Minimal CW-Complexes for Complements of

Reﬂection Arrangements of Type A andBn−1 n

DISSERTATION

zur Erlangung des Doktorgrades

der Naturwissenschaften

(Dr. rer. nat)

dem Fachbereich Mathematik und Informatik

der Philipps-Universit¨at Marburg

vorgelegt von

Daniel Djawadi

aus Clausthal-Zellerfeld

Marburg (Lahn) 2009I would like to thank my academic advisor Prof. Dr. Volkmar Welker for

his excellent support.

Eingereicht im M¨arz 2009

Mu¨ndliche Pru¨fung am 16.04.2009

Erstgutachter: Prof. V. Welker

Zweitgutachter: Prof. F. W. Kn¨ollerContents

1 Overview 1

2 Preliminaries 5

2.1 The reﬂection groups A and B . . . . . . . . . . . . . . . 5n−1 n

2.2 CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Cellular Homology . . . . . . . . . . . . . . . . . . . . 7

2.3 Hyperplane Arrangements . . . . . . . . . . . . . . . . . . . . 9

3 Discrete Morse Theory 13

3.1 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 A 20n−1

4.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 B 35n

5.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Matching 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Matching 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4 Describing Paths . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.2 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Details 71

6.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.1 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2.2 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Deutsche Zusammenfassung 103

References 1071 Overview

An arrangement of hyperplanes (or just an arrangement)A is a ﬁnite collec-

tionoflinearsubspacesofcodimension1inaﬁnitedimensionalvectorspace.

Each hyperplane H is the kernel of a linear function α , which is unique upH

to a constant.

WhentheunderlyingﬁeldisRtherearisequitenaturalquestionswhichhave

been studied in detail over the last century. The problem of counting regions

formed by an arbitrary arrangement of n lines in the plane already occurred

in the late 19th century. The general research on the properties of complex

hyperplane arrangements started in the late 1960’s with the groundbreaking

work of Arnold and Brieskorn.

Eventhoughtheseobjectsareeasilydeﬁned,theyyieldniceanddeepresults.

Thestudyofarrangementsrepresentsaninterestinginterfaceofdiverseﬁelds

of mathematics, such as algebra, algebraic geometry, topology and combina-

torics.

In this work we examine combinatorial properties of the complements of cer-

tain classical hyperplane arrangements.

R nA denotes the braid arrangement inR , consisting of the hyperplanesn−1

nH :={x∈R |x =x }, for 1≤i<j≤n.i,j i j

R nB denotes the arrangement inR which in addition to the hyperplanes Hi,jn

of the braid arrangement consists of the hyperplanes

nH := {x ∈R | x = −x }, for 1 ≤ i < j ≤ n and the coordinate-i,−j i j

nhyperplanes H :={x∈R |x = 0}, for i = 1,...,n.i i

nA complexiﬁcation of a real hyperplane arrangement inR is deﬁned to

nbe the hyperplane arrangement inC which is deﬁned by the same linear

forms.

We omit the indexC and denote byA andB the complexiﬁcations ofn−1 n

R Rthe real arrangements A and B , respectively. The notation is chosenn−1 n

according to the respective reﬂection groups of type A and B .n−1 n

For an arrangement of hyperplanesA we denote by M(A) the complement

of the union of all hyperplanes ofA. The complementsM(A ) andM(B )n−1 n

of the complexiﬁcations of the two arrangements above are the objects of our

study.

The topology of such complements have been the subject of studies since the

early 1970’s. The development started in 1972, when P. Deligne proved that

thecomplementofacomplexiﬁedarrangementisK(π,1)whenthechambers

nof the subdivision ofR induced by the hyperplanes are simplicial cones [7].

1With regard to this thesis one result of M. Salvetti from 1987 is of great im-

portance. He proved that the complement of a complexiﬁed real hyperplane

arrangement is homotopy equivalent to a regular CW-complex [18].

i i i−1Since the groups H (X ,X ) of the cellular cochain complex of a

CW-complex X are free abelian with basis in one-to-one correspondence

with the i-cells of X, we call a CW-complex minimal if its number of cells

iof dimension i equals the rank of the cohomology group H (X,Q).

Taking the regular CW-complexes, which are based on Salvetti’s work, as

a starting point, we derive minimal CW-complexes Γ and Γ for theA Bn−1 n

n ncomplements M(A ) ⊂C and M(B ) ⊂C of the complexiﬁcations ofn−1 n

the two arrangements above. Hence, we deduce CW-complexes which are

homotopy equivalent toM(A ) orM(B ) and which have a minimal num-n−1 n

ber of cells.

