Chapter I. Hyperplane Systems and Galleries

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 May 2013

§1. Pregalleries

(1.1) Equivalence relations on wordsets. We need terminology used in connection with free semigroups, relations, presentations, etc.

Let Al and Ob be sets, called the alphabet and the objects respectively. Let two maps be given: beg, end:AlOb. A word is a finite sequence a1,,an with aiAl and end(ai-1)=beg(ai) for i=2,,n.

Let W be the set of all words. Mostly we add to W for each COb an empty word C. Define beg, end:WOb by beg(a1,,an)beg(a1), end(a1,,an)end(an) and beg(C)end(C)C. If w,w are words and end(w)=beg(w), then by juxtaposition we can define wwW. This operation is associative and has, if empty words are added left and right neutral elements. This way we get a category with the property that each map from Al to the morphisms together with a map from Ob to the objects such that the image of beg(a) and end(a) are the domain and range respectively of the image of aAl, extends uniquely to a functor. Therefore we call this construction the free category generated by the quadruple (Al, Ob; beg, end). If |Ob|=1, then this is nothing else then the usual free semigroup on (or generated by) Al.

Now let R be any subset of W×W such that each pair (a,b)R satisfies beg(a)=beg(b) and end(a)=end(b). Define an equivalence relation R by: wRw iff there exists a sequence w=w0,,wn=w such that wi-1=xiaiyi and wi=xibiyi for some xi,yi,ai and biW with (ai,bi)R or (bi,ai)R. If wRw, then we say that w and w are equivalent by means of the operations (or relations) in R. Remark that in that case beg(w)=beg(w) and end(w)=end(w). This is the smallest equivalence relation with the property (a,b)RxayRxby and hence we call R the equivalence relation generated by R.

(1.2) A fundamental topological theorem.

Let X be a topological space and 𝒮 a cover of X consisting of open and simply connected sets. We choose in each S𝒮 a point aS and now we want to give a description of the groupoid Π, which has as objects the set 𝒮, and ΠST for S,T𝒮 consists of the homotopy classes of paths from aS to aT. This subgroupoid of the fundamental groupoid of X essentially describes the latter, because of the simply connectedness of the sets in 𝒮.

(1.3) Definition. A pregallery G consists of a sequence of elements of 𝒮: S0,,Sk such that Si-1Si for i=1,,k, together with a sequence K1,,Kk where Ki is a connected component of Si-1Si for i=1,,k. We set beg(G)S0 and end(G)Sk. This pregallery is denoted:

S0 K1 S1 K2 S2 Sk-1 Kk Sk.

If end(G)=beg(G), then obviously we can define the pregallery G.G. This way we get a category Pgal with objects 𝒮 and whose morphisms are the pregalleries.

A pregallery of the form SKT is called direct. Remark that the category Pgal is in fact the free category on the direct pregalleries, so we can use the terminology of (1.1).

S T K d σ γ γ1 γ γ1 d aS aT Figure 1.

Now we assign to each direct pregallery SKT an element of ΠST in the following way: Choose dK and a path γ in S from aS to d and a path γ in T from d to aT. Now the homotopy class of γγ doesn't depend on the choice of d,γ and γ. For let also d1,γ1 and γ1 be chosen. Choose also a path σ in K from d to d (K is connected). Because γσ and γ1 stay inside the simply connected set S, we have γσγ1. Also, σ-1γγ1. So γγγσσ-1γγ1γ1. The class defined this way is denoted by φ(SKT). Because of freeness φ extends in a unique way to a functor φ:PgalΠ.

(1.4) Definition. Consider the pairs

R { ( SKULT, SMT ) Pgal×Pgal |KLM= } { ( SSS, C ) Pgal×Pgal| S𝒮 }

and let Rbe the equivalence relation on Pgal generated by R.

The following fundamental theorem says that Π can be described as the quotient category Pgal/R. This observation is the basis for all other presentations of fundamental groupoids and groups in the sequel.

(1.5) Theorem. The functor φ:PgalΠ is surjective and φ(G)=φ(G) iff GRG.

For the proof it is convenient to have the following:

(1.6) Definition. Let γ:[0,1]X be a continuous curve. We call a pregallery G= S0 K1 S1 Kk Sk linked with γ by a division D: 0=t0t1tk+1=1, notation: GLγ(D), if:

(i) γ([ti,ti+1]) Si for i=1,,k
(ii) γ(ti)Ki for i=1,,k.

If there exists a division D such that GLγ(D), then we say that G is linked with γ, notation GLγ.

(1.7) Remark. If S0,,Sk is a sequence of elements of 𝒮 and γ a curve and D a division of [0,1] such that condition (i) of (1.6) is satisfied, then it follows that γ(ti)Si-1Si, so γ(ti)Ki for some component Ki of Si-1Si. Of course then GLγ(D), where G= S0 K1 S1 Kk Sk.

Proof of (1.5).

First the surjectivity of φ. Let γ:[0,1]X be a continuous curve from aS to aT. The collection {γ-1(S)|S𝒮} forms an open cover of [0,1], so by compactness of [0,1] there is a division D:0=t0t1tk+1=1 such that [ti,ti+1] is contained in γ-1(Si) for some Si𝒮. As remarked in (1.7) we now know that there is a pregallery G= S0 K1 S1 Kk Sk such that GLγ. The surjectivity of φ follows now from the

Claim. If GLγ then φ(G)={γ}.


Let GLγ and suppose G,γ and D as above. Put diγ(ti) for i=1,,k+1 and choose for i=1,,k a curve γi in Si-1 from aSi-1 to di and a curve γ1 in Si from di to aSi. Furthermore γ0 is the constant curve on aS0d0 and γk+1 is the constant curve on aSkdk+1. Let γiγ[ti,ti+1] (see notational remark 3). Now for i=0,,k, γi as well as γiγi+1 are curves in Si from di to di+1, so they are homotopic. So γγ0 γ1 γk γ0γ1 γ1γ2 γk γkγk+1 i=1k γiγi. So φ(G)= { i=1k γiγi } ={γ}.

