 1) facets Gk+1 withF k ≺ Gk+1 there exists a unique one such that

F k+1 C F k.

If F k ∩ Vk 6= ∅ then

F k C Gk+1,∀Gk+1 with F k ≺ Gk+1.

Proof. Let’s G be a facet in the closure of F k if P (G) 6= P (F k) from the definitionof polar order 2.3.5 we have that F k C G. Instead if P (G) = P (F k) again from2.3.5 we see that to which one between F k and G is bigger we can reduce tothe case where F k is of dimension one (after ε− deforming) and in this case theassertion is obvious.

If F k ∩ Vk 6= ∅ then from the definitions we have that F k ∩ Vk = P (F k). Let’stake G a k + 1−facet with F k ≺ G i.e. in the closure of F k. From 2.3.1 we havethat iG ≥ k + 1, in particular P (G) /∈ Vk, so F k C G.

We are finally able to define a gradient field over S and then prove that weobtain a perfect matching.

21


Definition 2.3.7 (Polar Gradient). Let S be the Salvetti complex, the (j+1)−thcomponent φj+1 of the polar gradient field φ is given by the pairs

(〈C,F j〉, 〈C,F j+1〉), F j ≺ F j+1

such that F i+1 C F j and ∀F j−1 ≺ F j the pair

(〈C,F j−1〉, 〈C,F j〉) /∈ φj

Theorem 2.3.8. The following are true:

(i) φ is an acyclic matching on S.

(ii) The pair(〈C,F j〉, 〈C,F j+1〉), F j ≺ F j+1

belong to φ if the following conditions hold:

(a) F j+1 C F j,

(b) ∀F j−1 such that C ≺ F j−1 ≺ F j, one has F j−1 C F j.

(iii) For each chamber C such that exists F j−1 with

C ≺ F j−1 ≺ F j, F j C F j−1

the pair (〈C, F j−1〉, 〈C,F j〉) ∈ φ, where F j−1 is the maximum (j − 1)−facetsatisfying the previous condition.

(iv) The set of k−dimensional singular cells is given by

Singk(S) = 〈C,F k〉 | F k ∩ Vk 6= ∅, F j C F k, ∀C ≺ F j F k

Equivalently, F k ∩ Vk is the maximum among all facets of C ∩ Vk.

Proof. iiiLet F j−1, C, F j satisfying the conditions. Let’s also suppose that exist F j−2

such that C ≺ F j−2 ≺ F j−1 and F j−1 C F j−2. But then there exists anotherfacet Gj−1, with F j−2 ≺ Gj−1 ≺ F j, moreover, by theorem 2.3.6 it follows thatF j−2 C Gj−1, which implies that F j−1 C Gj−1, contradicting the maximality ofF j−1. Then, by the definitions of polar gradient we have that

(〈C, F j−1〉, 〈C,F j〉) ∈ φ (2.3.2)

ii

22

2.3. Minimality

Indeed, if conditions iia and iib hold for a triplet C ≺ F j ≺ F j+1, then surely

(〈C,F j〉, 〈C,F j+1〉) ∈ φ

because we are asking more restrict conditions. On the other side if a pair asabove, belongs to φ then iia is clearly verified. Assuming iib does not hold, thenthere exist F j−1 such that C ≺ F j−1 ≺ F j and F j C F j−1. But we are in theconditions of iii and we have already proved that there exist F j−1 such that 2.3.2,which contradicts our assumption that the pair

(〈C,F j〉, 〈C,F j+1〉) ∈ φ

ivA cell 〈C,F k〉 does not belong to φ, according to ii and iii if and only if

F k C F k+1, ∀F k ≺ F k+1 (2.3.3)

andF k−1 C F k, ∀C ≺ F k−1 ≺ F k (2.3.4)

By theorem 2.3.6 conditions 2.3.3 holds if and only if P = F k ∩ Vk 6= ∅. Bygenericity P is a 0−dimensional facet in Vk and 2.3.4 holds iff P is the maximumfacet of the chamber C ∩ Vk that is exactly what we want to prove.

iFirst we show that no cell 〈C,F j+1〉 belongs to two different pairs of φ. The

only non obvious part is that it is not the end of two different pairs. This howevercan be seen ε−deforming till we reduce F j+1 to a 0−dimensional facet and herethe uniqueness follows easily from the convexity of the chamber C. What we thenneed to prove, according to discrete Morse theory is that φ has no closed loop. Todo this let us consider a path

(〈C1, Fj1 〉, 〈C1, F

j+11 〉, . . . , 〈Cm+1, F

jm+1〉

First of all we remember the conditions of the boundary and of the polargradient and we have the following inequalities at the k−th step:

F jk+1 ≺ F j+1

k , F jk+1 ≺ F j+1

k+1 , F j+1k+1 C F j

k+1

Moreover by theorem 2.3.6, for the uniqueness, if F j+1k C F j

k+1 then F j+1k+1 =

F j+1k . If this is false we have necessary that F j+1

k+1 C F j+1k . Then if the path is

closed than all the j + 1−facets are equal to a unique F j+1.What we want now to show is that at every step we necessarily have that

F jk C F j

k+1 which will conclude the theorem.

23


At this point, up to ε−deforming, we can assume that the path is in someVi−1(θ). By iii we have that F j

k+1 is the maximum vertex of Ck+1. But, by thedefinitions of boundary we have that Ck and Ck+1 belong to the same chamber ofAF jk+1

. Then all the facets of Ck are lower than F jk+1 as we want.

Theorem 2.3.9. The singular cells of the polar gradient are in one-to-one corre-spondence with the set of all the chambers of A, so the matching is perfect.

Proof. The second part follow from what we have said in the previous sectionbecause we already know that the number of chambers is equal to the sum of theBetti numbers. Since we have as many cells as chambers we have then as manycells as the Betti numbers, so every cell must be a generator in the right homologygroup and all the boundary map must be zero, otherwise the dimension of thehomology group will be strictly less that the number of cells.

So we need only to prove the correspondence. From iv of theorem 2.3.8 wehave that, watching the arrangement Ak = A ∩ Vk, the singular k−cells of Scorresponds to pair (C, v) in Ak where C is a chamber and v its maximum vertex.Of course v can also be seen as the minimum vertex of the chamber Copp of Akthat is opposite to C with respect to v. Since v is the minimum of Copp∩Vk−1 = ∅and for the genericity condition every chamber in Ak comes from a chamber in A,we have then proved that there is a one-to-one correspondence between Singk(S)and the chamber of A that intersect Vk but not Vk−1,in particular this gives acorrespondence between all the chambers and all the singular cells.

2.3.2 Adiprasito and 2-arrangements

In the article [Adi14] minimality is proved in a more general case.We are going to work with arrangement in Sd and treat then the case of Rd as

a special one. Here for a i−dimensional subspace in Sd we mean the intersectionof some (i+ 1)− dimensional linear subspace in Rd+1 with the sphere Sd.

Definition 2.3.10. If H is a hyperplane in Sd, then H is in general position withrespect to a polyhedron if H intersects the span of any face of the polyhedrontransversally. This naturally extends to collection of polyhedra. A hemisphere isin general position if its boundary is.

Definition 2.3.11. A 2−arrangement in Sd (resp. in Rd) is a finite collectionof distinct affine subspace of codimension 2 such that the codimension of everynon-empty intersection is even.

Clearly any collection of hyperplanes in Cd can be viewed as a 2−arrangementin Rd but this family is actually bigger.

24

2.3. Minimality

What we want now to do is associate to any 2−arrangement a complex ina similar way as what we have done for the complexified case with the Salvetticomplex.

Definition 2.3.12. A sign extension Aσ of a 2− arrangementA = hi : i ∈ [1, n]in Sd is any collection of hyperplanes Hi ⊂ Sd : i ∈ [1, n] such that for each i,we have that hi ⊂ Hi.

A hyperplane extension Ae of A in Sd is a sign extension of a together with anarbitrary finite collection of hyperplanes in Sd.

Any hyperplane extension of Sd, and in general any 1−arrangement gives astratification s of Sd.

Definition 2.3.13. Let A be a 2−arrangement. An extension Ae of A is fineif it gives rise to a stratification s, called combinatorial, that together with thecanonical attaching maps is a regular CW−complex.

We need some notations to go on

Notation. Given a CW−complex X in , a subcomplex Y and a subset M wedenote by

• R(X,M) the maximal subcomplex of X all whose faces are contained in M ,

• X∗ the dual block complex, meaning the complex with same set of faces butopposite inclusion,

• R∗(Y ∗,M) is the minimal subcomplex of Y ∗ containing all those faces of Y ∗

that are dual to faces of X intersecting M .

With the above notations we are able to define the space that we are going tostudy.

Definition 2.3.14. Let A be a 2−arrangement in Sd and s a combinatorial strat-ification induced by A. The complement complex of A with respect to s is theregular CW - complex

K(A, s) = R∗(s∗, Sd\A)

If instead A is a 2−arrangement in Rd, given a radial projection ρ of Rd into anopen hemisphere O in Sd, calling A′ the extension to a 2−arrangement in Sd of theimages ρ(h) for h ∈ A and s a combinatorial stratification of A′. The complementcomplex of A is R∗(s∗, O\A′).

Lemma 2.3.15. [BZ92, Proposition 3.1] Let A be a 2−arrangement in Rd (resp.Sd). Then every complement complex K(A, s) of A is a model for the complementM(A).

25


Let from now on s denote a combinatorial stratification of the sphere Sd. Ourgoal is to obtain a perfect acyclic matching on K(A, s). The idea is to studyinstead acyclic matching on s and see how they relate.

Notation. Let φ be an acyclic matching on a regular CW−complex C and D asubcomplex.

We will denote with φD the restriction of φ to D, that is, the collection ofmatching pairs in φ involving two faces of D.

We will denote with φ∗ the matching on the dual complex C∗ that is exactlythe same matching pairs of φ.

Let now φ be a matching on s. We will denote with φ∗K(A,s) the complementmatching induced by φ on K(A, s). That is the restriction to K(A, s) of the dualmatching.

The following theorem, whose proof is easy, gives us a clear correlation betweenthe two matchings on s and K.

Theorem 2.3.16. Consider an acyclic matching φ on s. Then the critical i−facesof φ∗K are in one to one correspondence with the union of

• the critical (d− 1)−faces of φ that are not faces of R(s,A),

• the outwardly matched (d− i− 1)−faces with respect to the pair (s, R(s,A)).

If M is furthermore an open subset of Sd such that all noncritical faces of φintersect M , then the critical i−faces of φ∗R∗(K,M) are in bijection with the unionof

• the critical (d−1)− faces of φ that are not faces of R(s,A) and that intersectM ,

• the outwardly matched (d− i− 1)−faces with respect to the pair (s, R(s,A).

We will now focus to the study of acyclic matching on s.Let first consider the special case of the empty arrangement and then the

general case.

Lemma 2.3.17. Let s be a combinatorial stratification of a fine extension of theempty arrangement in Sd. Let F be a closed hemisphere that is in general positionwith respect to s. Then R(s, F ) is collapsible.

Theorem 2.3.18. Let A be a non-empty 2−arrangement in Sd. Let s be a com-binatorial stratification of a fine extension Ae of A, and F a closed hemispherein general position with respect to s. Then, for any k−dimensional subspace H ofF(Aσ) extending an element of A, we have the following:

26

2.3. Minimality

1. The pair (R(s, F ∩H), (R(s, F ∩ A ∩H)) is out-ι(d) collapsible.

2. If A is non-essential, then (R(s, F ∩H), (R(s, F ∩ A ∩H)) is a collapsiblepair.

The previous theorem is the crucial step to prove the following that will be oneof the two results needed to construct the matching.

Theorem 2.3.19. [Adi14, Corollary 4.2] Let F denote a closed hemisphere in Sd,let O denote its open complement. Let A be a 2−arrangement w.r.t. O, and sthe associated combinatorial stratification. If H is a hyperplane in Sd in generalposition with respect to Ae and δF then:

(R(s, Sd\(O ∩H)), (R(s,A ∩ Sd\(O ∩H)))out−ι(d) (R(s, F ), R(s, F ∩ A))

Before being able to construct the matching on s we just need one more lemma,that is an easy consequence of the Goresky-MacPherson formula.

Lemma 2.3.20. Let A denote a subspace arrangement in Sd, O an open hemi-sphere, and let H be a hyperplane. Then we have the following

1. If O is in general position w.r.t. A, then for all i, βi(Sd\A) ≥ βi(O\A).

Where the βi are the Betti numbers.

2. If H is in general position w.r.t. A and O, then for all i, βi(O\A) ≥ βi((O∩H)\A).

Theorem 2.3.21. Let F be a closed hemisphere of Sd, O = F c, A a 2− arrange-ment with respect to O and s a combinatorial stratification induced by A, and Kthe associated complement complex.

Then, there exists an acyclic matching φ on s whose critical faces are thesubcomplex R(s, F ) and some additional facet such that the restriction of the com-plement matching to R ∗ (K,O) is perfect.