In order to decrease the number of cells, discrete Morse Theory provides our

basis tool. It was developed by R. Forman in the late 1990’s. Discrete Morse

TheoryallowstodecimatethenumberofcellsofaregularCW-complexwith-

out changing its homotopy type.

Parallel to our work, a general approach to ﬁnding a CW-complex homo-

topic to the complement of an arrangement using discrete Morse theory was

developed in [19]. Our approach is diﬀerent for the cases studied and leads

to a much more explicit description than the statement in [19].

iIt is well known that the rank of the cohomology groups H (M(A ),Q)n−1

iandH (M(B ),Q)ofthecomplementsM(A )andM(B )equalsthenum-n n−1 n

Bber of elements of length i in the underlying reﬂection groups S and S ,n n

Brespectively [1]. Here, S is the symmetric group and S is the groupn n

of signed permutations, consisting of all bijections ω of the set [±n] :=

{1,...,n,−n,...,−1} onto itself, such thatω(−a) =−ω(a) for alla∈ [±n].

Indeed, the numbers of cells of the minimal complexes Γ and Γ areA Bn−1 n

Bequal to the numbers of elements in S and S , respectively.n n

The cell-order of a CW-complex X is deﬁned to be the order relation on the

cells of X with σ≤τ for two cells σ,τ of X if and only if the closure of σ is

contained in the closure of τ. The poset of all cells of X ordered in this way

is called the face poset of X.

A main part of this thesis is devoted to the cell-orders of the minimal

CW-complexes. In case of the complex Γ the face poset turns out toAn−1

have a concise description.

ThecombinatoricsofthefaceposetofΓ seemstobetoocomplicatedtobeBn

described through a concise and explicit rule. Thus we formulate a descrip-

tionintermsofmechanismswhichallowtoconstructthecellsBwithA<B

fromagivencellA. Eventhoughthisdescriptionisrelativelycompact, there

2is still a lot of combinatorics included that has yet to be discovered.

This thesis is organized as follows:

In Section 2 we provide the mathematical background. We start Section

2 by introducing the real reﬂection groups A and B . Afterwards wen−1 n

brieﬂy present the main deﬁnitions concerning CW-complexes. For an in-

depth overview of the theory of CW-complexes we refer to [13]. After a brief

introduction to hyperplane arrangements, we give a short summary of the

constructionofSalvetti’scomplex, whichisbasedontheworkofBj¨ornerand

Ziegler [4].

Section3isanintroductiontodiscreteMorseTheory. Afterapresentationof

Forman’s approach we give a reformulation of the theory in terms of acyclic

matchings, which for our purpose is more applicable. Indeed, a large part of

this thesis is concerned with ﬁnding appropriate matchings.

We deduce a minimal CW-complex for M(A ) in Section 4. For this wen−1

deﬁne a representation of the cells of the initial complex in terms of certain

partitions of [n] := {1,...,n} and adapt the original cell-order to the new

representations. Afterwards the number of cells is decreased by deﬁning an

appropriate matching and applying the methods of discrete Morse Theory to

the initial complex. The resulting minimal complex Γ has as many cellsAn−1

as elements of the symmetric group S .n

At the end of Section 4 we examine the cell-order of Γ and present aAn−1

description. Finally we present the face poset of the minimal CW-complexes

Γ and Γ .A A2 3

In Section 5 we construct a minimal CW-complex for M(B ). Comparedn

to the A -case, this requires much more eﬀort. We deﬁne a representa-n−1

tion of the cells of the initial complex in terms of symmetric partitions of

[±n] :={1,...,n,−n,...,−1} and adapt the original cell-order to the new

representations. Afterwards, we apply the methods provided by discrete

Morse Theory twice, in order to decimate the number of cells. Hence, we

deﬁne two matchings and we prove that after the removal of the cells of the

ﬁrst matching, the methods are still applicable to the second matching.

The minimal CW-complex Γ has as many cells as elements of the groupBn

Bof signed permutations S . The remainder of Section 5 is needed to specifyn

a description of the cell-order of Γ . We give a counterexample showingBn

that, in contrast to the complex Γ , the relations A <B of cells of ΓA Bn−1 n

3in general have no representation as a chain of facets

A<A < <A <B .1 m

Due to the complexity we derive a description of the cell-order in terms of

mechanisms, which can be applied to the partition corresponding to a cell.

Therefore, a main part of the section is concerned with the translation of the

structure of the face poset of Γ into mechanisms.Bn

In Section 6 we discuss the relations A<B, i.e. A < B and dim(B) =

dim(A) + 1, in detail. We present a description of all cells B, such that

A<B. This description is given in terms of algorithms which can be ap-

plied to the partition corresponding to a cell of Γ and allow to determineBn

thecellsBwithA<BfromAeﬀectively. Itprovidesaninsighttothestruc-

tural details of Γ but also to its complexity. We present some examplesBn

at the end of Section 6 which illustrate that compared to their complicated

formulation, these algorithms are easily applicable.