S0 S1 S2 aS0=γ(t0) aS1 aS2=γ(t3) γ(t1) γ(t2) γ1 γ2 γ1 γ2 Figure 2.

Now we prove: GRGφ(G)=φ(G). It is sufficient to show that φ(SKULT) =φ(SMT) for each pair from R i.e. KLM. Choose then a point dKLM. Let γ,γ,γ be curves from aS, aU and aT respectively to d inside S,U and T respectively. Then φ(SKULT)= { γ(γ)-1 γ(γ)-1 } = { γ(γ)-1 } =φ(SMT).

d S T U aS aT aU γ γ γ Figure 3.

We also have to remark that φ(SSS)=φ(S)1, but this is true by the simply connectedness of S𝒮.

We prove the converse: φ(G)= φ(G) GRG, in four steps:

Step 1. If G,GPgalAB and G,GLγ(D), then GRG.

A=S0 S1 S2 S3 S4=B A=T0 T1 T2 T3 T4=B Figure 4.

Let G,γ and D be as before and let G be the pregallery G= T0L1 T1 Lk Tk. We have T0=S0=A and Tk=Sk=B. For i=0,,k we know γ([ti,ti+1])SiTi. Let Wi be the component of SiTi such that γ([ti,ti+1])Wi. Furthermore we know that γ(ti+1)SiTi+1, so let Wi+1 be the component of SiTi+1 such that γ(ti+1)Wi+1. Now define for i=1,,k:

Gi S0 K1 S1 Si-1 Wi Ti Li+1 Ti+1 Lk Tk Gi S0 K1 S1 Si-1 Ki Si Wi Ti Li+1 Ti+1 Lk Tk So: Gi S0 K1 S1 Si-1 Ki Si Wi+1 Ti+1 Lk Tk

where you get this last line by substituting i+1 for i in the first line for i=0,,k-1. Now γ(ti) WiKi Wi, so GiRGi and also γ(ti+1) Wi+1 WiLi+1, so GiR Gi+1, so GiRGi+1 for i=1,,k-1. So G1RGk. Now G1= S0 W1 T1 L2 Lk Tk, but because S0=T0 we have W1=L1, so G1=G. Analogously Gk=G. So GRG.

Step 2. If GLγ(D) and D is a refinement of D, then there exists a G, such that GLγ(D) and GRG.


Let again G, γ and D be as before. Of course it is sufficient to prove this for the case that D has one more division point than D, so D= { 0=t0 tisti+1 tk+1=1 } . Now we can take simply G= S0 K1 S1 Si Si Si Ki+1 Si+1 Kk Sk.

Step 3. If G,GLγ, then GRG.


Let GLγ(D) and GLγ(D). Denote by D the common refinement of D and D. According to step 2 there are G1 and G2 such that GiLγ(D) for i=1,2 and G1RG and G2RG. But step 1 says that also G1RG2 so GRG.

Step 4. Let G,GPgalAB. If GLγ,GLγ, and γγ, then GRG.

aA aB γ γ γ γj+1 (i,j) Ii,j Figure 5.

Since γγ, there is a homotopy F:[0,1]×[0,1]X such that F(t,0)= γ(t), F(t,1)= γ(t), F(0,s)= aA and F(1,s)= aB. The collection {F-1(S)|S𝒮} is an open cover of [0,1]2 and so by compactness of the latter there is a division D={0=t0t1tk+1=1} such that for each pair (i,j){0,,k}2 there is a Si,j𝒮 such that F(Ii,j)Si,j where Ii,j[ti,ti+1]×[tj,tj+1]. For j=0,,k+1 define γj(t)F(t,tj). Now fix an index j. F[(ti,tj),(ti,tj+1)] (see notational remark (0.3)) is a curve in Si-1,jSi,j for i=1,,k, so F(ti,tj) and F(ti,tj+1) lie in the same component of Si-1,jSi,j; call this component Ki. From F(Ii,j)Si,j it follows that γj([ti,ti+1])Si,j for i=0,,k and also γj+1([ti,ti+1])Si,j. We already saw γj(ti), γj+1(ti)Ki for i=1,,k. Then it follows from remark (1.7) that GjLγiLγj+1, where Gj S0,j K1 S1,j K2 S2,j Sk-1,j Kk Sk,j. So for all j=0,,k we have GjLγj+1 and Gj+1Lγj+1 so by step 3 we have GjRGj+1. Moreover γ0=γ,GLγ and G0Lγ0, so GRG0 (step 3). Analogously Gk+1RG. So GRG0RG1RG2RRGk+1RG. So GRG.

To finish the proof of theorem (1.5), suppose φ(G)=φ(G). For the defining curves γ and γ (i.e. φ(G)={γ} and φ(G)={γ}) certainly GLγ and GLγ, holds. And from φ(G)=φ(G) it follows that γγ, so by step 3 (Suggested correction: 4) we have GRG.

§2. Hyperplane Systems

Let 𝕍 be a real finite dimensional vector space.

(2.1) If M is a hyperplane and the set A𝕍 is contained in one of the two open halfspaces determined by M (i.e. components of 𝕍-M), then we denote this halfspace by DM(A). For singletons {x} we also write DM(x).

Let H𝕍 be an open set (so "open in H" and "open in 𝕍" is the same thing) and a collection of hyperplanes of 𝕍 (not necessarily containing the origin), which is locally finite on H. Recall that this means, chat each xH has a neighbourhood UH such that UM for only finitely many M.

(2.2) Proposition. Let 𝕍,H and be as above. Then H-MM is open.


Let xH-MM. There is a neighbourhood U of x such that {M|UM} is finite. Then UMDM(x) is a neighbourhood of x inside H-MM.