Proof. We prove it by induction on the dimension, where the case d = 0 is clearlytrue. Let us assume now that d ≥ 1 and let H be a generic hyperplane in Sd.Calling sH the restriction of the combinatorial stratification to H we have byinduction hypothesis a matching ϕ on it that respects all the hypothesis. We canlift this matching to a matching φ on s of the faces intersecting H, this will stillbe an acyclic matching, moreover, using Theorem 2.3.19 we have that:

(R(s, Sd\(O∩H)), (R(s,A∩Sd\(O∩H)))out−ι(d) (R(s, F ), R(s, F ∩A)) (2.3.5)

We now define ψ as the union of the matching φ and the out-ι(d) sequenceassociated to the equation above and claim that ψ is the acyclic matching that wewant.

27


By construction ψ has the desired critical faces so we need only to show thatit is perfect.

The equation 2.3.5 also tells us that if we consider the matching ψ∗R∗[K,O] all

the critical faces that are not in R∗[O∩H] have dimension e = d− ι(d)− 1. ThenTheorem 1.3.5 tells us that R∗[K,O] is obtained from the complex R∗[K,O ∩H]by attaching cells of dimension e. We want to show that each of these cells adda generator in homology, but if that’s not the case one of them should delete agenerator in the (e− 1)− homology and then

βe−1(O\A) = βe−1(R∗[K,O]) < βe−1(R∗[K,O ∩H]) = βe−1((O ∩H)\A),

in contradiction with Lemma 2.3.20. We have then proved that the matchingis perfect.

As an easy consequence of the previous theorem we then see that we are ableto construct perfect acyclic matching on every 2−arrangement.

Corollary 2.3.22. [Adi14, Theorem 5.4] Any complement complex of any 2−arrangement A in Sd or Rd admits a perfect acyclic matching.

Proof. Let us first solve the case of 2−arrangement in Rd. Let then A be a2−arrangement in Rd, ρ a radial projection of Rd into an open hemisphere Oin Sd and A′ the corresponding arrangement in Sd. Then the acyclic matchingconstruct in the previous theorem with respect to R∗[K(A′, s), O] is perfect.

Let then study now the case of a 2−arrangement A in Sd. Let F be a genericclose hemisphere, O = F c and s a combinatorial stratification. Again using theprevious theorem we are able to construct an acyclic matching ψ on s such thatthe complement matching restricted to R∗[K(A, s), O] is perfect.

By theorem 2.3.18 R∗(s, F ) is out-ι(d) collapsible. If we now consider thematching ψ together with this collapsment we can see using again Lemma 2.3.20that the obtained acyclic matching is perfect on K(A, s).

2.3.3 Delucchi and the central case

The article [Del08] main aim is to answer a question asked in [SS07] about acompletely combinatorial formulation of the polar ordering. The approach usedhere is that of constructing an acyclic matching in the context of oriented matroids,so the results won’t hold for affine arrangement but it has the big advantage thatit does not require the choice of a generic flag in the ambient space.

From now on we will used, unless explicitly stated, the notation for hyperplanearrangement, since it’s more geometrically intuitive and closer to the one used in

28

2.3. Minimality

the previous subsection. However everything written is true for oriented matroids,even if not realizable.

Let A be a finite central arrangement of hyperplanes in Rn.The first thing we need is the definition of an order of the chambers.

Definition 2.3.23. Let B ∈ C(A) a base chamber, a valid order a on C(A) is alinear extension of the partial order <B. Given C,C ′ ∈ C(A) we will denote bys(C,C ′) the set of hyperplanes that separates the two chambers, then:

C <B C′ ⇔ s(B,C) ⊂ s(B,C ′)

We will denote by CB(A) the set of chambers endowed with this partial order.

In the following we will suppose that we have a fixed base chamber B and avalid order a and we can define one of the most important object of our study.

Definition 2.3.24. For every C ∈ C(A) we let

J (C) = X ∈ L(A) | supp(X) ∩ s(C,K) 6= ∅ for every K a C

It is easily seen that J (C) is an upper ideal in L(A) but what we are reallyinterested in is that it is principal. This is not obvious and follow from a series ofproposition that can be found in [Del08, p. 16-18]. We will here just lay down thelast one, together with some notation.

Notation. Given H ∈ A, let A′ = A\H and similarly, given a chamber C wewill denote with C ′ the unique chamber in A′ that contains C. This inclusioninduce an order preserving map:

ϕ : C(A′)→ C(A)

where we sent C ′ to the minimum chamber C, with respect to a valid order a onC(A), that is contained in C ′. This let us build a valid order on C(A′) that isthe pullback of a along ϕ and that we will call a′. Similarly we are able to buildJ ′(C ′) for any chamber C ′.

The inclusion A′ → A induces an order preserving injection

ι : L(A′)→ L(A), X →⋂

supp(X).

We will identify J ′(C ′) with its image under this map.

Lemma 2.3.25. Given a chamber C different from −B (the chamber opposite toB), choose H ∈ A\s(B,C) and let A′ = A\H. For every Y ∈ J (C) we have⋂

(supp(Y )\H) ∈ J ′(C ′)

29


Given this lemma, that we are not going to prove, we are then able to provethe theorem.

Theorem 2.3.26. Given a valid order a with respect to a chamber B, for everyC ∈ C(A), J (C) ⊂ L(A) is a principal upper ideal.

Proof. We will argue by induction on the size ofA. The base step, whenA containsonly one hyperplane, is trivial, so we can prove the inductive step, assuming that|A| > 1. We will prove that J (C) is closed under the meet operator, which willimmediately imply the claim.

If C = −B, then clearly J (C) contains only the intersection of all the hyper-planes, so is principal. We can then suppose that C 6= −B, in particular there isan hyperplane H ∈ A\s(B,C). We call A′ = A\H and by induction hypothesisA′ satisfies the theorem.

We introduce now the order preserving map

λ : L(A)→ L(A′), Y →⋂

(supp(Y )\H)

that by lemma 2.3.25 satisfies λ(J (C)) ⊆ J ′(C ′). Moreover the inclusion ι ofJ ′(C ′) in J (C) is well defined. Now that we have introduced the instrumentslet’s consider two elements Y1, Y2 ∈ J (C), as already said we want to prove thatY1 ∧ Y2 exists and ∈ J (C). The induction tells us that λ(Y1) ∧ λ(Y2) ∈ J ′(C ′).Applying ι to this element we have then an element ι(λ(Y1) ∧ λ(Y2)) ∈ J (C) and≤ Y1 ∧ Y2 because ιλ(Y ) ≤ Y for every Y ∈ J (C). But now we already now thatL(A) is a lattice, meaning that Y1 ∧ Y2 surely exists in L(A), The fact that J (C)is an upper ideal concludes the proof.

Thanks to the theorem, we are now able to define the following object.

Definition 2.3.27. For every C ∈ C(A) let

XC = minJ (C)

Corollary 2.3.28. If we define FC = XC ∩ C, we have |FC | = XC.

Proof. As in the proof of the theorem before we can assume that C 6= −B becauseotherwise the claim is trivial and proceed by induction. Again the base step when|A| = 1 is easy so we can assume that |A| > 1. We will call with WC the setof hyperplanes adjacent to C. Since C 6= −B there exist H ∈ WC ∩ s(C,−B)and call A′ = A\H. If we call H+ the closed half-space bounded by H andcontaining B we have that

C = C ′ ∩H+.

30

2.3. Minimality

Moreover by induction hypothesis we have that |F ′C′ | = X ′C′ which implies that

dim(X ′C′ ∩ C ′) = dim(X ′C′).

λ(XC) = X ′C′ .

By definition of λ we have that

λ(XC) =⋂

(supp(XC)\H

then there are two possible cases:

• XC = X ′C′

dim(C ∩XC) = dim(C ′ ∩H+ ∩X ′C′) = dim(X ′C′ ∩H+) = dim(XC)

where the last equality is true because XC ⊂ H+.

• XC = X ′C′ ∩H

dim(C ∩XC) = dim(C ′ ∩H+ ∩X ′C′ ∩H) =

= dim(C ′ ∩X ′C′ ∩H) = dim(X ′C′ ∩H) = dim(XC)

In both cases we have then prove that dim(XC ∩C) = dim(XC) that is equiv-alent to our thesis.

From everything written above is then clear the following lemma

Lemma 2.3.29. XC is uniquely determined by the following properties:

1. s(K,C) ∩ supp(XC) 6= ∅ for all K a C,

2. For every Y ∈ L(A) such that Y does not contain XC there is a chamberK a C such that s(K,C) ∩ supp(Y ) = ∅.

At this point we have all the instruments and we should understand how toapply them to the Salvetti complex S of A to construct a perfect acyclic matching.The idea is to use the patchwork lemma and, instead of constructing directly thematching for all the complex, subdivide the complex in subcomplexes N(C), oneper chamber C ∈ C(A) and construct for each of this subcomplexes an acyclicmatching with exactly one singular cell. Let’s then see what this subcomplexesare.

31


Definition 2.3.30. Let S be the Salvetti complex of a linear arrangement A withthe partial order given by “be a face of“. Then, for every chamber C, we willdenote by

S(C) = S≤〈C,P 〉

where P is the unique minimal element of F(A). In other words S(C) is thesmaller subcomplex of S that contains the cell 〈C,P 〉. Moreover we define

N(C) = S(C)−⋃C′aC

S(C ′)

The definition of S(C) is slightly different from the one given in [Del08] but itis easy to see that the definitions of N(C) are the same.

Lemma 2.3.31. Under the above definitions, we have for every C ∈ C(A)

N(C) ' F(AXC )

Proof. First we note that from the definitions follows that

N(C) = 〈D,F 〉 | D = C.F and C.F 6= K.F for all K a C

but then D is uniquely determined by F so we have a correspondence betweenN(C) and F(A), we now want to say that this correspondence is actually one-to-one with F(AXC ).

Let us then suppose that F ∈ F(AXC ). Then, for all K ∈ C(A) we have thats(C.F,K) ∩ supp(F ) = s(C,K) ∩ supp(F ). But then from the lemm 2.3.29 wehave that for all K a C s(C,K) ∩ supp(F ) 6= ∅, which implies K.F 6= C.F sinceobviously s(K.F,K) ∩ supp(F ) = ∅.

For the other inclusion, suppose 〈C.F, F 〉 ∈ N(C) and F /∈ F(AXC ). But then,again by lemma 2.3.29, there is K a C with s(K,C) ∩ supp(F ) = ∅ which impliesK.F = C.F and thus a contradiction.

Now, we need to study F(AXC ) and in general F(A) for a linear arrangementA. We recall that F(A) is a stratification of a certain Rd, the stratification canalso be seen as a CW−decomposition of Rd. Using the shellability can then beproved

Theorem 2.3.32. Every valid order a on C(A) defines an acyclic matching of theface poset F(A) such that the only critical element is the chamber opposite to B.

Proof. Every valid order on C(A) induces a recursive coatom ordering of F(A)([BLVS+99, Proposition 4.3.2]). In theorem 1.4.12 we have seen that this inducesan acyclic matching with the critical faces corresponding to the spanning faces ofthe given shelling. It is now easy to see that the definition of valid order impliesthat the only spanning chamber is the last one.

32

2.3. Minimality

This is the last piece needed to be able to construct a perfect acyclic matchingon all the Salvetti complex of A.

Theorem 2.3.33. Let A be an arrangement of real hyperplanes in a real spaceand fix any B ∈ C(A). To any valid order a corresponds a family of perfect acyclicmatchings of the associated Salvetti complex S which critical cells are in naturalbijection with the chambers of A.

Proof. Given a valid order a with respect to a chamber B and thanks to theprevious theorem and lemma we are able to construct on N(C) an acyclic matchingwith exactly one critical cell. In particular we also know that this cell is the onein correspondence with the chamber −B in AXC .

Since the N(C) are a partitions of S we are then able, attaching this matching,to construct a matching on S. Let f be the function that sends a cell 〈D,F 〉 tothe chamber C if and only if 〈D,F 〉 ∈ N(C). f is a map of posets that preservesthe order, since by construction given σ, τ ∈ S, σ < τ and τ ∈ N(C) then surelyσ ∈ S(C) which implies that f(σ) ≤ C.

We are then on the hypothesis of the patchwork lemma that assures us thatthe induced matching on S is still acyclic.

The critical cells are the union of the critical cells of the N(C)s which imme-diately implies that we have a bijection between critical cells and chambers.

33


34

Chapter 3

Local Abelian Homology

The aim of this chapter is to use what done in the previous pages to study theabelian local homology of the hyperplane arrangements. In the first part we willsee what local homology is and its principal property, as well as how it behaveswith Discrete Morse theory. In the second section instead we will see how theacyclic matching, in particular the polar matching proposed by [SS07], can helpus in compute abelian local homology.

In the last section, eventually, we will talk about the Braid arrangements. Wewill see a particular matching on them and use it to define an algorithmic way tocompute the boundary of the reduced complexes.

3.1 Homology with local coefficients

Homology with local coefficients, or local homology, is a homology theory firstintroduced by Steenrod in [Ste43].

In the first part of this section we will then follow the article to see the basicdefinitions before then moving to its correlations with Discrete Morse theory.

Let R be an arcwise connected topological space, x a point of R and π1(R, x)the fundamental group with base point x.

The fundamental idea of this theory is that we are goint to compute the ho-mology where the coefficients will be a group G with an action of π1(R, x).