Section 7 is a translation of this overview into German.

42 Preliminaries

2.1 The reﬂection groups A and Bn−1 n

As a start of this thesis we provide a short description of the real reﬂection

groups A and B . Since we do not need the details of the theory of ﬁniten−1 n

reﬂection groups, we brieﬂy list the facts concerning these two special cases.

For a deeper insight into the theory we refer to [11] and [3].

Let V be a ﬁnite dimensional real vector space endowed with a positive

deﬁnite symmetric bilinear-form h,i. A reﬂection in V is a linear function

s which sends some nonzero vector α to its negative while ﬁxing pointwiseα

the hyperplane H which is orthogonal to α. It is easy to see that s can beα α

written as follows:

2hγ,αi

s (γ) =γ− α.α hα,αi

The bilinearity of h,i implies that s is an orthogonal transformation, i.e.α

hs (γ),s ()i =hγ,i.α α

A ﬁnite real reﬂection group is a ﬁnite subgroup of O(V) generated by re-

ﬂections.

Example 2.1. (A ,n≥ 2): This reﬂection group is the symmetric groupn−1

S , i.e. the group of all permutations of [n] :={1,...,n}. It can be thoughtn

nof as a subgroup of O(R ) by assigning to each transposition (ij) ∈ Sn

nthe reﬂection s . Then S acts onR by permuting the basis vectorse −e nj i

e ,...,e . Since S is generated by transpositions, it is a reﬂection group.1 n n

nThe set of ﬁxed points of the action of S onR equals the line whichn

is spanned by e + + e . Furthermore it leaves stable the orthogonal1 n

complement which consists of the points with coordinates summing up to 0.

Thus S also acts on an (n−1)-dimensional vector space. This accounts forn

the subscript n−1.

nExample 2.2. (B ,n≥ 2): LetthesymmetricgroupS actonR asabove.n n

Deﬁne additional reﬂections s for i = 1,...,n and s for 1≤i<j≤n.e e +ei i j

These reﬂections together with the reﬂections s generate the reﬂectione −ej i

BgroupB . ItcanbeconsideredasthegroupofsignedpermutationsS whichn n

isthegroupofallbijectionsω oftheset[±n] :={−n,...,−1,1,...,n}, such

Bthat ω(−a) =−ω(a) for all a∈ [±n]. The number of elements of S equalsn

n2 n! .

52.2 CW-complexes

In this section we provide a short outline of the main deﬁnitions and some

important facts concerning CW-complexes. For more details see [13].

n nLet B be the unit ball inR .

Deﬁnition 2.3 (CW-complex). A CW-complex is a space X constructed in

the following way:

0(1) Start with a set X , equipped with the discrete topology, whose points

are regarded as 0-cells of X.

n n−1(2) Inductively, form the n-skeleton X from X by attaching n-cells

n n−1 n−1 nσ via continuous maps ϕ : S → X . This means that X isαα U

n−1 n n−1the quotient space of the disjoint union X B of X with aα α

ncollection of n-balls B under the identiﬁcations x ∼ ϕ (x) for x ∈αα U

n n n−1 n n∂B . Thus, as a set, X = X σ where each σ is an openα α α α

n-ball.

(3) One can either stop this inductive process at a ﬁnite stage, setting

nX = X for some n < ∞, or one can continue indeﬁnitely, settingS

nX = X . In the latter case X is given the weak topology: A set

n

nA⊂ X is open (or closed) if and only if A∩X is open (or closed) in

nX for each n.

All CW-complexes in this paper are ﬁnite.

A CW-complex for which all the attaching maps ϕ are homeomorphisms isα

called a regular CW-complex.

nExample2.4. (compareFigure1,page8)Then-ballB hasaCW-structure

0 n−1 nwith just three cellsσ , σ andσ where the (n−1)-cell is attached by the

n−2 0 nconstant map S → σ obtaining an (n−1)-sphere. The cell σ is then

n n−1attached by the identity map sending an element x∈∂B =S to itself.

n nEach cell σ in a cell complex X has a characteristic map Φ : D → Xαα α

which extends the attaching mapϕ and is a homeomorphism from the inte-α

n nrior of B onto σ . One can take Φ to be the compositionαα αU

n n−1 n nB ֒→ X B → X ֒→ X where the middle map is the quotientα αα

nmap deﬁning X . A subcomplex of a cell complex X is a closed subspace

A⊂X that is a union of cells of X. Since A is closed, the image of the char-

acteristic map of each cell in A is contained in A. In particular the image of

the attaching map of each cell in A is contained in A. Thus A is itself a cell

complex.

6