(2.3) Corollary. The components of H-MM are open.

(2.4) Definition. These components are called chambers in H with respect to . We denote the collection of chambers by K(,H) or by K() if no confusion is caused by the ambiguity.

If AK(), then DM(A) is defined for all M. Let B also be a chamber. If M, then there are two possibilities:

  1. DM(A)DM(B)= in which case we say that M separates A and B or
  2. DM(A)=DM(B).

(2.5) Definition. (A,B) { M|M separatesAandB } .

After these preliminaries we come to the main definitions of this paragraph.

(2.6) Definition. Let 𝕍,H and be as in (2.1). If, moreover, H is convex then the triple (𝕍,H,) is called a hyperplane configuration.

(2.7) Definition. Let (𝕍,H,) be a hyperplane configuration and let MppM be given, where pM is a locally finite collection of hyperplanes in 𝕍. Then we call the quadruple (𝕍,H,,p) a hyperplane system. Let us summarize the defining properties of a hyperplane system (𝕍,H,,p):


(i) 𝕍 is a finite dimensional real vector space.
(ii) H is an open convex subset of 𝕍
(iii) is a collection of hyperplanes of 𝕍 locally finite on H.
(iv) p is an assignment MpM, where pM is a locally finite collection of hyperplanes of 𝕍.

We associate a space Y(𝕍,H,,p) (or briefly Y) to a hyperplane system (𝕍,H,,p) as follows:

(2.9) Definition. Y { x+-1y𝕍 |yHand (xNpMyM) } .

In the sequel the complex structure of 𝕍 is not relevant. We merely use the notations 𝕍 for 𝕍×𝕍 and Re and Im for the projections as it is convenient. Note that we can assume that if M then MH and pM, for otherwise M doesn't occur in the definition of the space Y. Remark also that {N+-1M|M,NpM} is locally finite on 𝕍+-1H=Im-1(H), hence Y is open in 𝕍.

(2.10) Definition. Together with each hyperplane configuration (𝕍,H,) there is a canonical simple hyperplane system defined, also denoted by the triple (𝕍,H,), by setting pM{M}. For instance, if the origin of 𝕍 is contained in all M, then Y can be considered as a complement of complex hyperplanes: Y=Im-1(H)- MM.

The theory we are going to develop has two main applications

  1. The case of simple hyperplane systems.
  2. Generalized root systems.

In the latter case, the collection pM consists of hyperplanes all parallel to M, hence the use of the letter p.

Now we construct a cover of Y such that the result of §1 is applicable.

(2.11) Definition. For CK() we define:

YC { x+-1y|yH andxNpM yDM(C) } .

Note that YCY.

(2.12) Proposition. YC is open in 𝕍.


Let x+-1yYC. Choose cC and put M={M|M[y,c]=}. Due to the compactness of [y,c] and locally finiteness of on H, - is finite. Then pM-pM is locally finite on 𝕍, since the finite union of a locally finite collection is again locally finite. If xNpM, then yDM(C). Thus [y,c]M=, so M, which means Np. Therefore x lies in a chamber UK(p). Choose also a chamber VK(,H) such that [y,c]V. This is possible, because [y,c] is connected and for any M we have [y,c]MM. Now we have U+-1VYC, for if UN and NpM, then Np, so MM which implies that VDM(V)=DM(c)=DM(C). So U+-1V is a neighbourhood of x+-1y inside YC.

(2.13) Proposition. Y=CK()YC.


Let x+-1yY. Choose an open neighbourhood U of y such that {M|UM} is finite. By deleting from UyMM (which is closed because of the finiteness of ) we see that we can suppose that ={M|yM}. Moreover we may assume that U is connected. Then DM(U) is defined for all M and DM(U)=DM(y). Because the interior of MM is empty there has to be a CK() such that UC. For this C we have DM(C)=DM(U) for all M. Now let xNpM. Then yM, so M. Then yDM(y)=DM(U)=DM(C). So we have proved that x+-1y𝕍.

(2.14) Corollary (from 2.12) and (2.13)). The space Y is open in 𝕍 and {YC|CK()} is an open cover of Y.

In order to prove that YC is simply connected (even contractible) and to study the intersections we consider now the space XYC1YC2YCn for certain C1,,CnK(). First observe:

X= { x+-1y|y HandxNpM yi=1n DM(Ci) } .

In particular, if x+-1yX and xNpM then i=1nDM(Ci). The latter is equivalent to DM(C1)= DM(C2)== DM(Cn) and this means that M(Ci,Cj) for ij. So if we give the following:

(2.15) Definition. Let A,BK(). p(A,B)M(A,B)pM and

p(C1,,Cn) ijp (Ci,Cj).

($) Then we have Np(C1,,Cn). Now note first that p(C1,,Cn) is locally finite on 𝕍, for we have the following:

(2.16) Lemma. If A,BK(), then |(A,B)|<.


By compactness of S[a,b], where aA and bB, and locally finiteness of on H, we see that {M|MS} is finite. But it is easy to see that =(A,B).

From ($) we see that the projection Re:𝕍𝕍 maps the set X into the union of chambers with respect to p(C1,,Cn). In fact we have the following:

(2.17) Proposition. The projection Re:𝕍𝕍 maps each connected component of X onto a chamber w.r.t. p(C1,,Cn). This sets up a bijective correspondence between the connected components of X and K(p(C1,,Cn)). Furthermore, these components are contractible.