Definition 3.1.1. A system of local groups in the space R is a family of groupsGyy∈R such that for each class of path αyz from y to z there is a group iso-morphism between Gy and Gz and the composition of isomorphism αyzβzw is theisomorphism corresponding to the path from y to w.

Theorem 3.1.2. If G is a group with an action of π1(R, x) for some x ∈ R, thenthere is a system Gy of local groups in R such that the operations are determinedby π1(R, x).

35

Chapter 3. Local Abelian Homology

Proof. For each point y ∈ R we choose a class of path λxy from x to y, withλxx =identity. Let now Gy be a group isomorphic to G for each y ∈ R and weassociate this isomorphism with λyx = λ−1

xy .If we have a path αyz, we obtain a isomorphism from Gy to Gz in the following

way

αyz(g) = λxz[λxyαyzλzx)(λyx(g))]

where the interiors λ are paths, while the exteriors group isomorphisms.It is now easy to control that what defined above is a system of local groups.

We now suppose that our space R has a CW−decomposition ∆ that we supposefinite and see how, from a system of local groups, we can create our homologytheory.

For each cell σ we choose a representative point x(σ) and we call Gx(σ) = Gσ.

Definition 3.1.3. A q-chain of ∆ is a function f attaching to each oriented q−cellσ of ∆ an element f(σ) ∈ Gσ such that f(−σ) = −f(σ). Chains are added byadding functional values, so they form a group isomorphic to the direct sum of thegroups Gσ for all q−cells σ.

We want now to define the boundary maps between chains.If σ′ < σ we choose a path in the closure of σ joining x(σ) to x(σ′). We obtain

an isomorphism Gσ → Gσ′ which is denoted by hσ′σ. We postulate that the closureof each cell is simply connected in order for hσ′σ to be independent of the path.

Using h we are now able to define the boundary ∂ and co-boundary δ of aq−chain f .

∂f(σq−1) =∑σq

[σq−1 : σq]hσq−1σq(f(σq)),

δf(σq+1) =∑σq

[σq : σq+1]h−1σqσq+1(f(σq)).

It is a simple check that δδf = 0 and ∂∂f = 0 so we are able to define homologyand cohomology groups as usual.

We have now to talk a bit about algebraic Discrete Morse theory, in order tostudy this and in general all kind of homology theory, again following [Koz08].

We will denote in the following with C∗ a chain complex of modules over somecommutative ring. We will call it free if each Cn is a finitely generated free module.A basis Ω of C∗ is simply a set of free generators Ωn for each Cn. Given b ∈ Ωn anda ∈ Ωn−1 we denote by [b : a] the coefficients of ∂(b) with respect to a, if a /∈ Ωn−1

by convention we set [b : a] = 0.

36

3.1. Homology with local coefficients

Remark 3.1.4. Since the following theory is valid only for modules over commuta-tive ring from now on we will work with a group G with an action of the abelian-ization of π1(R, x) that is, an action of H1(R).

Clearly a free chain complex with a basis (C∗,Ω) can be represented as a rankedposet, called P (C∗,Ω), where the weights are the incidence numbers defined above.

Definition 3.1.5. Let C∗,Ω) be a free chain complex with a basis. A partialmatching M ⊆ Ω × Ω on (C∗,Ω) is a partial matching on P (C∗,Ω) such that, if(a, b) ∈M then [b : a] is invertible.

The notion of acyclic matching is the same as before.We want to make clear that we are exactly in this situation since, given a

CW−complex and a local homology chain on in, we have that the chain is freeand we can take as a basis the cells of ∆. The above definition then asks us thatevery incidence number is invertible but this is always true because our modules areover a group. Then in our particular case the above definition of partial matchingis actually the same as the one given at the beginning of the thesis.

Given an acyclic matching M we let Cn(Ω) be the set of critical elements ofΩn, meaning the elements that are not in any pair.

Given two elements b ∈ Ωn and a ∈ Ωn−1, an alternating path between themis a path of the form:

p = b a1 b1 · · · bn a

where (ai, bi) ∈M for each i = 1, . . . , n. Its weight is

ω(p) = (−1)n[b : a1] · [b1 : a2] · · · [bn : a]

[b1 : a1] · [b2 : a2] · · · [bn : an]

Comparing the formula with the one in 1.2.2 we see that they are really similar,but now we are taking into account the possibility that the incidence numbers aredifferent from ±1.

Definition 3.1.6. Given a free chain complex with a basis (C∗,Ω) and an acyclicmatchingM. The Morse complex CM∗ is defined as follows. CMn is freely generatedby the elements of Cn(Ω) and the boundary operator is defined by

∂M(s) =∑p

ω(p) · p•

where the sum is taken over all alternating paths p starting in s and p• is theending point of the path.

37


We are nearly ready to set out the main theorem of algebraic Morse theory, wejust need one more definition.

Definition 3.1.7. The chain complex where the only nontrivial modules are inthe dimensions d and d− 1, both free of dimension one and the boundary map isthe identity is called atom chain complex and denoted by Atom(d).

Theorem 3.1.8 ([Koz08, Theorem 11.24]). Assume that we have a free chaincomplex with a basis (C∗,Ω), and an acyclic matching M. Then C∗ decomposesas a direct sum of chain complexes CM∗

⊕T∗ where T∗ '

⊕(a,b)∈MAtom(dim b).

In particular, since the atom chain complexes are all acyclic by construction,we obtain immediately the corollary

Corollary 3.1.9. Assume that we have a free chain complex with a basis (C∗,Ω),and an acyclic matching M. Then the homology of the chain complex is the sameas that of the Morse complex.

To conclude this section we simply want to underline that we are exactly inthis situations with the abelian local homology. Theorem 3.1.8 can be appliedto the abelian local homology defined above, so, studying the alternating path inan acyclic matching, in principle we are then able to use them to study the localhomology.

3.2 Local homology of hyperplane arrangement

In this section we will focus on the study of local homology of hyperplane arrange-ment.

Let A be a complexified real arrangement and choosing a base chamber C0 anda base point O ∈ C0 we define, for each hyperplane Hi, a positive oriented looparound this hyperplane as an element of π1(M(A), O), calling it ti.

Given now an abelian local system L over M(A) we have a homomorphism

Z[π1(M(A, O)]→ Z[H1(M(A)]→ Z[t±1i ]Hi∈A ⊆ End(L)

The basepoint O ∈ C0 can be taken as the unique 0−cell of S contained in C0,namely equal to 〈C0, C0〉. We can do the same for each cell of S, and associate to〈C,F 〉 the point 〈C,C〉. This will be the point that we consider in the constructionof our chain complex, moreover since all the points are in the 0−skeleton of S upto homotopy we can consider only paths in the 1−skeleton of S.

A sequence, or galleries, of adjacent chambers uniquely correspond to a specialkind of combinatorial paths in the 1−skeleton of S, which we call positive path.

38

3.3. Local homology of the Braid arrangements

Two galleries with the same ends and of minimal length determine two homotopicpaths [Sal87]. This implies that when we are going from a cell to another we areonly interested in the hyperplanes that separate the corresponding chambers.

Example 3.2.1. To make things a bit clearer we want to see explicitly how wecan calculate the boundary. Let A = H1, . . . , Hn, C,D ∈ C such that s(C,C0) =H1, . . . , Hi, s(D,C0) = H1, . . . Hi−1, s(C,D) = Hi.

The incidence coefficients between two cells 〈C,F 〉 and 〈D,G〉 is given by titimes the incidence number with integral coefficients.

Let us take the polar order and the corresponding acyclic matching definedin [SS07] Studying the alternating paths between critical cells we are then able,thanks to 3.1.8, to study the local homology of the arrangement. This has beendone in the second part of [SS07] for the general case and in [GS09] for a morespecific approach to the case of linear arrangement.

3.3 Local homology of the Braid arrangements

In this section we want to talk about a new matching on the Braid arrangementsthat help us find the alternating path in a somewhat easier way. In the first partwe then present our matching and in the second we will see how to compute theboundary in local homology and a program in Python written to do so.

The idea is to use the description of the Braid arrangement given at the endof [SS07] that uses the concept of tableau.

We recall that the braid arrangement of dimension n is the arrangement An =Hij = xi = xj, 1 ≤ i < j ≤ n + 1 in Rn+1 where the xi is a system ofcoordinates.

Proposition 3.3.1. A k−cell 〈C,F 〉 ∈ S(An) is represented by a tableau withn + 1 boxes and n + 1 − k rows filled with all the integers in 1, . . . , n + 1 suchthat (x1, . . . , xn+1 is a point in F if and only if:

• i and j belong to the same row if and only if xi = xj,

• i belongs to a row less than the one containing j if and only if xi < Xj.

The chamber C belong to the half-space xi < xj if and only if

• The row that contains i is less than the one containing j or

• i and j belong to the same row and the column which contains i is less thanthe one containing j.

39


Example 3.3.2. The tableau in A2

2 1

3represents the cell 〈C,F 〉 with C = x2 < x1 < x3

and F = x2 = x1 ∧ x2 < x3.

Using the tableaux there is an easy way to see when a cell is in the boundaryof another.

Lemma 3.3.3. Given a tableau T all the tableaux in its boundary can be obtainedby T by taking a row and spitting it in two rows while preserving the order (if i, jwhere in the row and i was before j than or i and j now belong to different rowsor still i is before j).

The matching that we are now going to discuss was first proposed by GiovanniPaolini. Later on we found out that was already been studied in [Dja09] butwithout our focus on the boundary in local homology.

Given a tableau T we will call with Tij the element in row i and column j ofT , we give to the elements of a tableau T an order, saying that Tij is before Ti′j′if i < i′ or i = i′ and j < j′.

Proposition 3.3.4. The following matching M on An is acyclic for all n. Givena tableau T let i be the minimum such that

• Ti1 6= maxj Tij In this case the tableau T is matched with the tableau obtainedby T by splitting the row i taking maxj Tij and all the following elements inthe row i and moving them up.

• Ti1 = maxj Tij but Ti1 > maxj T(i+1)j then T is matched with the tableauobtained by gluing together the rows i and i + 1 putting the row i after therow i+ 1.

The critical tableaux are in correspondence with the permutations of the num-bers between 1 and n + 1, where given a permutation we associate a tableau suchthat the elements are in the same order of the permutation, the first column isincreasing and on each row the maximum is on the first column.

Example 3.3.5. The tableau 2 1 3 5 4 is matched with the tableau

5 4

2 1 3

To the permutation (4, 3, 5, 6, 2, 1) we associate the critical tableau

4 3

5

6 2 1

40


Proof. The fact that the one described above is a partial matching and that thecritical tableaux are the one written is clear, so we only have to show that thematching is acyclic.

Let us suppose that there is a cyclic alternating path, namely

T = T0 T ′1 T1 . . . T ′n Tn = T

where, for each k, (T ′k, Tk) ∈M and T ′k < Tk−1.Let suppose that T1 is obtained by T ′1 by gluing together the rows i and i+ 1,

this implies that T ′2 is obtained by T1 by splitting the row i. To prove this we needto divide in two cases:

• If T ′2 is obtained by T1 by splitting a row j > i then our matching will splitthe row i in two and so the path is not alternating.

• If T ′2 is obtained by T1 by splitting a row j < i we can repeat the sameargument and see that at all the following step we will work only in a rowsmaller or equal to j and then at the end we cannot return to T .

The same argument used above can be repeated for each boundary operation, evenfor the one between T0 and T ′1 so in our path the only things that we can do is tosplit a certain row i in two and then glue them together. Let Til be the maximumelement in the row i of T .

Let us now consider T ′1. Til must become the first element of the row i becauseotherwise the matching does not glue the rows i and i + 1 together. If we recallwhen a tableau is in the boundary of another then in the i−row of T ′1 there areonly elements of the row i of T and after Til. If there are all said elements thenT1 = T , otherwise in T1 the maximum in the row i is in a column bigger than l. Itis now immediate to see that there is no way to move the maximum in a smallercolumn so to obtain again T in the path and this concludes the proof.

We need now to give a definition to some special elements of T . An elementTij of T is a block maximum if it is bigger than all the elements in the row beforeit (Tij > Tik ∀0 < k < j). A block is a sequence of elements in a row of T startingwith a block maximum and ending just before the following block maximum.

Let now consider the following map:

β : S(An)→ P1, . . . , n+ 1

Where with P1, . . . , n+1 we mean the set of ordered partitions of 1, . . . , n+1,that sends a tableau T to the sequence of its blocks. The first block of the sequenceis the one starting with T11, and we go on from left to right, from the first row tothe last.

41


Example 3.3.6. The tableau7 4

5 2 8 9 1 3

is sent to ((7, 4), (5, 2)), (8), (9, 1, 3).

It is immediate to see that if (T, T ′) ∈ M then β(T ) = β(T ′) and that if atableau is critical then the blocks correspond to the rows.

We can now try to describe the alternating paths between critical cells.

Proposition 3.3.7. Given two critical tableaux T and T ′, there exist an alternat-ing path between them if and only if:

• The block maximums of T ′ are the block maximums of T plus one,

• All the blocks of T included one block of T ′ (the one with same block maxi-mum) preserving the order of the elements.

