Chapter IV. The K(π,1)-Problem

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 10 May 2013

§1. Introduction

Let (𝕍,H,,p) be a hyperplane system. In this chapter we consider the question under which circumstances one can prove that the space YY(𝕍,H,,p) is a K(π,1). If this is the case we call the system (𝕍,H,,p) a K(π,1)-system.

(1.1) Definition. A hyperplane system is called simplicial, if the cone on a chamber is a simplicial cone, i.e. of the form i=1kϕi-1((0,)) for some independent set {ϕ1,,ϕk}(𝕍×)*. The cone on a set A𝕍 is defined to be {(v,t)𝕍×|t>0,vtA}. Cf. figure 32 (simplicial) and figure 24 (not simpl.).

The result of Deligne [Del1972] can be stated as (see (3.2) for central):

(1.2) Theorem A simple, central and simplicial hyperplane system is K(π,1).

(1.3) Conjecture. This theorem is still true if one drops the condition that the system has to be central.

After some preliminaries in §2 we construct in §3 an open cover 𝒰^ of the universal covering space Y of Y. To ensure that the abstract simplicial complex Σ^nerf(𝒰^) has simplices with a finite number of vertices we suppose from now on that the configuration satisfies:

(1.4) Assumption. For each facet FF() the number of facets contained in F is finite.

Assuming that the system is locally-K(π,1) (3.7), by which we mean that the star subsystems (2.9) are K(π,1)'s, we prove that this cover is simple i.e. that intersections are contractible. Then the geometric realisation |Σ^| of Σ^ has the same homotopy type as Y (Weil [Wei1952]). So the K(π,1)-problem is reduced to a contractibility problem for a simplicial complex, if one knows that the system is locally K(π,1). But this "locally-K(π,1)" problem is often solved at the same time one solves the contractibility problem for the complex in a positive way, because one can then apply induction on the dimension of 𝕍.

We investigate the simplicial complex. In §4 we do this in the general situation and give a description in terms of galleries. In §5 we describe the complex in terms of (extended) Artin groups in the cases that the system is associated to a linear Coxeter group or an extended Coxeter group (generalised root system). The construction is then of the type of the "Coxeter complex" of a Coxeter group.

As example (3.5) shows, not every hyperplane system is a K(π,1)-system. In the central case the simpliciality takes care for a nice triangulation of a sphere S around the origin. The simplicial complex Σ^min (defined in (3.6)), which is the complex used by Deligne, can be described in terms of this triangulation and the galleries. But in this context the use of the simpliciality condition is not essential. Because we wanted to know where the use of the condition was crucial, we have tried to postpone the use as far as possible. This is a main reason why we construct in §6 a space Z^(𝕍,H,,p) homotopic to |Σ^|. Besides this the space Z^ is more related to the structure of the facets and the galleries.

Also in §6 we sketch the way how certain generalisations of the theorem of Deligne (1.2), for instance conjecture (1.3), perhaps can be proved. One of the conjectures we need (namely "simpliciality implies the direct pieces property (5.23)) can be seen as a generalisation of the "key lemma" of Deligne ([Del1972] prop. 1.19). Strengthening somewhat this conjecture gives an interesting and simple solution conjecture about the word problem for Artin groups. In general for the proposed kind of proof of the K(π,1) conjecture it seems necessary to have knowledge about the minimal representatives of galleries. Hence the connection with word problems.

§2. Stars and their hyperplane systems

Let again (𝕍,H,,p) be a hyperplane system. We need some more notions and properties; first of the configuration (𝕍,H,).

(2.1) Definition. Let FF(), then the star of F, St(F) is defined by St(F)GFG.

(2.2) Proposition. We have St(F)=(i) pr(-F)(F) (ii)K(-F).

Proof.

(ii) Since FM for all M-F, F is contained in a chamber with respect to -F.

(i) By (ii) The right hand side of (i) equals M-FDM(F). Now let FG, then by criterion (I.3.5) for each M-F we have GDM(F), so G is contained in the right hand side. And conversely.

(2.3) Corollary. St(F) is open in 𝕍.

(2.4) Lemma. Let F1,F2F(). Then the following properties are equivalent:

(i) F1F2
(ii) St(F2) St(F1)
(iii) For each AK() we have ASt(F2)ASt(F1).

Proof.

(i) (ii) Let EF() and ESt(F2). So F2E. Then F1F2E, so ESt(F2).

(ii) (iii) is trivial.

(iii) (i) Let M-F1, By criterion (I.3.5) we have to show that F2DM(F1). Suppose not. Then F2DM, where DM is the other halfspace. So St(F2)DM is an open nonempty set so there exists a chamber A which intersects it and since it is also a union of facets, A is even contained in it. By (iii) ASt(F1), so ADM(F1), contradiction with ADM.

(2.5) Lemma. Let FG, F,GF(), then:

(i) LFLG, so dim(F)dim(G).
(ii) LF=LGF=G.
(iii) FGdim(F)< dim(G).

Proof.

(i) If GM then FM, so LF=FMM GMM= LG.

(ii) If LF=LG, then FMGM for all M. And with (I.3.5): FMGMGDM(F). It follows by definition of facet that F=G.

(iii) follows from (i) and (ii).

(2.6) Proposition. Let F1,F2F() and F1F2. Then there is a facet FF() such that F1F2=F.

Proof.