The projection Re maps each component of X into a component of the union of the chambers w.r.t. p(C1,,Cn) i.e. into a chamber. Choose bC1, then DM(b)= DM(C1)== DM(Cn) for each M(C1,,Cn)ij(Ci,Cj). Then for each UK(p(C1,,Cn)), U+-1bX holds. So ($$) U+-1bXRe-1(U) and the latter is nonempty. We want to show that it is contractible. If u+-1yXRe-1(U); then [u+-1y,u+-1b] XRe-1(U). For if uNpM, then yDM(Ci) for i=1,,n and so also yDM(b) for M(C1,,Cn). So we have [y,b]DM(b), i.e. u+-1[y,b]XRe-1(U). From this we can see that U+-1b is a deformation retract of XRe-1(U). U+-1b itself is contractible (for so is U), so XRe-1(U) is indeed contractible. Hence for each UK(p(C1,,Cn)) there is one and only one connected, even contractible, component of X, namely XRe-1(U), that is mapped onto (this follows from ($$)) U by the projection Re.

(2.18) Corollary (n=1). If CK() then YC is contractible, in particular simply connected.

(2.19) Corollary (n=2). If A,BK() then the components of YAYB are in a bijective correspondence with the elements of K(p(A,B)). This correspondence is given by K(p(A,B)) UYAYB Re-1(U).

Denote such a component YAYBRe-1(U) for the moment by Co(U). From (2.18) it follows that we have the situation of (1.2). A pregallery is now given by a sequence C0,C1,,Cn,CiK(), together with (by (2.19)) a sequence U1,,Un, where UiK(p(Ci-1,Ci)). This pregallery YC0 Co(U1) YC1 Co(Un) YCn is denoted by C0 U1 C1 Un Cn and from now on all the pregalleries we are considering, are of this type. We need one more corollary of (2.17):

(2.20) Corollary (n=3). Let A,L,BK() and UK(p(A,L)), VK(p(L,B)) and WK(p(A,B)). Then UVWCo(U)Co(V)Co(W).


() is easy. For () choose xUVW. Then for all hyperplanes Np(A,L,B) we have xN, so there is a OK(p(A,L,B)) with xO. According to proposition (2.17) the component YAYBYLRe-1(O) is mapped onto O by Re, so it contains a d with Re(d)=x. Now we have dCo(U)Co(V)Co(W).

Combining the results of §1 and §2 we now get the main result of this section:

(2.21) Theorem. Let (𝕍,H,,p) be a hyperplane system. Let Pgal be the free category on the direct pregalleries AUB where A,BK(), UK(p(A,B)), beg(AUB)A and end(AUB)B. Then there is a surjective functor φ:PgalΠ, where Π is the subgroupoid of the fundamental groupoid of Y(𝕍,H,,p), consisting of all homotopy classes of paths from and to points of the form aC where in each YC there is chosen a point aC.

Two pregalleries have the same φ-image if and only if they are equivalent by means of the relations

R { ( AU LV B, AW B ) Pgal×Pgal |UVW } { ( A𝕍A, A ) |AK() } .


This is a restatement of theorem (1.5) for the case of hyperplane systems. By (2.20) the equivalence relations are indeed the same.

§3. Tits Galleries

Given a path in the space Y (defined in §2), then by a small homotopy we can make the imaginary part avoid intersections of distinct hyperplanes of . For the set of such points has codimension bigger then one. This leads to the idea, that it is always possible to choose the sequence C1,,Cn of chambers of a pregallerv in such a way, that Ci-1 and Ci are separated by only one hyperplane. Such a sequence we will call a Tits gallery. Tits himself and Deligne call this a gallery, but we prefer to use this word for the pregalleries that arise this way. Much of this paragraph can be found in Bourbaki V §1 [Bou1968] and in Deiigne [Del1972]. But our situation is slighty more general, because we consider only an open part H of 𝕍. Let (𝕍,H,) be a hyperplane configuration as defined in (2.6).

(3.1) Definition. Two points of H are said to be on the same facet, if for each M there are only two possibilities: (i) Both points are contained in M. (ii) The points are not contained in M and they are moreover on the same side of M i.e. DM(x)=DM(y), if x and y are these two points. This is an equivalence relation and the classes are called facets of H with respect to . The collection of facets is denoted by F(,H) or for short F().

For each FH (in particular for FF()) we define: F{M|MF} and LFMF the support of F. Furthermore (co)dim(F) (co)dim(LF).

From the definition it is clear that for FF() we have

(3.2) F=LF D-F(F) H

(In general, if A𝕍 is on one side of M, for instance if A is convex and AM=, for all M, then D(A) M DM(A)).

(3.3) Proposition. FH=LF M-F ( DM(F) H ) .

Remark. FH is just the closure of F as subset of H. We denote this in the sequel by F.

Proof of (3.3).

From (3.2) it follows at once that the left hand side is contained in the right hand side. Conversely, let y be contained in the right hand side and choose xF, then [x,y]LF and [x,y)DM(F) for all M-F. So [x,y)F. But that means that yF.

(3.4) Corollary. The closure (in H) of a facet is a (disjoint) union of facets.


For LFH and each DM(F)H are so.

We also have a useful criterion to see if a facet is part of this union:

(3.5) Criterion. Let F,GF(). Then FF if and only if for all M FMGDM(F) holds.


Let FG and FM, then FM, for if GM, then GM, and so FM, a contradiction. So G is contained in one of both open half spaces of M. Now we have FGDM(G), so FDM(G), which also means GDM(F). Conversely, suppose that the condition holds. Then F is contained in LG M-G ( DM(G) H ) and so by (3.3) also in G, which is to be proved). For if GM, then FM, by contraposition of the condition, so FLG. If GM then two possibilities arise: (1) FM and then certainly FDM(G). (2) FM and then by the condition GDM(F), so FDM(G)DM(G).

(3.6) Remark. K()= { FF()| codim(F)=0 } .

(3.7) Definition. Let CK() and FF() with FC.
If codim(F)=1, then F is called a face of C.
If codim(F)=2, then F is called a plinth of C.
If M is support of a face of C, M is called a wall of C.
MC { M|M is a wall ofC } .