• Let i be the row such that T ′i1 is not a block maximum of T and let k be thelength of the row i of T ′. Then for every 1 ≤ l < m ≤ k the element T ′il in Tis in a row bigger or equal to that that contains the element T ′im. Moreoverany elements of this row must be in T in a row with block maximum biggerthat T ′i1.

Proof. Let us suppose then that we have an alternating path between criticaltableaux

T = T0 T ′1 T1 . . . T ′n = T ′

Since the number of blocks of T ′ is one plus the number of blocks of T and wecannot remove a block with a boundary operation then the only time we are addinga block is between T ′1 and T (here we are obliged). After this, every other times wedo a boundary operation we have to split the only row with two blocks otherwisewe will create an additional block. This row during the path can only decreaseand at any time after the first passage the block that is not in T is always the firstof the row. Then with a boundary operation if we don’t want to create new blockswe can only add elements at the end of this block. This implies that all the threeconditions are necessary.

We show now a way to construct an acyclic path for every pair of tableauxthat respect all the conditions.

T ′1 is obtained by T by splitting the row that contains T ′i1 in two parts, wherethe part that goes below has the single elements T ′i1. Then T1 is equal to T withthe exception that in the i−row it contains as first element T ′i1. After this weproceed by induction, let us suppose that we have done j step. In the followingstep we check if the row of Tj that contains two blocks contains in the secondblock some element that should go in the first one in T ′, in this case we split the

42


row by putting below the first block together with the smaller of this elements.Otherwise we split the row but putting above the first block. It is now easy to seethat this construction always arrives at T ′ and we stop there. As we have alreadynoticed the row that contains two blocks can only decrease then whichever pathwe consider between two critical cells the first time that we have two blocks in thei−rows the intermediate tableau is exactly the same because we have taken all theelements in higher rows in the right order even if maybe in different ways.

Remark 3.3.8. Watching closely the proof above we see that in the construction ofthe alternating path we do not have a variety of choice in what to do. First of all,we recall that we have to work row per row, so here we can only study the caseof a single row, or even better of a single inductive step. This is because when wechange the row with two blocks we cannot go back. At a certain step a certainrow must have two blocks and if the second block does not contains any elementthat should go in the first one we are obliged to split as in the proof. If it containsat least one, we should check if they are already ordered. Here two things couldhappen, first of all if there are at least two consecutive elements in the right orderwe can take instead of only one at times any subsequence of them and add themtogether to the first block. Moreover if all the elements are ordered (for exampleif there is only one remaining) we can add it to the first block and put the blockabove instead of below.

This description of the alternating paths is the idea behind our algorithm.Before arriving to it we need yet another tool, an explicit description of the localhomology boundary of two tableaux.

Lemma 3.3.9. Given two tableaux T and T ′, we define the set Q as the set ofpairs (a, b) of elements in 1, . . . , n + 1 such that a < b and in T a is before b andafter in T ′. Then, the boundary in local homology is given by:

[T : T ′]∏

(a,b)∈Q

tab

Where tab is the loop around the hyperplane xa = xb and [T : T ′] is the incidencenumber with integer coefficients, in particular can be taken equal to

[T : T ′] = sgn(nm − nm+1)(−1)k

where nj is the maximum in the row j of T ′, m,m + 1 are the row of T ′ that arejoined in T and k = ]nj | nj < minnm, nm+1.

Proof. The first part of the lemma follows from the definition of abelian localhomology and of the tableau. For the second part it is an easy calculation to

43


verify that the incidence numbers given derives from an appropriate orientation ofthe cells of the Salvetti complex, see [Sal87] for more details of how the complexis actually built.

Thanks to the previous Lemma and Theorem 3.1.8, using the previous descrip-tion of the alternating paths between critical tableaux we are able to write analgorithm that explicitly do the calculations. The program, written in Sage, isdivided in two subfunctions to make everything more readable.

The first function that we are listing takes the following as input:

• a critical tableau (tab), written as a permutation of 1, . . . , n+ 1.

• The block maximums of tab (heads) as an array whose elements are thepositions of the block maximums in the permutation. In particular since thetableau is critical heads[i] gives us the position of the first element in thei−row of tab.

• A certain row of tab, called line, the number actually identifies the firstelement of the row in the list.

• the number of block maximums of tab.

The output are all the critical tableau T for which there exist an alternatingpath between T and tab and whose block maximums are the ones of tab minusthe one in the row specified. The idea of the algorithm, is pretty easy, we followProposition 3.3.7 and we do a recursion after having moved the first element ofthe row and modified every input accordingly.

de f f a c e s ( tab , heads , l i n e , f i n p o s ) :t =[ ]i f heads [ l i n e ]==heads [ l i n e +1] :

r e turn [ tab ]e l s e :

f o r k in range ( l i n e +1, f i n p o s +1):headsK=heads [ : ]f o r l in range ( l i n e +1,k+1):

headsK [ l ]=headsK [ l ]−1f o r h in range ( heads [ k ] , heads [ k +1 ] ) :

tabbb=tab [ : heads [ l i n e ] ]+ tab [ heads [ l i n e ] + 1 : ]tabbb . i n s e r t (h , tab [ heads [ l i n e ] ] )f o r p in f a c e s ( tabbb , headsK , l i n e , k ) :

t . append (p)re turn t

44


The second function is the one that actually compute the boundary, given twocritical tableaux (tab1 and tab2 ), their block maximums (heads1 and heads2 ) andthe row (line) in the first tableau that is not contained in a block of the second;return us the local boundary.

The idea behind the algorithm is the same described above, i.e. to work rowper row. Putting together 3.3.8 and 3.3.9 we start with the row that contains theblock maximums, we compute the partial boundary of each paths that sends allthe elements in the special block of this row in the row below and then we continuetill we arrive at our critical cell. The idea is that even if there may be more thanone path between two critical cells in every path the first times that we have twoblocks in a certain row the tableau is always the same so we can actually divide thecomputations of the boundary row per row. On a specific row the algorithm studythe order in which are the element that we want to move and act accordingly,separating the case when they are already ordered and when they are not. Moredetails are given as comment in the algorithm.

Remark 3.3.10. In the algorithm is also used the following observation: let us callwith ai1, . . . , aik the elements of line in tab1 contained in the row i of tab2 withthe order given by tab1 (meaning that aij is before ail in line whenever j < l).Then the boundary is not zero if and only if there exist m ∈ [1, k] such that in therow i of tab2 the elements ai1, . . . , aim are in the reversed order while the elementsaim, . . . , aik are ordered.

The idea of the proof is simply to do the calculation and see that there are alot of paths with same incidence number apart from the sign.

de f bord ( tab1 , tab2 , heads1 , heads2 , l i n e ) :#f=p o s i t i o n in tab2 o f the e lements in l i n e o f tab1f =[ ]f o r i in range ( heads1 [ l i n e ] , heads1 [ l i n e +1 ] ) :

j=heads1 [ l i n e ]whi l e tab1 [ i ] != tab2 [ j ] :

j=j+1f . append ( j )

R = PolynomialRing (GF( 5 ) , ( n+2)∗∗2 ,” z ” ) . gens ( )

#f i n d i n g the row that conta in s f [ 0 ]g=len ( heads2)−2whi le heads2 [ g]> f [ 0 ] :

g=g−1

#i n i t i a l i z i n g the boundary de l t a

45


de l t a =(−1)∗∗( l i n e +1)i=0

#Sta r t i ng to work row per row t i l l we a r r i v e at l i n ewhi l e g>=l i n e :

#We cons id e r the e lements o f f in the row g#that are in the wrong orderwhi l e ( i<l en ( f )−1)and ( f [ i +1]>heads2 [ g ] ) and ( f [ i ]> f [ i +1 ] ) :

d e l t a=−de l t af o r z in range ( heads2 [ l i n e ] , f [ i ] ) :

i f tab2 [ f [ i ] ]> tab2 [ z ] :d e l t a=de l t a /R[ ( n+2)∗( tab2 [ z ]−1)+tab2 [ f [ i ] ] ]

i=i+1

#The f i r s t i f o r which f [ i ] and the f o l l o w i n g e lements#are in the r i g h t order so we have two d i f f e r e n t pathsi f f [ i ]>heads2 [ g ] :

p2=−1f o r z in range ( heads2 [ g ] , f [ i ] ) :

i f tab2 [ f [ i ] ]> tab2 [ z ] :p2=p2/R[ ( n+2)∗( tab2 [ z ]−1)+tab2 [ f [ i ] ] ]

p1=1f o r z in range ( heads2 [ g ] , f [ i ] ) :

k=0whi le z != f [ k ] and k<i :

k=k+1i f ( k==i ) and tab2 [ f [ i ] ]< tab2 [ z ] :

p1=p1∗R[ ( n+2)∗( tab2 [ f [ i ]]−1)+ tab2 [ z ] ]d e l t a=de l t a ∗( p1+p2 )

f o r z in range ( heads2 [ l i n e ] , heads2 [ g ] ) :i f tab2 [ f [ i ] ]> tab2 [ z ] :

d e l t a=de l t a /R[ ( n+2)∗( tab2 [ z ]−1)+tab2 [ f [ i ] ] ]

i=i+1

#We check the l a s t e lements in the row g ,

46


#f o r the remark they must be orderedwhi le ( i<l en ( f ) ) and ( f [ i ]>heads2 [ g ] ) and ( f [ i−1]< f [ i ] ) :

f o r z in range ( heads2 [ g ] , f [ i ] ) :k=0whi le z != f [ k ] and k<i :

k=k+1i f ( k==i ) and tab2 [ f [ i ] ]< tab2 [ z ] :

d e l t a=de l t a ∗R[ ( n+2)∗( tab2 [ f [ i ]]−1)+ tab2 [ z ] ]

f o r z in range ( heads2 [ l i n e ] , heads2 [ g ] ) :i f tab2 [ f [ i ] ]> tab2 [ z ] :

d e l t a=de l t a /R[ ( n+2)∗( tab2 [ z ]−1)+tab2 [ f [ i ] ] ]i=i+1

i f ( i>l en ( f )−1):g=l i n e −1

e l i f ( f [ i ]<heads2 [ g ] ) :g=g−1

e l s e :d e l t a=0g=l i n e −1

re turn de l t a

Finally, we use the two previous function to compute the boundary of an Anwith n given as input.

n=input (” Choose the value f o r n : ”)

tab =[ ]f o r x in range (1 , n+2):

tab . append ( x )

f o r A in i t e r t o o l s . permutat ions ( tab ) :

#d e f i n i t i o n o f headsAA = l i s t (A)headsA =[0]f o r i in range (1 , n+1):

i f A[ i ]>A[ headsA [ l en ( headsA ) −1 ] ] :headsA . append ( i )

47


2 1

3

4

2

3 1

4

2

3

4 1

1

3 2

4

1

3

4 2

1

2

4 3

1

2

3

4

1− t12 1− t13 1− t14 −1 + t23 −1 + t24 1− t34

Table 3.1: The boundary ∂1 of A3

headsA . append (n+1)

l ineA=0f o r l ineA in range (0 , l en ( headsA )−1):

i f l en ( headsA)>2:f o r B in f a c e s (A, headsA , l ineA , l en ( headsA )−2):

#d e f i n i t i o n o f headsBheadsB =[0]f o r i in range (1 , n+1):

i f B[ i ]>B[ headsB [ l en ( headsB ) −1 ] ] :headsB . append ( i )

headsB . append (n+1)

C=bord (A,B, headsA , headsB , l ineA )p r i n t (”0 i s a f a c e o f 1 with de l t a =2”) . format (A,B,C)

As an example we see the boundary ∂2 of A3 in 3.1,3.2,3.3.

Using this tables one can then compute the local homology of A3 and in generalof An for small n. The same calculations have already been done in literature inthe case all the ti equal to t for example in [DL16] or in [Set09] for n ≤ 7. Doingthe calculations in our example again with a single t using Sage to compute the

48


Smith forms of the matrices we obtain the known result:

H0(M(A3),Q[t±1]) ∼=Q[t±1]

(t− 1)∼= Q

H1(M(A3),Q[t±1]) ∼=(Q[t±1]

(t− 1)

)4

⊕ Q[t±1]

(t3 − 1)

H2(M(A3),Q[t±1])

(Q[t±1]

(t− 1)

)3

⊕ Q[t±1]

(t3 − 1)⊕(Q[t±1]

(t6 − 1)

)2

H3(M(A3),Q[t±1]) ∼= 0

This method seems slightly different from the previous approaches but it is stillnot fast enough to be used for large n.

49

Chap

ter3.