Let also a configuration (𝕍,H,) be given and let and HH. If FF(,H), then there is one and only one FF(,H) such that FF. For if xF, then xF for some FF(,H) and if y is on the same facet w.r.t. and H (definition(3.1)), then x and y are also on the same facet w.r.t. and H so then FF. Uniqueness follows from the fact that F(,H) is a partition of H.

(3.8) Definition. The map F(,H) F(,H) obtained above is denoted by pr(,HH) or if H=H, then pr(). For FF() we define: prF pr (F,𝕍H) :F(,H) F(F,𝕍).

(3.9) Theorem. (cf. [Del1972] Deligne 1.5(ii)).
The map prF restricted to {EF()|FE} is bijective.


Injectivity. Suppose FEi (i=1,2) and prF(E1)=prF(E2). For MF we have EiM iff prF(Ei)M; So E1M iff E2M. And if E1M, then E1 DM(prF(E1)) = DM(prF(E2)) = DM(E2), so then E1 and E2 on the same side of M.

If MF, then FM and by criterion (3.5) and the fact that FEi for i=1,2 we see that EiDM(F). So also in this case E1 and E2 on the same side of M. We have seen that for all M, EiM for i=1,2 or E1 and E2 on the same side of M, holds, so E1=E2.

Surjectivity. Let GF(F,V), Then G=LGGMFDM(G). It follows that LFG, so each neighbourhood of a point fF has points in common with G. Choose a neighbourhood U of f such that UDM(F)H for all MF and choose a point uUG. Let EF(,H) be the facet such that uE. Then FE and prF(E)=G (what was to be proved). The latter is clear from EG (uEG). We prove the fact that FE holds, by criterion (3.5): Suppose FM, because UDM(F) we have uDM(F), so EDM(F).

G G G U F u f Figure 6.

(3.10) Remark. pr(,HH) (K(,H))K(,H) and in particular (by (3.9)) prF(K())=K(F).

If H=𝕍 and MM, then it doesn't harm to suppose that the origin 0𝕍 is contained in this intersection. This way we can define in this case for each EF(): -E {x𝕍|-xE} F().

(3.11) Definition. Let E,FF(,H) and EF. Then we define the reflection of the facet E in the facet F by: spF(E) prF-1 (-prF(E)). Remark again spF(E)K() if EK().

(3.12) Lemma. If CK(), then MC iff CD-{M}(C).


See Bourbaki [Bou1968] V §1.4.

(3.13) Theorem. If CK(), then C=DC(C).


See Bourbaki [Bou1968] V §1.4 prop.9.

(These proofs also work in our (more general) situation)

(3.14) Corollary. If A,BK() and (A,B), then A(A,B).


If A(A,B)=, then BDM(A) for all MA, so B MA DM(A) =DA(A)= A, so A=B conflicting (A,B).

We need some propositions about separating hvperplanes.

(3.15) Proposition. If A,L,BK(), then

(i) (A,B)= (A,L)Δ (L,B) (Δ: symmetric difference of sets)
(ii) |(A,B)|= |(A,L)|+ |(L,B)|-2 (A,L) (L,B)


(i) For any M consider DM(A), DM(B) and DM(L). Three cases:
1. DM(A)= DM(B)= DM(L) then M an element of neither side of (i)
2. DM(A)= DM(B) DM(L) then also M an element of neither side of (i)
3. DM(A)= DM(L) DM(B) then M an element of both sides of (i)
(ii) follows from (i).

(3.16) Proposition. If A,B,LK(), then the following are equivalent.

(i) (A,L) (L,B)=
(ii) (A,B)= (A,L) (L,B)
(iii) |(A,B)|= |(A,L)|+ |(L,B)|


(i) (ii) follows from (3.15)(i). (i) and (ii) (iii) is obvious. (iii) (i) follows from (3.15)(ii).

(3.17) Proposition. If C0,,Cn is a sequence of elements of K() then

(i) (C0,Cn) i=1n (Ci-1,Ci)
(ii) Equivalent are:
  1. (Ci-1,Ci) (Cj-1,Cj) = for ij
  2. |(C0,Cn)|= i=1n |(Ci,1,Ci)|
and in this case we have equality in (i) and if we have a hyperplane system: p(C0,Cn)= i=1n p(Ci-1,Ci) .

Remark. For n3 the implication (C0,Cn)= i=1n (Ci-1,Ci) (ii)a. need not hold (it does hold for n=2 by proposition (3.16) (ii) (i)), consider for instance the configuration of figure 7.

C0 C1 C2 C3 Figure. 7

Proof of (3.17).

Trivial for n=1. Suppose proved for n-1. Then by (3.15) (i) we have (C0,Cn) (C0,Cn-1) (Cn-1,Cn) and by induction we know that the latter is contained in i=1n-1 (Ci-1,Ci) (Cn-1,Cn). This proves (i) for n also. We always have:
($) |M(C0,Cn)| i=1n (Ci-1,Ci) i=1n |(Ci-1,Ci)|
The first inequality follows from the already proved fact (i) and the second is obvious. Furthermore the last inequality is an equality iff (ii)a. holds. This proves the implication b. a. Suppose now that a. holds. If we can show now that
($$) (C0,Cn)= i=1n (Ci-1,Ci)
then we are done, for then (besides the second) the first inequality of ($) is also an equality, so b. is verified then. Now by induction we have:
($$$) (C0,Cn-1)= i=1n-1 (Ci-1,Ci)
so also (C0,Cn-1) (Cn-1,Cn)= ( i=1n-1 (Ci-1,Ci) ) (Cn-1,Cn) and by a. the latter is empty, so it follows from (3.16) (i) (ii) that (C0,Cn)= (C0,Cn-1) (Cn-1,Cn) and this together with ($$$) gives the desired result ($$).

(3.18) Definition. We call a sequence C0,,Cn of chambers a Tits gallery from C0 to Cn if |(Ci-1,Ci)|=1 for i=1,,n. The number n is called the length of the Tits gallery. The only element of (Ci-1,Ci) is often denoted by Mi.