Local

Ab

elianH

omology

2 14 3

24 1 3

24 3 1

14 2 3

14 3 2

3 1 24

3 2 14

3 24 1

3 14 2

34 1 2

34 2 1

124 3

1− t12 1− t14 1− t13t14 −1 + t24 −1 + t23t24 0 0 0 0 0 0

13 24

0 0 0 t−123 (1− t24) t24(t34 − 1) 1− t13 1− t12t13 1− t14 0 0 0

134 2

0 0 0 t34 − t−123 t34 − 1 0 0 0 1− t13 1− t14 1− t12t14

2 134

t34 − 1 0 0 0 0 t−112 (t13 − 1) t13(1− t23) 0 t−1

12 (1− t13)(t24 − 1) t−112 (t14 − 1) t14(1− t24)

23 14

0 t−113 (1− t14) t14(t34 − 1) 0 0 t−1

12 − t23 1− t23 0 t−112 (1− t24) 0 0

234 1

0 t34 − t−113 −1 + t34 0 0 0 0 1− t23 0 t−1

12 − t24 1− t24


50


4 1 2 3 4 2 1 3 4 2 3 1 4 1 3 2 4 3 1 2 4 3 2 1

14 2 3

1− t14 1− t12t14 1− t12t13t14 0 0 0

14 3 2

0 0 0 1− t14 1− t13t14 1− t12t13t14

2 14 3

t−112 (t14 − 1) t14(1− t24) t13t14(1− t24) t−1

12 (t14 − 1) t−112 (t13t14 − 1) t13t14(1− t23t24)

24 1 3

t−112 − t24 1− t24 0 t−1

12 − t23t24 0 0

24 3 1

0 0 1− t24 0 t−112 − t23t24 1− t23t24

3 1 24

t24(t13t23)−1(t14 − 1) (t13t23)−1(1− t24) 0 t−113 t24(t14 − 1) t14t24(1− t34) 0

3 2 14

(t12t13t23)−1(1− t14) t14(t13t23)−1(t24 − 1) t−123 t14(t24 − 1) 0 0 t14t24(1− t34)

3 24 1

(t13t23)−1(t24 − t−112 ) (t13t23)−1(t24 − 1) t−1

23 (t24 − 1) t24(t−113 − t34) t24(1− t34) t24(1− t34)

3 14 2

(t13t23)−1(t14 − 1) (t13t23)−1(t12t14 − 1) t12t14(t−123 − t34) t−1

13 (t14 − 1) t14(1− t34) t12t14(1− t34)

34 1 2

t−113 t−123 − t34 0 0 t−1

13 − t34 1− t34 0

34 2 1

0 t−113 t−123 − t34 t−1

23 − t34 0 0 1− t34


51


52

Chapter 4

Minimality of infinite affinearrangement

As described in the previous chapter, in [Del08] Delucchi constructs a perfectacyclic matching in the central case using an order of the chambers and the shella-bility. This chapter is a joint work with Giovanni Paolini in which we will try tounderstand how much is still valid in the affine case, even with infinite hyperplane.In the first section we will see how to modify some definitions to make them stillwork and in particular that we must ask a more rigid requirement for an orderto be valid. Following this we will prove that if we have a valid order we areable to build an acyclic matching with as many singular cells as chambers in thearrangements in the case of locally finite affine hyperplane arrangements. Finally,in the last section we will show a valid ordering of the chamber that we will calleuclidean order and prove that even in the infinite case all the boundary maps arezero.

4.1 Decomposition of the Salvetti complex

We are going to construct an acyclic matching on the Salvetti complex of a lo-cally finite affine arrangement A, with critical cells in explicit bijection with thechambers of A. Here with locally finite we mean that each compact set intersecta finite number of hyperplanes and also that each chamber has a finite numberof walls. Following the ideas of Delucchi [Del08], we want to decompose the Sal-vetti complex into “pieces” (one piece for every chamber) and construct an acyclicmatching on each of these pieces with exactly one critical cell. More formally, weare going to decompose the poset of cells S(A) as a disjoint union

S(A) =⊔C∈C

N(C),

53

Chapter 4. Minimality of infinite affine arrangement

so that every subposet N(C) ⊆ S(A) admits an acyclic matching with one criticalcell.

Definition 4.1.1. Given a chamber C ∈ C, let S(C) ⊆ S(A) be the set of all thecells 〈C ′, F 〉 ∈ S(A) such that C ′ = C.F . In other words, a cell is in S(C) if allthe hyperplanes in supp(F ) do not separate C and C ′.

Notice that the cells in S(C) form a subcomplex of the Salvetti complex (usingposet terminology, S(C) is a lower ideal in S(A)). This subcomplex is dual to thestratification of Rn induced by A. Also, the natural map S(C)→ F which sends〈C ′, F 〉 to F is an order-preserving bijection. This yields a poset isomorphismS(C) ∼= F .

Now fix a total order a of the chambers:

C = C0 a C1 a C2 a . . .

(when C is infinite, the order type is that of natural numbers).

Definition 4.1.2. Let N(C) ⊆ S(C) be the subset consisting of all the cells notincluded in any S(C ′) with C ′ a C.

The subsets N(C), for C ∈ C, form a partition of S(C). All the 0-cells arecontained in N(C0) = S(C0). Therefore, for C 6= C0, the cells of N(C) do notform a subcomplex of the Salvetti complex. If A is a (finite) central arrangement,this definition of N(C) coincides with the one given in [Del08, Section 4].

We want to choose the total order a of the chambers so that each N(C) admitsan acyclic matching with exactly one critical cell. In [Del08] this is done taking anylinear extension of the partial order ≤C0 , for any base chamber C0. Such a totalorder works well for central arrangements but not for general affine arrangements,as we see in the following two examples.

Example 4.1.3. Consider a non-central arrangement of three lines in the plane,as in Figure 4.1 on the left. Choose C0 to be one of the three simplicial unboundedchambers. In any linear extension of ≤C0 , the last chamber C6 must be the non-simplicial unbounded chamber opposite to C0. However S(C6) ⊆

⋃C 6=C6

S(C), soN(C6) is empty, and therefore it does not admit an acyclic matching with onecritical cell. Figure 4.2 shows the decomposition of the Salvetti complex for oneof the possible linear extensions of ≤C0 .

Example 4.1.4. Consider the arrangement of five lines depicted on the right ofFigure 4.1. For every choice of a base chamber C0 and for every linear extensionof ≤C0 , there is some chamber C such that N(C) is empty.

54

4.2. Construction of the acyclic matching

C0 C6

Figure 4.1: Two line arrangements.

Now we are going to state the condition on the total order a on C that producesa decomposition of the Salvetti complex with the desired properties. First recallthe following definition from [Del08].

Definition 4.1.5. Given a chamber C and a total order ≤ on C, let

J (C) = X ∈ L | supp(X) ∩ s(C,C ′) 6= ∅ ∀C ′ a C.

Notice that J (C) is an upper ideal of L, and coincides with L for C = C0. In[Del08, Theorem 4.15] it is proved that, if A is a (finite) central arrangement anda is a linear extension of ≤C0 (for any choice of C0 ∈ C), then J (C) is a principalupper ideal for every chamber C ∈ C. This is the condition we need.

Definition 4.1.6 (Valid order). A total order a on C is valid if, for every chamberC ∈ C, J (C) is a principal upper ideal generated by some flat XC = |FC | ∈ Lwhere FC is a face of C.

The total orders of Example 4.1.3 are not valid, because J (C6) is empty. Avalid order that begins with the same chamber C0 is showed in Figure 4.3.

4.2 Construction of the acyclic matching

Throughout this section we assume to have an arrangement A together with avalid order a of C (as in Definition 4.1.6). Using the decomposition

S(A) =⊔C∈C

N(C)

of Section 4.1 (associated to the valid order a), we are going to construct a properacyclic matching on S(A) with critical cells in bijection with the chambers. In moredetail, we are going to construct an acyclic matching on every N(C) with exactly

55


C0

C1

C2

C3

C4

C5

C6

(a) Total order of the cham-bers

C0

(b) N(C0)

C1

(c) N(C1)

C2

(d) N(C2)

C3

(e) N(C3)

C4

(f) N(C4)

C5

(g) N(C5)

C6

(h) N(C6) is empty

Figure 4.2: A non-central arrangement of three lines in the plane, with a linearextension of ≤C0 . Here N(C5) and N(C6) do not admit an acyclic matching withone critical cell.

56


C0

C1

C2

C3

C5

C6

C4

(a) Total order of the cham-bers

C0

(b) N(C0)

XC1C1

(c) N(C1)

XC2

C2

(d) N(C2)

XC3

C3

(e) N(C3)

XC4

C4

(f) N(C4)

XC5

C5

(g) N(C5)

XC6

C6

(h) N(C6)

Figure 4.3: A non-central arrangement of three lines in the plane, with a validorder of the chambers. For every chamber C except C0, the generator XC of J (C)is highlighted.

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one critical cell, and then attach these matchings together using the PatchworkTheorem (Theorem 1.1.4). This strategy is the same as the one employed in[Del08], but our proofs are different since we deal with affine and possibly infinitearrangements.

Lemma 4.2.1. Suppose that a is a valid order of C, in the sense of Definition4.1.6. Then

N(C) = 〈D,F 〉 ∈ S(C) | F ⊆ XC.

Proof. To prove the inclusion ⊆, assume by contradiction that there exists somecell 〈D,F 〉 ∈ N(C) with F * XC . By minimality of XC in J (C) we have that|F | /∈ J (C). This means that there exists a chamber C ′ a C such that supp(F )∩s(C,C ′) = ∅. Then C and C ′ belong to the same chamber of A|F |, which impliesC ′.F = C.F . By definition of S(C), we have that C.F = D. Then C ′.F = D, so〈D,F 〉 ∈ S(C ′). This is a contradiction since 〈D,F 〉 ∈ N(C) and C ′ a C.

For the opposite inclusion, consider a cell 〈D,F 〉 ∈ S(C) with F ⊆ XC . Then|F | ∈ J (C), so for every chamber C ′ a C there exists an hyperplane in supp(F )∩s(C,C ′). By the same argument as before we can deduce that D = C.F 6= C ′.Ffor all C ′ a C, which means that 〈D,F 〉 /∈ S(C ′) for all C ′ a C. Therefore〈D,F 〉 ∈ N(C).

Now that we have a good description of every N(C), we can begin the con-struction of the matching. For every chamber C ∈ C consider the map

ηC : S(C)→ C

that sends a cell 〈D,F 〉 to the chamber opposite to D with respect to F . If weendow C with the partial order ≤C , then ηC becomes a poset map.

Lemma 4.2.2. The map ηC : S(C)→ (C,≤C) is order-preserving.

Proof. Let 〈D,F 〉, 〈D′, F ′〉 ∈ S(C) and suppose that 〈D′, F ′〉 ≤ 〈D,F 〉. ThenF ′ F and therefore supp(F ′) ⊆ supp(F ). Call E and E ′ the images of thesetwo cells under the map ηC . By definition of ηC and S(C) we have that s(C,E) =s(C,D)∪ supp(F ) and s(C,E ′) = s(C,D′)∪ supp(F ′). In addition F ′ F impliesthat s(D,D′) ⊆ supp(F ) \ supp(F ′). Since s(C,D′) ⊆ s(C,D) ∪ s(D,D′), weconclude that

s(C,E ′) = s(C,D′) ∪ supp(F ′) ⊆ s(C,D) ∪ s(D,D′) ∪ supp(F ′)

⊆ s(C,D) ∪ supp(F ) = s(C,E).

Therefore E ′ ≤C E.

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Consider the restriction ηC = ηC |N(C) : N(C)→ C. The matching on N(C) willbe obtained as a union of acyclic matchings on each fiber η−1

C (E) of ηC . Lemma4.2.2, together with the Patchwork Theorem, will ensure that the matching onN(C) is acyclic. We now fix two chambers C and E, and study the fiber η−1

C (E).

Lemma 4.2.3. Let a be a valid order of C, and let C,E be two chambers. A cell〈D,F 〉 ∈ S(A) is in the fiber η−1

C (E) if and only if D is opposite to E with respectto F , F ⊆ XC and supp(F ) ⊆ s(C,E).

Proof. Suppose that 〈D,F 〉 ∈ η−1C (E). In particular 〈D,F 〉 ∈ N(C), thus by

Lemma 4.2.1 we have that F ⊆ XC . By definition of ηC , D is the unique chamberopposite to E with respect to F . Finally supp(F ) ⊆ s(D,E) by definition of ηC ,and supp(F )∩s(C,D) = ∅ by definition of S(C), so supp(F ) ⊆ s(D,E)\s(C,D) ⊆s(C,E).

We want now to prove that a cell 〈D,F 〉 that satisfies the given conditions isin the fiber η−1

C (E). Since D is opposite to E with respect to F , we deduce thatsupp(F ) ⊆ s(D,E). Then, using the hypothesis supp(F ) ⊆ s(C,E), we obtainsupp(F )∩s(C,D) = ∅. This means that C.F = D, i.e. 〈D,F 〉 ∈ S(C). By Lemma4.2.1, we conclude that 〈D,F 〉 ∈ N(C). The fact that ηC(〈D,F 〉) = E followsdirectly from the definition of ηC .

A cell 〈D,F 〉 in the fiber η−1C (E) is determined by F , because D is the unique

chamber opposite to E with respect to F . Then we immediately have the followingcorollary.

Corollary 4.2.4. The fiber η−1C (E) is in order-preserving (and rank-preserving)

bijection with the set of faces F E such that F ⊆ XC and supp(F ) ⊆ s(C,E).

Assume from now on that the fiber η−1C (E) is non-empty. By Lemma 4.2.3,

this means that E ∩ XC 6= ∅. Consider the restricted arrangement AXC , and letE ′ = E∩XC and C ′ = C ∩XC be the restrictions of our chambers to AXC . Noticethat C ′ 6= ∅ by definition of a valid order, so both C ′ and E ′ are chambers of AXC .With this notation we can restate the above corollary as follows.

Corollary 4.2.5. Suppose that the fiber η−1C (E) is non-empty. Then C ′ = C ∩XC

and E ′ = E ∩XC are chambers of the arrangement AXC , and η−1C (E) is in order-

preserving bijection with the set of faces F E ′ such that supp(F ) ⊆ s(C ′, E ′) inAXC .

Proof. By Definition 4.1.6, XC = |FC | for some face FC of C. Then C ′ = C∩XC =FC is a chamber of AXC .

Consider now any cell 〈D,F 〉 ∈ η−1C (E), and let D′ = D∩XC . If we prove that

D′ is a chamber of AXC then the same is true for E ′ since they are opposite with

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respect to F and F ⊆ XC (by Lemma 4.2.1). Let F ′C = FC .F in the arrangementAXC (so F ′C is a chamber of AXC ), and consider the chamber D = C.F ′C in A.Then D = C.F = D (the first equality holds because F ′C F , and the secondequality because D ∈ S(C)). Therefore D′ = D∩XC = D∩XC = F ′C is a chamberof AXC .

The second part is mostly a rewriting of Corollary 4.2.4, but some care shouldbe taken since we are passing from the arrangement A to the arrangement AXC .To avoid confusion, in AXC write supp′ and s′ in place of supp and s. Givena face F ⊆ XC , we need to prove that supp(F ) ⊆ s(C,E) in A if and only ifsupp′(F ) ⊆ s′(C ′, E ′) in AXC . This is true because

supp′(F ) = H ∩XC | H ∈ supp(F ) and H + XC;s′(C ′, E ′) = H ∩XC | H ∈ s(C,E) and H + XC.

Constructing an acyclic matching on η−1C (E) is then the same as constructing

one on the set of faces of E ′ given by Corollary 4.2.5. We start by considering thespecial case E ′ = C ′.

Lemma 4.2.6. Suppose that the fiber η−1C (E) is non-empty. Then E ′ = C ′ if and

only if E is the chamber opposite to C with respect to XC. In this case, η−1C (E)

contains the single cell 〈C,FC〉.

Proof. If E is opposite to C with respect to XC , then clearly E ′ = C ′. Conversely,suppose that E ′ = C ′ = FC . Let 〈D,F 〉 be any cell in η−1

C (E). As in the proof ofCorollary 4.2.5, we have that D∩XC = F ′C where F ′C = FC .F in AXC . Notice thatF ⊆ E ∩XC = E ′ = FC , so F ′C = FC .F = FC . In other words, the chambers C,D and E all contain the face FC . Since F ⊆ FC ⊆ C ∩D, we have that s(C,D) ⊆supp(F ). But D ∈ S(C) implies that D = C.F , i.e. s(C,D) ∩ supp(F ) = ∅.Therefore s(C,D) = ∅, so C = D. Now E is the opposite of D with respect to F ,and E ∩XC = D ∩XC = FC , so F = FC . This means that E is the opposite of Cwith respect to XC . The previous argument also shows that η−1

C (E) contains thesingle cell 〈C,FC〉.

In particular, for every chamber C there is exactly one fiber η−1C (E) for which

E ′ = C ′. This special fiber contains exactly one cell, which is going to be criticalwith respect to our matching.

Consider now the case E ′ 6= C ′. In view of Corollary 4.2.5, we work withthe restricted arrangement AXC in XC . Until Lemma 4.2.8, all our notations (forexample, supp(F ) and s(C ′, E ′)) are intended with respect to the arrangementAXC . In what follows we make use of the definitions and facts of Section 1.4.

Lemma 4.2.7. Let yC′ be a point in the interior of C ′. The faces F E ′ suchthat supp(F ) ⊆ s(C ′, E ′) are exactly the faces of E ′ that are visible from yC′.

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E ′ C ′

Figure 4.4: The faces of E ′ that are visible from a point in the interior of C ′.

Proof. Suppose that supp(F ) ⊆ s(C ′, E ′). In particular, for every facet G ⊇ F ofE ′, the hyperplane |G| separates C ′ and E ′ and so G is visible from yC′ . Then Fis visible from yC′ .

Conversely, suppose that F is visible from yC′ . Denote by B ⊆ supp(F ) theset of hyperplanes |G| where G ⊇ F is a facet of E ′. All the facets G ⊇ F of E ′

are visible from yC′ , so the hyperplanes |G| separate C ′ and E ′. In other words,B ⊆ s(C ′, E ′). In the central arrangement AXC|F | = supp(F ), the chambers π|F |(C

′)

and π|F |(E′) are therefore opposite to each other, and B is the set of their walls.

Then every hyperplane in supp(F ) separates C ′ and E ′.

Fix an arbitrary point yC′ in the interior of C ′. By the previous lemma, thefaces F given by Corollary 4.2.5 are exactly the faces of E ′ that are visible fromyC′ . See Figure 4.4 for an example.

The idea now is that, if E ′ is bounded, the boundary of E ′ is shellable and wecan use a shelling to construct an acyclic matching on the set of visible faces. Wefirst need to reduce to the case of a bounded chamber (i.e. a polytope).

Lemma 4.2.8. There exists a finite set A′ of hyperplanes in XC, and a boundedchamber E ⊆ E ′ of the hyperplane arrangement A′ ∪ AXC , such that the poset offaces of E that are visible from yC′ is isomorphic to the poset of faces of E ′ thatare visible from yC′.

Proof. Let XC∼= Rk. Let Q be a finite set of points which contains yC′ and a

point in the (relative) interior of each visible face of E ′. For i = 1, . . . , k, defineqi ∈ R as the minimum of all the i-th coordinates of the points in Q, and qi as themaximum.

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E ′ C ′yC′

Figure 4.5: Construction of the bounded chamber E ⊆ E ′ in the proof of Lemma4.2.8. The points of Q are colored in blue, the hyperplanes of A′ are the dashedlines.

ChooseA′ as the set of the 2k hyperplanes of the form xi = qi−1 and xi = qi+1,for i = 1, . . . , k. See Figure 4.5 for an example.

Let E be the chamber of AXC ∪ A′ that contains Q \ yC′. By construction,E is bounded.

The walls of E ′ and of E are related as follows: WE = WE′ ∪ A′′ for someA′′ ⊆ A′. The hyperplanes in WE′ separate yC′ and E, whereas the hyperplanesin A′′ do not. This means that a facet G of E is visible if and only if |G| ∈ WE′ .

There is a natural order-preserving (and rank-preserving) injection ϕ from theset V of the visible faces F of E ′ to the set of faces of E, which maps a face Fto the unique face F of E such that F ∩ Q ⊆ F ⊆ F . We want to show that theimage of ϕ coincides with the set of visible faces of E.

Consider a facet G of E. Then G is in the image of ϕ if and only if |G| 6∈ A′′,which happens if and only if G is visible.

Consider now a generic face F of E. If F = ϕ(F ) for some F ∈ V , thenQ∩F ⊆ F and so F is not contained in any hyperplane of A′′. Then all the facetsG ⊇ F of E are visible, and so F is visible. Conversely, if F is not in the imageof ϕ, then F is contained in some hyperplane of A′′ and therefore also in somenon-visible facet G. Then F is not visible.

We now show that the poset of visible faces of a polytope admits an acyclicmatching such that no face is critical. We will use this result on the polytope E,in order to obtain a matching on the fiber η−1

C (E).

Theorem 4.2.9. Let X be a k-dimensional polytope in Rk, and let y ∈ Rk be apoint outside X that does not lie in the affine hull of any facet of X. Then there

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exists an acyclic matching on the poset of faces of X visible from y, such that noface is critical.

Proof. By [Zie12, Theorem 8.12] and [Zie12, Lemma 8.10], there is a shellingG1, . . . , Gs of ∂X such that the facets visible from y are the last ones. Supposethat Gt, Gt+1, . . . , Gs are the visible facets. Notice that there is at least one visiblefacet and at least one non-visible facet. In particular, the first facet G1 is notvisible and the last facet Gs is visible. In other words, we have 2 ≤ t ≤ s.

In [Del08, Proposition 1] it is proved that a shelling of a regular CW complexY induces an acyclic matching on the poset of cells (P,<) of Y (augmented withthe empty face ∅), with critical cells corresponding to the spanning facets of theshelling. In our case, Y = ∂X is a regular CW decomposition of a sphere, so theonly spanning facet of a shelling is the last one (see for example [Del08, Lemma2.13]).

Let M be an acyclic matching on ∂X induced by the shelling G1, . . . , Gs, asin [Del08]. We claim that the construction of [Del08] produces a matching whichis homogeneous with respect to the grading ϕ : (P,<)→ 1, . . . , s given by

ϕ(F ) = mini ∈ 1, . . . , s | F ≤ Gi.

To prove this, we need to briefly go through the construction ofM. The first step[Del08, Lemma 2.10] is to construct a total order @i on each Pi (the set of faces ofcodimension i). The order @0 is simply the shelling order of the facets. It followsfrom the recursive construction of @i that each ϕ|Pi : (Pi,@i)→ 1, . . . , s is order-preserving. Then the linear extensionC of P constructed in [Del08, Definition 2.11]is such that ϕ : (P,C)→ 1, . . . , s is also order-preserving. By construction of thematching [Del08, Lemma 2.12], if (p, q) ∈ M (with p ≥ q) then p C q. From thiswe obtain ϕ(p) ≥ ϕ(q) and ϕ(p) ≤ ϕ(q), so ϕ(p) = ϕ(q). Therefore the matchingis homogeneous with respect to ϕ.

The set of visible faces of X is ϕ−1(t, . . . , s) ∪ X. Notice that the emptyface ∅ belongs to ϕ−1(1), so it does not appear in ϕ−1(t, . . . , s) because t ≥ 2.

LetM′ be the restriction ofM to ϕ−1(t, . . . , s). This is an acyclic matchingon ϕ−1(t, . . . , s) with exactly one critical face, the facet Gs. ThenM′∪(X,Gs)is an acyclic matching on the poset of visible faces of X such that no face iscritical.

We are finally able to attach the matchings on the fibers η−1C (E), putting all

the results of this section together.

Theorem 4.2.10. Let A be a locally finite hyperplane arrangement, and let a bea valid order of the set of chambers C. For every chamber C ∈ C, there exists aproper acyclic matching on N(C) such that the only critical cell is 〈C,FC〉. The

63


union of these matchings forms a proper acyclic matching on S(A) with criticalcells in bijection with the chambers.

Proof. Consider the mapη : S(A)→ C × C

defined as〈D,F 〉 7→

(C, ηC(〈D,F 〉)

),

where C ∈ C is the chamber such that 〈D,F 〉 ∈ N(C).Corollary 4.2.5 provides a description of the non-empty fibers η−1(C,E), since

by definition η−1(C,E) = η−1C (E). By Lemma 4.2.6 we know that for every C ∈

C there is exactly one non-empty fiber such that E ∩ XC = C ∩ XC , and thisfiber contains the single cell 〈C,FC〉. By Lemma 4.2.7 and Lemma 4.2.8, everyother non-empty fiber η−1(C,E) is isomorphic to the poset of visible faces of somepolytope in XC (with respect to some external point not lying on the affine hull ofthe facets). Finally, by Theorem 4.2.9 this poset admits an acyclic matching withno critical faces.

We want to use the Patchwork Theorem (Theorem 1.1.4) to attach these match-ings together. To do so, we first need to define a partial order on C ×C that makesη a poset map. The order ≤ on C × C is the transitive closure of:

(C ′, E ′) ≤ (C,E) if and only if C ′ C and E ′ ≤C E(we denote by the “less than or equal to” with respect to the total order a).

To prove that η is a poset map, suppose to have 〈D′, F ′〉 ≤ 〈D,F 〉 in S(A).Let η(〈D′, F ′〉) = (C ′, E ′) and η(〈D,F 〉) = (C,E). Since S(C) is a lower idealof S(A), we immediately obtain that 〈D′, F ′〉 ∈ S(C) and C ′ a C. Then Lemma4.2.2 implies that E ′ ≤C E. Therefore (C ′, E ′) ≤ (C,E).

By the Patchwork Theorem, the union of the matchings on the fibers of η formsan acyclic matching on S(A), with critical cells in bijection with the chambers.

We now have to prove that the obtained matching is proper. To do so, weprove that the (C × C)-grading η is compact. Since every fiber η−1(C,E) is finiteby Lemma 4.2.3, we only need to show that the poset (C × C)≤(C,E) is finite forevery pair of chambers (C,E).

We prove this by double induction, first on the chamber C (with respect to theorder a) and then on n = |s(C,E)|. The base case, C = C0 and n = 0, is trivialsince E = C0.

We want now to prove the inductive step. Given a pair (C, n) ∈ C×N, supposethat the claim is true for every pair (C ′, n′) such that either C ′ a C, or C ′ = Cand n′ < n. For every chamber E with |s(C,E)| = n we have that

(C × C)≤(C,E) =⋃C′ CE′≤CE

(C′,E′) 6=(C,E)

(C × C)≤(C′,E′) ∪ (C,E)

64


This is a union over a finite number of sets, and by induction hypothesis every set(C × C)≤(C′,E′) is finite. Therefore the set (C × C)≤(C,E) is finite.