(3.19) Remark. This Tits gallery is completely determined by its first chamber C0 and the sequence M1,,Mn.


Let A,BK() and (A,B)={M}. Then ABD-{M}(A) and so A and B are both unequal to the latter. By lemma (3.12) this means that MAB. Then M is the support of a common face F of A and B. So B is determined by A and M by the rule B=spF(A). The remark follows now.

(3.20) Corollary of (3.17). For Tits gallery C0,,Cn are equivalent:

  1. MiMj for ij
  2. |(C0,Cn)| =n

and in that case we have (C0,Cn)= {M1,,Mn}.

(3.21) Definition If the conditions a. and b. of (3.20) are fulfilled then we call such a Tits gallery a direct Tits gallery. (Deligne uses the word minimal but we want to use that word for other things)

(3.22) Theorem (Deligne [Del1972 1.3]). If A,BK(), then there exists a direct Tits gallery from A to B.


Induction on |(A,B)|. If (A,B)=, then A=B, then then trivial. Now let |(A,B)|>0 and let the theorem be proved in cases where the number of separating hyperplanes is less. By corollary (3.14) we know that (A,B)A. Let M(A,B)A and let F be the face of A supported by M. Set CspF(A). Then we have by (3.15)(i) (C,B)= (C,A)Δ (A,B)= (A,B)- {M}, since (C,A)= {M} (A,B). By induction there is a direct Tits gallery C=C1,,Cn=B from C to B. Then A,C1,,Cn is a direct Tits gallery from A to B.

(3.23) Theorem. Let A,BK(). If B=spF(A) for some plinth F of A then there are exactly two direct Tits galleries from A to B.


First we remark that, if C and C are chambers with a common plinth P, then (C,C)P. For let MP, so PM. Also PCC, so by criterion (3.5) both C and C are contained in DM(P), hence M(C,C). It follows that C0,,Cn where CiP is a (direct) Tits gallery iff prp(C0),,prp(Cn) is a (direct) Tits gallery. The theorem follows from

(3.24) Lemma. If 0LMM and codim(L)=2, then there are exactly two direct Tits galleries from a chamber A to the chamber -A.


The quotient space 𝕍/L has dimension two, so essentially the situation is now two-dimensional and the lemma is clear from a picture (Although an exact proof is rather length; it uses for instance remark (3.19)).

A -A Figure 8.

(3.25) Definition. The equivalence relation in the set (or better: the free category) of all Tits galleries, generated by the pairs PL{(G,G)|G and G are the two direct Tits galleries from A to spP(A), where P is a plinth of some chamber A} is called plinth equivalence and denoted PL.

(3.26) Theorem. Let A,BK() and suppose that G and G are the two direct Tits galleries from A to B. Then GPLG.

Remark. Both theorem and proof are essentially due to Deligne [Del1972] 1.12. We don't want to use his condition that the chambers are simplicial cones. Therefore we need the simple but technically somewhat complicated lemma (3.28). For the proof we first give a definition:

(3.27) Definition. If F1 and F2 are two faces of a chamber A, then we call F1 and F2 plinthed if there exists a plinth P of A such that PF1F2. The supporting walls are also said to be plinthed w.r.t. A.

Proof of (3.26).

If |(A,B)|=0, then there is nothing to prove. Else, let G be the direct Tits gallery A=C0,C1,,Ck=B and G:A=D0,D1,,Dk=B. Suppose (C0,C1)={M} and (D0,D1)={M}.

Case 1. If M=M, then C1=D1 (see remark (3.19)) and then C1,,Ck and D1,,Dk are two direct Tits galleries from D1=C1 to B=Ck=Dk. By induction they are plinth equivalent. But then also GPLG.

Case 2. Let MM and M and M are plinthed walls of A by a plinth P. Let G1 be the unique direct Tits gallery from A to spP(A) that starts with C0,C1 and G2 the one that starts with D0,D1. Let E be a direct Tits gallery from spP(A) to B. By (3.15)(i) (spP(A),B)= (A,B)ΔP and the latter equals (A,B)=P since P(A,B). So by (3.20) G1E and G2E are direct Tits galleries from A to B. Because G1E and G (respectively G2E and G) start with C0,C1 (respectively D0,D1) by step 1 G1EPLG (respectively G2EPLG) holds. It is clear that G1EPLG2E, so GPLG.

Case 3. General case. By lemma (3.28) (proved in the sequel) there is a sequence M=M0,M1,,Mm=M such that Mi-1 and Mi are plinthed walls of A for i=1,,m. For i=1,,m-1 there exists a direct Tits gallery Gi from A to B starting with A,spMi(A). Let G0G and GmG. By step 2 we have Gi-1PLGi for i=1,,m. So GPLG. There remains:

(3.28) Lemma. If M,MA(A,B), where A,BK(), then there is a sequence M=M0,Mm=M such that Mi-1 and Mi are plinthed walls of A for i=1,,m and Mi(A,B).

A B M M a b x a Figure 9.


Let F and F be the faces of A with M and M as support respectively and choose points aF and aF. Choose also bB. By choosing a,a and b in general position we may assume that the affine hyperplane spanned by them intersects transversaly each intersection of hyperplanes of (A,B) inside the compact triangle Δ(a,a,b) By compactness and locally finiteness Δ(a,a,b) meets only a finite number of facets and these have codimension less than or equal to 2. Furthermore for each x[a,a] [b,x]A is of the form [ax,x] for some unique axA (A is convex). It follows that [a,a]x axA Δ(a,a,b) is a piecewise linear arc passing only faces and plinths of A.

Theorem (3.26) is proved.

(3.29) Corollary of (3.22) and (3.26). Let CK() be fixed. Then Gend(G) induces a bijection between the plinth equivalence classes of direct Tits galleries G with beg(G)=C and K().