By the Patchwork Theorem, the matchings on the fibers η−1(C,E) can beattached together to form a proper acyclic matching on S(A). By constructionthis matching is a union of proper acyclic matchings on the subposets N(C) forC ∈ C, each of them having exactly one critical cell 〈C,FC〉.

We end this section with a few remarks. We are not going to use them inthe following, but they are interesting by themselves (especially in relation with[Del08]).

The first remark is that, without the need of a valid order, the results of thissection allow to obtain a proper acyclic matching on S(C0) (for any chamberC0 ∈ C) with the single critical cell 〈C0, C0〉. This is because N(C0) = S(C0), andin the construction of the matching on N(C0) we do not use the existence of avalid order that begins with C0. As noted in Section 4.1, there is a natural posetisomorphism S(C0) ∼= F for every chamber C0 ∈ C. Then the existence of anacyclic matching on S(C0) can be stated purely in terms of F , without speakingof the Salvetti complex. This result is stated in [Del08, Theorem 3.6] in the caseof the face poset of an oriented matroid.

Theorem 4.2.11. Let A be a locally finite hyperplane arrangement. For everychamber C ∈ C(A) there is a proper acyclic matching on the poset of faces F(A)such that C is the only critical face.

The second remark is that, given a valid order a of C and a chamber C ∈ C,the poset N(C) is isomorphic to F(AXC ).

Lemma 4.2.12 (cf. [Del08, Lemma 4.20]). Suppose that a is a valid order of C.For every chamber C ∈ C there is a poset isomorphism

N(C) ∼= F(AXC

).

Proof. The isomorphism in the left-to-right direction sends a cell 〈D,F 〉 ∈ N(C)to the face F , which is in F(AXC ) by Lemma 4.2.1. The inverse map sends a faceF ∈ F(AXC ) to the cell 〈C.F, F 〉, which is in N(C) by definition of S(C) and byLemma 4.2.1. These maps are order-preserving.

Together, Lemma 4.2.12 and Theorem 4.2.11 give an alternative (but equiva-lent) construction of our matching on S(A), closer to the approach of [Del08].

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4.3 Euclidean orders

In this section we are going to construct a valid order aeu of the set of chambersC, for any locally finite arrangement A, using the Euclidean distance in Rn. Thenwe are going to prove that the matching induced by this order (given by Theorem4.2.10) yields a minimal Morse complex.

Denote by d the Euclidean distance in Rn. Also, if K is a closed convex subsetof Rn, denote by ρK(x) the projection of a point x ∈ Rn onto K. The point ρK(x)is the unique point y ∈ K such that d(x, y) = d(x,K).

The first step is to prove that there exist a lot of generic points with respect tothe arrangement A. For this, we need the following technical lemma. By measurewe always mean the Lebesgue measure in Rn.

Lemma 4.3.1. Let K1 and K2 be two closed convex subsets of Rn. Let

S = x ∈ Rn | d(x,K1) = d(x,K2) and ρK1(x) 6= ρK2(x).

Then S has measure zero.

Proof. This proof was suggested by Federico Glaudo. Let di(x) = d(x,Ki) fori = 1, 2. Each function di : Rn → R is differentiable on Rn \Ki by [GM12, Lemma2.19], and its gradient in a point x 6∈ Ki is the versor with direction x− ρKi(x).

Let f(x) = d1(x)−d2(x). Denote by A the open set of points x ∈ Rn\(K1∪K2)such that ρK1(x) 6= ρK2(x). On this set, the function f is differentiable and itsgradient does not vanish. It is known that the gradient of f must vanish almosteverywhere on A∩f−1(0) [EG92, Corollary 1 of Section 3.1], hence A∩f−1(0) hasmeasure zero.

It is easy to check that the points in K1 ∪ K2 cannot belong to S. ThenS = A ∩ f−1(0) has measure zero.

Lemma 4.3.2 (Generic points). Given a locally finite hyperplane arrangement Ain Rn, let G ⊆ Rn be the set of points x ∈ Rn such that:

(i) for every C,C ′ ∈ C with d(x,C) = d(x,C ′), we have ρC(x) = ρC′(x) ∈ C∩C ′;

(ii) for every L,L′ ∈ L with L′ ( L, we have d(x, L′) > d(x, L).

Then the complement of G has measure zero. In particular, G is dense in Rn.

Proof. Given C,C ′ ∈ C, let SC,C′ be the set of points x ∈ Rn such that d(x,C1) =d(x,C2) and ρC1(x) 6= ρC2(x). By Lemma 4.3.1, every SC,C′ has measure zero.

Similarly, for every L,L′ ∈ L with L′ ( L, denote by TL,L′ the set of pointsx ∈ Rn such that d(x, L′) = d(x, L). We have that TL,L′ is an affine subspace ofRn of codimension at least 1, and in particular it has measure zero.

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4.3. Euclidean orders

The complement of G is the union of all the sets SC,C′ for C,C ′ ∈ C and TL,L′for L,L′ ∈ L with L′ ( L. This is a finite or countable union of sets of measurezero, hence it has measure zero.

We call generic points the elements of G, as defined in Lemma 4.3.2. Noticethat, by condition (ii) with L = Rn, a generic point must lie in the complement ofA.

We are now able to define Euclidean orders.

Definition 4.3.3 (Euclidean orders). A total order aeu of the set of chambersC is Euclidean if there exists a generic point x0 such that C aeu C

′ implies thatd(x0, C) ≤ d(x0, C

′). The point x0 is called a base point of the Euclidean orderaeu.

In other words, a Euclidean order is a linear extension of the partial order onC given by C < C ′ if d(x0, C) < d(x0, C

′), for some fixed generic point x0 ∈ Rn.In particular, for every generic point x0 there exists at least one Euclidean orderwith x0 as a base point. Since the set of generic points is dense, we immediatelyget the following corollary.

Corollary 4.3.4. For every chamber C0 ∈ C, there exists a Euclidean order aeu

that starts with C0.

Proof. It is enough to take the base point x0 in the interior of the chamber C0.

Theorem 4.3.5. Let aeu be a Euclidean order with base point x0. For everychamber C, let xC = ρC(x0) and let FC be the smallest face of C that contains xC.Then J (C) is the principal upper ideal generated by XC = |FC |. Therefore aeu isa valid order.

Proof. First we want to prove that XC ∈ J (C). This is equivalent to proving thatfor every chamber C ′ aeu C there exists a hyperplane H ∈ supp(XC) ∩ s(C,C ′).We have that ρXC (x0) = xC because FC is the smallest face that contains xC .Then it is also true that ρπXC (C)(x0) = xC . Given a chamber C ′ aeu C, we havetwo possibilities.

• d(x0, C′) < d(x0, C). Then C ′ * πXC (C), because all the points of πXC (C)

have distance at least d(x0, C) from x0. This means that there exists ahyperplane H ∈ supp(XC) = AXC which separates C and C ′.

• d(x0, C′) = d(x0, C). Since x0 is a generic point, we have that xC = xC′ ∈

C ∩ C ′. Then FC is a common face of C and C ′, and every hyperplane ins(C,C ′) contains FC .

67


Now we want to prove that X ⊆ XC for every X ∈ J (C). Suppose bycontradiction that X * XC for some X ∈ J (C). In particular, XC 6= Rn andthus x0 6= xC . We first prove that supp(XC ∪X) is non-empty.

Let C ′ be the chamber of A such that x0 ∈ πXC (C ′) and C ′ ≺ FC . SincexC ∈ XC ⊆ πXC (C ′), the entire line segment ` from x0 to xC is contained inπXC (C ′). Then there is a neighbourhood of xC in ` which is contained in C ′,hence d(x0, C

′) < d(x0, xC) and therefore C ′ aeu C. Since X ∈ J (C), there existsa hyperplane H ∈ supp(X) ∩ s(C,C ′). We also have that FC ⊆ C ∩ C ′, and thusXC ⊆ H.

Consider now the flat X ′ = ∩Z ∈ L | XC ∪X ⊆ Z, i.e. the meet of XC andX in L. The flat X ′ is contained in the hyperplane H constructed above, so inparticular X ′ 6= Rn. In addition, since X * XC , X ′ is different from XC . Then thepoint y0 = ρX′(x0) is different from xC , and we have d(x0, y0) < d(x0, xC), becausex0 is generic (see condition (ii) of Lemma 4.3.2). Let F be the smallest face thatcontains the line segment [xC , xC + ε(y0 − xC)] for some ε > 0. By construction,for every chamber C ′′ such that C ′′ F we have that C ′′ aeu C. This holds inparticular for C ′′ = C.F . Then we have supp(F ) ∩ s(C,C ′′) = ∅.

Since X ∈ J (C) and C ′′ aeu C, there exists a hyperplane H ∈ supp(X) ∩s(C,C ′′). By construction, xC ∈ C ∩ C ′′ and then XC is contained in everyhyperplane of s(C,C ′′). In particular, XC ⊆ H. Therefore XC ∪ X ⊆ H, whichmeans that H ∈ supp(XC ∪ X) ⊆ supp(X ′). Both xC and y0 belong to X ′,hence F ⊆ X ′. Putting everything together, we get H ∈ supp(X ′) ∩ s(C,C ′′) ⊆supp(F ) ∩ s(C,C ′′) = ∅. This is a contradiction.

Remark 4.3.6. For a given generic point x0, there might be more than one Eu-clidean order aeu with base point x0. Nonetheless, all Euclidean orders with agiven base point produce the same faces FC (by Theorem 4.3.5) and the samecritical cells (by Theorem 4.2.10). The decomposition

S(A) =⊔C∈C

N(C)

also depends only on x0 (by Lemma 4.2.1), and therefore the construction of thematching is not influenced by the choice of aeu (once the base point x0 is given).

Since Euclidean orders are valid, we are able to construct an acyclic matchingon the Salvetti complex of any arrangement. We also prove that this matchingyields a minimal Morse complex.

Theorem 4.3.7. Let A be a locally finite hyperplane arrangement in Rn. Let Mbe the matching on S(A) given by Theorem 4.2.10, induced by a Euclidean orderwith base point x0. Then the associated Morse complex S(A)M is minimal (all theincidence numbers vanish).

68

4.3. Euclidean orders

Proof. If the arrangement A is finite, we know that the sum of the Betti numbersof S(A) is equal to the number of chambers [OS80, Zas97]. By Theorem 4.2.10,the critical cells ofM are in bijection with the chambers. Then the Morse complexis minimal.

Suppose from now on that A is infinite. Fix a chamber C ∈ C, and consider theassociated critical cell 〈C,FC〉 ∈ N(C). Recall from the proof of Theorem 4.2.10the poset map η : S(A)→ C ×C, and let (C,E) = η(〈C,FC〉). Since the matchingis proper, the set η−1((C × C)≤(C,E)) is finite.

Consider now the finite set of chambers

C ′ = D ∈ C | D ∩ F ′ 6= ∅ for some cell 〈C ′, F ′〉 ∈ η−1((C × C)≤(C,E))∪ D ∈ C | d(x0, D) ≤ d(x0, C).

Notice that C ∈ C ′, and every chamber D aeu C is also contained in C ′. Let A′ ⊆ Abe the finite subarrangement of A consisting of the hyperplanes that intersect atleast one chamber in C ′. By construction, every chamber D in C ′ is also a chamberof the arrangement A′. The point x0 is generic also with respect to A′, thusit induces a Euclidean order a′eu of the chambers of A′. Choose a′eu so that itcoincides with aeu until the chamber C.

The subcomplex S = η−1((C × C)≤(C,E)) of S(A) can be also regarded as asubcomplex of S(A′), since for every cell 〈C ′, F ′〉 ∈ S we have C ′ ∈ C(A′) andF ′ ∈ F(A′). In addition, the restriction ofM to S can be extended to a matchingM′ on S(A′) induced by a′eu as in Theorem 4.2.10.

Consider now an M-critical cell 〈D,G〉 ∈ S(A) such that there is at least onealternating path from 〈C,F 〉 to 〈D,G〉. Since M is homogeneous with respectto η, every alternating path starting from 〈C,F 〉 is entirely contained in S. Inparticular, 〈D,G〉 ∈ S. Then the alternating paths from 〈C,F 〉 to 〈D,G〉 are thesame in S(A) (with respect to the matchingM) and in S(A′) (with respect to thematching M′). In particular, the incidence number between 〈C,F 〉 and 〈D,G〉is the same in the two Morse complexes. Since A′ is finite, the Morse complexS(A′)M′ is minimal and all incidence numbers vanish. Therefore the incidencenumber between 〈C,F 〉 and 〈D,G〉 in S(A)M also vanishes.

What we now want to prove is a particular version of the Brieskorn’s Lemma,whose first proof can be found in [Bri73].

Lemma 4.3.8 (Brieskorn). Let A be a nonempty complex arrangement and < theEuclidean order with respect to a base point x0. Let Lk = X ∈ L(A) | codim(X) =k and <X the Euclidean order with respect to x0 in the subarrangement AX . Thenthere is a one-to-one correspondence between the critical k−cells of S(A) and thecritical k−cells of S(AX) for each X ∈ Lk.

69


x0

C0 =F0

C1

C2

C3

C4

C5

C6

C7

C8

C9

F1

F2 F3

F4

F5

F6

F7

F8 = F9

Figure 4.6: Euclidean order with respect to x0

Proof. Let 〈C,FC〉 be a critical k−cell of S(A), XC = |FC |. This means thatcodim(XC) = k. We want to say that 〈πXC (C), XC〉 is a critical k−cell of S(AXC ).This is equivalent to say that, called xC the point of C of minimum distance fromx0, xC is still the point of minimum distance of πXC (C). This is clearly truebecause xC is the point of minimum distance of XC from x0 otherwise xC wouldnot have been in the interior of XC .

We have then defined the map, we have now to prove that this map is injectiveand surjective.

• Injectivity

Let 〈C,FC〉 and 〈C ′, FC′〉 two critical k−cell of S(A) with |FC | = |FC′ | = XC .We need to show that πXC (C) 6= πXC (C ′). This is obvious because by howwe have defined the Euclidean order FC = C ′ ∩ XC = FC′ . Then the twochambers C and C ′ have a face in common that is in XC , so at least onehyperplane of supp(XC) separate them.

• Surjectivity

Let X ∈ Lk and 〈D,X〉 a critical k−cell of S(AX). Let’s call with xC thepoint of minimum distance of X from x0 and with C the chamber in F(A)adjacent to xC and such that πX(C) = D. xC is also clearly the point ofminimum distance of C from x0 since it is for D by hypothesis and thisimplies that 〈C,X ∩ C〉 is a critical k−cell of S(A).

70

4.4. Local homology of line arrangement

4.4 Local homology of line arrangement

The first step to study local homology is that of studying alternating path betweencritical cells. We will now focus only in the case of line arrangements. In the entiresection we suppose that we have fixed a Euclidean order with base point x0.

The alternating paths between a critical 1-cell and the only critical 0-cell areparticularly easy, since all the zero cells are in N(C0) and it is immediate to seethat there are always the paths.

We want to see now that there is a correspondence between alternating pathsfrom critical 2-cells to critical 1-cells and a special kind of sequences in F1(A),which we call alternating sequences, where F ∈ F1(A) if and only if |F | ∈ L1(A).

First of all we notice that, given an alternating path between critical cells ofthe form

〈D, p〉〈C1, F1〉〈D1, p1〉〈C2, F2〉 · · · 〈Cn, Fn〉, (4.4.1)

the starting cell plus the sequence (F1, . . . , Fn) completely determines the alter-nating path. This is because for each i there are only two cells with Fi and one ofthem is in N(C0). By construction of the matching, if this cell is in an alternatingpath then all the following cells in the path are in N(C0) and so the last one cannotbe a critical 1-cell. Then Ci is uniquely determined by Fi for every i. Each cell〈Di, pi〉 is also uniquely determined, since it is matched with 〈Ci, Fi〉.

We now describe which alternating sequences in F1(A) give rise to an alternat-ing path. Given a face F ∈ F1(A), let H be the unique line in supp(F ), if ρH(x0)is not in the interior of F we denote by p(F ) the endpoint of F which is closer toρH(x0), otherwise p(F ) = ∅.

In addition, let C(F ) be the unique chamber such that 〈C(F ), F 〉 /∈ N(C0).

Definition 4.4.1. Given two different faces F, G ∈ F1(A), we say that F → G if

• F ∩G = p(F ) and p(F ) 6= ∅;

• Let HF = supp(F ) and HG = supp(G), then HF = HG or F and C0 are inthe same half-plane with respect to HG.

Lemma 4.4.2. The alternating paths between 〈D, p〉 and 〈C,F 〉 are in one toone correspondence with the alternating sequences in F1(A) of the form (F1 →F2 . . .→ Fn = F ) such that 〈C(F1), F1〉 < 〈D, p〉.

Proof. We have already said that an alternating path as in 4.4.1 is completelydetermined by the starting cell and the sequence (F1, . . . , Fn). Let us now see thatfor each i ∈ [1, n − 1] Fi → Fi+1 since the condition that 〈C(F1), F1〉 < 〈D, p〉 isobvious.

71


Let now Ei be the chamber opposite to C(Fi) with respect to Fi, then fromhow we have construct the matching it is immediate to see that the cell 〈C(Fi), Fi〉is matched with 〈D(Fi), p(Fi)〉 where D(Fi) is the chamber opposite to Ei withrespect to p(Fi). By hypothesis 〈C(Fi+1), Fi+1〉 < 〈D(Fi), p(Fi)〉 which implies thatFi∩Fi+1 = p(Fi) and thatD(Fi).Fi+1 = C(Fi+1). If we call withHi+1 = supp(Fi+1)since 〈C(Fi+1), Fi+1〉 /∈ N(C0) C0 and C(Fi+1) are in opposite semispace withrespect to Hi+1 but the same is true for Fi and C(Fi+1) because D(Fi) and Fi arein opposite semispace with respect to Hi+1 (unless of course Fi ⊂ Hi+1). Then wehave that Fi → Fi+1.

We need now to prove that given a sequence as in the hypothesis there existan alternating paths between 〈D, p〉 and 〈C,F 〉. This follows easily from what wehave said above and a simple induction on the length of the sequence.

The case n = 1 is easy since we have by hypothesis that 〈C(F1), F1〉 < 〈D, p〉.For the induction step we need only to prove that if we have F,G ∈ F1(A), F → Gthen 〈C(G), G〉 < 〈D(F ), p(F )〉. From the first conditions of Definition 4.4.1 wehave that G ≺ p(F ). We need to see that D(F ).G = C(G). This is equivalentto prove, by definition of C(G), that D(F ) and C0 are in opposite semispace withrespect to HG that again follows from the fact that F and C0 are in the samesemispace with respect to HG and the definition of D(F ).

Now that we have a simple description of the alternating paths, we want to useit to study the boundary of the Morse complex.

Definition 4.4.3. Given two different faces F, G ∈ F1(A), with F → G, let

[F → G] =[〈D(F ), p(F )〉 : 〈C(G), G〉][〈D(F ), p(F )〉 : 〈C(F ), F 〉]

,

where the incidence numbers on the right are taken in the Salvetti complex S(A),and D(F ) is defined as in the above lemma.

Theorem 4.4.4. Let A be a line arrangement in R2, and let 〈D, p〉 and 〈C,F 〉 betwo critical cells. Then their incidence number in the Morse complex is given by

[〈D, p〉 : 〈C,F 〉] =∑s∈Seq

ω(s)

where Seq is the set of sequence of Lemma 4.4.2 and for each sequence of the form

s = F1 → F2 → · · · → Fn = F

then

ω(s) = (−1)n[〈D, p〉 : 〈C(F1), F1〉]n−1∏i=1

[Fi → Fi+1]

72

4.4. Local homology of line arrangement

Proof. It follows directly from [Koz08, Definition 11.23] and Lemma 4.4.2.

Lemma 4.4.5. Given two different faces F, G ∈ F1(A), we have

[F → G] =

±1 if p(F ) = p(G)

±∏

t−1Hi

otherwise,

where tHi is a positive oriented loop around the line Hi and the product is over thelines through p(F ) that separate F and C0.

Proof. We need here some definitions given in [SS07, Chapter 5] about combinato-rial and positive paths in the 1−skeleton of S(A). By definition of local homology,given two cells 〈D, p〉, 〈C,F 〉 we have that

[〈D, p〉 : 〈C,F 〉] = [〈D, p〉 : 〈C,F 〉]Zu(D,C)

as an element of π1(M(A), x0) where u(D,C) = Γ(D)−1u(D,C)Γ(C) and Γ(C) =u(C,C0) and u(·, ·) is a positive path between the first and the second chamber.If we restrict ourselves to the case of abelian local homology, then as an elementof H1(M(A)) u(D,C) is equal to the product of the positive loops around thehyperplanes in s(C0, C) ∩ s(D,C).

Then in our special case we need to study the relation between u(D(F ), C(F ))and u(D(F ), C(G)) that is between s(C0, C(F ))∩s(D(F ), C(F )) and s(C0, C(G))∩s(D(F ), C(G)). By construction s(D(F ), C(F )) is equal to the set of hyperplanesthrough p(F ) minus HF = supp(F ).

Let us now suppose that p(G) = p(F ) and let H ∈ s(C(F ), C(G)). By hypoth-esis H does not separate F and C0 (because HG does not), so H /∈ s(C0, C(F ))and s(C0, C(G)) = s(C0, C(F ))ts(C(F ), C(G)). Moreover, since all the chambersC(F ), C(G) and D(G) are adjacent to p(F ) then

s(D(F ), C(G)) = s(D(F ), C(F ))\s(C(G), C(F ))

because HF /∈ s(D(F ), C(G)). Then

s(C0, C(F )) ∩ s(D(F ), C(F )) = s(C0, C(G)) ∩ s(D(F ), C(G)).

Let us now instead suppose that p(F ) 6= p(G). We want to prove thats(C0, C(G)) ∩ s(D(F ), C(G)) = ∅ and this will imply the thesis. Let then H ∈s(C0, C(G)) ∩ s(D(F ), C(G)). Since both D(F ), C(G) are adjacent to p(F ) thenp(F ) ∈ H. Given HG = supp(G), then there exist G′ ⊂ HG and p(G′) = p(F ).s(C(G), C(G′)) is the set of all hyperplanes trough p(F ) minus HG. If H 6= HG

then H /∈ s(C0, C(G′))∪s(D(F ), C(G′)) = s(D(F ), C(F ))∪s(C0, C(F )) so all thehyperplanes adjacent to p(F ), giving a contradiction. At the same time H cannotbe equal toHG because it does not separateD(F ) and C(G) sinceD(F ).G = C(G).

73


x0

C0

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

l1

l2

l3

l4

l5

Figure 4.7: Deconing A3

〈C4, p4〉〈C5, p5〉〈C7, p7〉〈C9, p9〉〈C10, p10〉〈C11, p11〉〈C1, F1〉 q4 − 1 q4(1− q2) 0 0 q1 − 1 q1(1− q5)〈C2, F2〉 q2q3 − 1 q2 − 1 1− q1 0 0 0〈C3, F3〉 q4 − 1 q4 − q−1

3 0 1− q5 0 0〈C6, F6〉 0 0 q4 − 1 0 q3q5 − 1 q5 − 1〈C8, F8〉 0 0 0 q2 − 1 q1 − 1 q1 − q−1

3

Table 4.1: The boundary ∂2 of the deconing of A3

Example 4.4.6 (Deconing A3). In the following example we explicitly computethe matrix associated to ∂2 for the Morse complex of the arrangement in Figure4.7. Given a chamber Ci we denote by 〈Ci, Fi〉 the associated critical cell if it is ofdimension 1 or 〈Ci, pi〉 if it is of dimension 2. (〈C0, C0〉 is the only critical 0−cell.

Since in our computations there are only negative oriented loops we will denotewith qi = t−1

i the negative oriented loop around the line li.Following theorem 4.4.4 and Lemma 4.4.5 we obtain the matrix 4.1.Specializing to the case where q1 = . . . = q5 = q we obtain that

H1(M(A),Q[q±1]) ∼=(Q[q±1]

q − 1

)3

⊕ Q[q±1]

q3 − 1

as already computed for example in [GS09].

74

Aknowledgements

La mia avventura quinquennale a Pisa e giunta al termine e questa tesi ne el’epilogo. Sono stati cinque anni molto densi, in alcuni momenti difficili, ma inmolti altri fantastici. Ne esco con un bagaglio non indifferente non solo di nozionimatematiche ma anche di lezioni di vita e per questo devo ringraziare tutti colorocon cui ho condiviso anche solo un attimo di questa avventura.

In particolare ringrazio Giove senza il quale questa tesi sarebbe molto diversa.Lo ringrazio per non avermi mai fatto mancare il suo aiuto e le sue conoscenzee per le molte decine di ore che abbiamo passato insieme a discutere. Ringrazioinoltre il mio relatore, Mario Salvetti, per i molti consigli e tutto il tempo che miha dedicato.

Ringrazio tutti i miei amici, di Pisa e non, per tutti i momenti di svago. Inparticolare ringrazio i miei compagni di corso in Normale con i quali ho condiviso lamaggior parte del mio tempo; coloro con cui ho condiviso il collegio e con cui sonoandato a mensa insieme, Alice, Benzo, Camilla, Claudio, Fabio e Tess rendendoil viaggio meno lungo; coloro con cui ho giocato a calcetto in queste calde seratepisane e coloro con cui ho giocato ad Hanabi, in particolare Sasha per tutte leottime cene a casa sua.

Ringrazio la mia dolce meta, Eli, per essermi sempre stata vicina. Per esseresempre stata pronta ad ascoltare tutti i miei dubbi e le mie paure ed avermi sempreincoraggiato a non farmi abbattere dalle difficolta.

E per ultima, ma certamente non per ultima, ringrazio la mia famiglia per nonavermi mai fatto mancare il loro supporto qualunque fossero le mie scelte e peravermi sempre dimostrato il loro affetto in ogni modo possibile.

75


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