§4. Galleries

In this section we combine the results of §2 and §3 to get the main result of this chapter. Let (𝕍,H,,p) be again a hyperplane system as in §2.

(4.1) Definition. We call a pregallery C0 U1 C1Un Cn a gallery if C0,,Cn is a Tits gallery. The number n is called the length. The subcategory of Pgal we obtain is denoted Gal. For GGal we denote the length by |G|.

(4.2) Proposition. Let A=C0,,Cn=B be a direct Tits gallery. Given WK(p(A,B)) denote by Ui the unique element of K(p(Ci-1,Ci)) which contains W. Then AWBR C0U1 C1Un Cn. Conversely, if UiK(p(Ci-1,Ci)) for i=1,,n and Wi=1nUi, Then WK(p(A,B)) and AWBR C0U1 UnCn.


(In fact the condition that C0,,Cn is a direct Tits gallery, is too strong. It is enough to require that condition (ii)a. of proposition (3.17) holds: (Ci-1,Ci) (Cj-1,Cj)= , if ij). Let WK(p(A,B)), define Wipr(p(Ci,Cn)p(A,B))(W) and let Ui be as in the proposition i.e. Ui pr ( p(Ci-1,Ci) p(A,B) ) (W) . Then WWi and WUi so we have Wi-1UiWi . So

AWS= C0WCn RC0U1 C1W1 CnR C0U1 C1U2 C2W2 CnRR C0U1 UnCn.

Second part: i=1nUi is convex, so i=1nUiW for some WK(p(A,B)). if xi=1nUi and xNMpM then Np(Ci-1,Ci) for i=1,,n. Then also Np(A,B) by proposition (3.17). On the other hand we have WUi for i=1,,n, because W is convex and if xW and xN, then Np(Ci-1,Ci). So W=i=1nUi. Moreover Ui= pr ( p(Ci-1,Ci) p(A,B) ) (W) and so as in the first part we conclude the equivalence of the pregalleries.

(4.3) Definition. Such a gallery C0U1 C1Un Cn with U1U2Un and C0,,Cn a direct Tits gallery we call a direct gallery.

(4.4) Corollary. Every direct gallery is pregallery-equivalent to a unique direct pregallery. Each direct pregallery is pregallery-equivalent to a direct gallery (which is in general not unique).


Clear from proposition (4.2). For the second statement we also need theorem (3.22).

In this case we call the gallery a direct repacing gallery of the given direct pregallery.

(4.5) Definition. Let AK() and B=spP(A) for some plinth P of A. Suppose WK(p(A,B)). According to theorem (3.23) there are exactly two direct Tits galleries from A to B. Let G and G be the two direct replacing galleries belonging to AWB by (4.2). Then we call the pair (G,G) a flip relation. The equivalence relation in Gal generated by these flip relations is called flip equivalence and denoted by .

(4.6) Corollary of proposition (4.2) and theorem (3.26): Let G and G be two direct replacing galleries of a direct pregallery. Then GG.

(4.7) Definition. Pairs of type ( AU ABA, A ) Gal×Gal we call cancel relations. The equivalence relation in Gal generated by the flip relations and the cancel relations is called gallery equivalence or simply equivalence and is denoted by (note that , see (5.5)).

(4.8) Proposition. If G,GGalAB, then GRGGG.

For the proof we need:

(4.9) Lemma. Let R { ( AU LV B, AW B ) R| |(L,B)| 1 } { ( A𝕍A, A ) |AK() } . Then for G,GPgal GRGGRG. (This means that if two pregalleries are pregallery-equivalent, then this equivalence can be realised by a sharpened sort of pregallery-equivalence relations, i.e. one can suppose that |(L,B)|1).


The () part is trivial. For the () part it is sufficient to show that AU LV B R AW B if UBW. We proceed by induction on |(L,B)|. If |(L,B)|1, then we are done. Let |(L,B)|>1. Choose a MB(L,B) (exists by (3.14)) and define LspM(B). Then |(L,B)|=|{M}|=1. Put:

V pr ( p(L,L) p(L,B) ) (V)V V pr ( p(L,B) p(L,B) ) (V)V

Let xUVW and suppose xNMpM. Then Np(A,L) (for xUK(p(A,L))) and Np(L,B) (for xVK(p(L,B))), so Np(L,L) (p(L,B)). It follows that Np(A,L)= p(A,L)Δ p(L,L). So xW for some WK(p(A,L)). So now we have AU LVB R AU LV LV B for VVV and |(L,B)|=1. Since |(L,L)|= |(L,B)-{M}|= |(L,B)|-1 the induction hypothesis says that AU LV L R AW L, for xUVW. Finally AW LV B R AW B for xWVW and |(L,B)|=1.

Proof of proposition (4.8).

Sufficient for the () part is: 1. AUBUA and A are pregallery-equivalent. Well, UU𝕍, so AUBUA RA𝕍AR A. 2. If G and G are as in the definition of flip relation (4.5), then GRG. The () part: Let G,GGalAB and GRG. To prove: GG. Let G=G0,G1,,Gn=G be a sequence of pregalleries such that we obtain Gi from Gi-1 by a pregallery-equivalence operation. By lemma (4.9) we may assume that these operations are in the there defined sharpened form. Choose now each Gi a gallery Gi in the following way: substitute for each direct piece a replacing direct gallery (4.4). Then automatically G0=G0=G and Gn=Gn=G, because these were already galleries.
Claim. Gi-1Gi. Sufficient for this is: Let (AULVB,AWB)R as in (4.9). If G (respectively G) is a replacing direct gallery of AUL (respectively AWB), then

(£) G.LVBG. This suffices, because all other replacing direct galleries are (even flip) equivalent by (4.6). For the proof of (£) let (L,B)={M} and consider two cases: (1) M(A,L). In this case G.LVB is direct because UV and so we have even G.LVBG. (2) M(A,L). Then G.BVL is a replacing direct gallery of AUL. So GG.BVL and we have G.LVBG.BVLVBG.

Summarising we have the main result of this chapter:

(4.10) Theorem. Let (𝕍,H,,P) be a hyperplane system. Let Gal be the category of all galleries i.e. the free category generated by the so-called one step galleries AUB where A and B are chambers separated by one common wall M, UK(pM), beg(AUB)A and end(AUB)B. Let in each CK() a point aC be chosen and let Π be the subgroupoid of the fundamental groupoid of this space Y(𝕍,H,,p) (2.9), where ΠAB for A,BK() consists of the homotopy classes of paths from -1aA to -1aB.

Then there exists a surjective functor φ:GalΠ. Moreover the equivalence relation on Gal "having the same φ-image" is generated by the

cancel relations: ( AU BUA, A ) Gal×Gal (4.7) flip relations: ( G,G ) Gal×Gal (4.5)

where G and G are the two direct replacing galleries of a direct pregallery AW spP(A), AK(), P plinth of A and WK(p(A,spP(A))).


The required functor is the restriction to Gal of the functor of theorem (2.21). This restriction remains surjective, because each pregallery is pregallery-equivalent to some gallery by (4.4). The assertion about the relations follows from the corresponding one in (2.21) and proposition (4.8).

This theorem describes the fundamental groupoid of the space Y(𝕍,H,,p) as a combinatorial object, namely the quotient category Gal/ which is even more related to the structure of the hyperplane system than Pgal/R (cf. (1.5) and (2.21)).

§5. Simple Hyperplane Systems and Signed Galleries

In the case of a simple hyperplane system we have pM={M} for all MK(). So if AUB is a one-step gallery with (A,B)={M}, then we have two possibilities: U=DMi(A) or U=DM(B). In the first case the one-step gallery is called positive and we denote it by A+B, otherwise it is called negative and denoted A-B. Therefore an arbitrary gallery is now of the form

C0t1 C1t2 tk Ck

where C0,C1,,Ck is a Tits gallery and ti{+,-}. For this reason we will call them sometimes signed galleries. Such a signed gallery is called positive (respectively negative), if ti=+ (respectively ti=-) for i=1,,k. There is a bijective correspondence between the Tits galleries and the positive galleries.

(5.1) Proposition. A positive gallery is direct if and only if the underlying Tits gallery is so.


To see the if-part of this statement, note that if C0,C1,,Ck is a direct Tits gallery and (Ci-1,Ci)={Mi}, then Mi (C0,Ci-1)= {M1,,Mi-1}, for all the Mi are different (def.(3.21)). So C0DMi(Ci-1) and thus DM1(C0) DM2(C2) DMk(Ck-1) . Hence C0+ C1+ Ck+ is direct as a gallery (def. (4.3)).

Corresponding to the two possibilities of UK((A,B)) the cancel relations are now the pairs of the type:

(5.2) ( A+ B-A, A ) or ( A- B+A, A ) forA,B K(), |(A,B)|=1.

The flip relations are the pairs:

(5.3) ( C0t1 C1 tk Ck, D0tk D1 t1 Dk )

where C0,C1,,Ck and D0,D1,,Dk are the two direct Tits galleries from C0=D0 to Ck=Dk=spP(C0) for some plinth P of C0 and in the sequence of signs t1,,tk there is at most one change of sign.

Proof of this statement.

If (5.3) is a plinth relation, then there is a UK((C0,Ck)) =K(P)= { C0,, Ck, D1,, Dk-1 } , where Ci=prP(Ci) and Di=prP(Di), that "prescribes signs" i.e. such that both galleries are replacing direct galleries of the pregallery C0UCk. Suppose for instance that U=Ci. Then t1=t2==ti=- and ti+1==tk=+, hence at most one change of sign (no change if i=0 or i=k). Conversely, if there is at most one change in sign, for instance ti=- and ti+1=+, then U=Ci is as above.

Example. To give examples we recall (3.19) that a Tits gallery is given by its first chamber and the sequence of separating walls. A signed gallery can be denoted by this sequence, underlining the elements of corresponding to the negative steps. For instance, if we number the elements of in Figure 10 from 1 to 4 then the galleries of the plinth relation ( C0- C1+ C2+ C3+ C4, D0+ D1+ D2+ D3- D4 ) can be denoted as 1_ 2 3 4 respectively 4 3 2 1_ (assuming that one knows what the first chamber is).

1 1 2 3 3 4 4 P C0=D0 C4=D4 U=C1 C2 C3 D1 D2 D3 Figure 10.

We depict a signed gallery by drawing an arrow for positive one step galleries and an arrow for negative one step galleries. The following sequence illustrates the next remark

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1_234 1_23411_ 1_14321_ 4321_ Figure 11.

(5.4) Remark. Two signed galleries are equivalent iff they are equivalent by means of the cancel relations and the positive (i.e. with t1==tk=+ in (5.3)) flip relations.


Consider an arbitrary flip relatlon (5.3) with for instance t1=t2=ti=- and ti+1==+. Then

C0- C1- Ci+ Ci+1 +Ck

is by means of cancel relations equivalent to

C0- C1- Ci+ Ci+1 + Ck+ Dk-1+ + Dk-i- Dk-i+1- -Dk

and this is by means of a positive flip relation equivalent to

C0- C1- Ci+ Ci-1 + C0+ D1+ + Dk-i- Dk-i+1- -Dk

and this is by means of cancel relations equivalent to

C0+ D1+ + Dk-i- Dk-i+1- -Dk.

(5.5) We conclude this paragraph with an example of two equivalent galleries which are not flip equivalent:

1 2 3 1 2 3 1221_ 3_223 Figure 12.

They are equivalent for

1221_ 123_3 21_ 3_213 21_ 3_21 1_23 3_223.

Notes and References

This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.

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