F1F2 is a union of facets (since Fi is so (I.3.4)). Take FF1F2 with maximal dimension. Suppose aF1F2. To prove: aF. Choose fF. If a=f, then finished. If af, then consider the line l through a and f.

Case 1: lLF contains only f. Since St(F) is open (2.3), St(F)l is open in l, so there is a e[a,f)St(F). Let E be the facet containing the point e. Then FE, but FE, for eF. So by (2.5)(iii) dim(E)>dim(F). But since F1F2 is convex, we have eF1F2, so EF1F2. Hence dim(E)dim(F), a contradiction.

Case 2: lLF. Since F is open in LF, lF is an open interval on l containing f. If (a,f]F, then aF and we are ready. If (x,f]F and xF for some x(a,f), then there is a M, such that xM and fM. So a and f are at different sides of M, but F1F2 is an intersection of hyperplanes in and closed halfspaces. DM(f) is one of them, so then aF1F2, contradiction.

(2.7) Definition. Let F1,F2F(). If F1F2, then we denote the unique facet FF() such that F=F1F2 as exists by (2.6) by F1F2. If F1 and F2 are such that there exists a GF() such that GF1F2, then by locally finiteness there are only finitely may such facets G. Then by (2.6) thre is a facet FF() such that F=GF1F2G. We denote this facet by F1F2.

Note that for i=1,2, F1F2Fi (respectively FiF1F2) and this "meet" (respectively "join") is maximal (respectively minimal) with this property, if it exists.

(2.8) Theorem. If F1,F2F(), then St(F1)St(F2) is nonempty, if and only if F1F2 is defined and in this case St(F1)St(F2)=St(F1F2).

Proof.

St(F1) St(F2)= (GF1G) (GF2G)= GF1F2G =($) GF1F2G St(F1F2). The equality ($) holds because F1F2G F1F2 GF1 F2G.

(2.9) Definition. (cf. def. (II.4.8)) Let again FF() and define

YF Y(𝕍,𝕍,F,p|F)and Y(F) Y(𝕍,St(F),F,p|F)

Note that Y(F)=YIm-1(St(F)) (or simply Im-1(St(F)) if Im is considered as a map YH). Furthermore note that Y(F)YYF.

(2.10) Proposition. (cf. prop. (II.4.9)). Let FF(). Then there exists a map r:YFY(F) with the properties:

(i) The map r is a homotopy inverse of the inclusion i:Y(F)YF.
(ii) For another FF() such that FF exists we have r(Y(F))Y(F).

Proof.

Choose fF and let B be a closed ball around f inside St(F) in some metric 𝕍 (St(F) is open!). For each y𝕍 there is a unique r(y)B, such that [f,r(y)]=[f,y]B. Now r is a continuous map 𝕍St(F). The rest of the proof is analogous to the proof of proposition (II.4.9) from the point ($). Including the proof of (ii) for which we have to remark that fFFFSt(F).

y f F B r(y) St(F) Figure 23.

(2.11) Corollary. The spaces Y(F) and YF are homotopy equivalent.

(2.12) Corollary. Let j be the inclusion Y(F)Y and aY(F), then the homomorphism j*:π(Y(F),a)π(Y,a) is injective.

(2.13) Lemma. Let F1,F2F() such that F1F2 exists and let aY(F1F2). Then

(2.14) π(Y(F1),a) π(Y(F2),a) =π(Y(F1F2),a)

where these fundamental groups are seen as subgroups of π(Y,a), which is done by (2.12).

Proof.

This follows from (2.10)(ii) in the same way as (II.4.12) follows from (II.4.9)(ii).

The following corollary is the key to an important result of this chapter (theorem (3.11)). It makes it possible to prove this result for so called non central (def. (3.2)) systems. This in contrast to Deligne [Del1972]. Compare remark (3.14).

(2.13) Corollary. For i=1,2 let FiF() and γi a curve in Y(Fi). Suppose γ1 is homotopic to γ2 in Y. Then F1F2 exists and the curves γi are homotopic in Y to a curve γ in Y(F1F2).

Proof.

It is easy to see that we may restrict ourselves to the case that γ1 and γ2 are loops, say from a point aY(F1F2). From γ1Yγ2 it follows that {γ1}={γ2} and so it is an element contained in the left hand side of (2.14). So it is also contained in the right hand side, hence of the form {γ} where γ is a curve in Y(F1F2).

(2.16) Definition. Let FF(). Define:

GalF { C0U1 C1Uk CkGal| FCi } and ΠF { xΠ|x {γ}for some curveγ withγ(t) Y(F)fort [0,1] }

(2.17) Lemma. We have φ(GalF)=ΠF.

Proof.

This is a simple inspection of the proof of (I.1.5).

(2.18) Corollary of (2.15). For i=1,2 let FiF(). If GiGalFi and GG2, then F1F2 exists and GiG for some gallery GGalF1F2.

Proof.

Let γi be a curve in Y(Fi) such that φ(Gi)={γi} as exists by (2.17). From G1G2 it follows that φ(G1)=φ(G2), hence γ1γ2. So by (2.15) there is a curve γ in Y(F1F2) such that γγi. Then there is by (2.17) a GGalF1F2 such that φ(G)={γ}. Then φ(G)=φ(Gi), hence GGi.

(2.19) Remark. It is possible and, since we have developed all of the necessary tools, perhaps more natural to prove (2.18) and a "gallery version" of (2.12) independent from (2.10) and its corollaries with the help of galleries alone. Here is a sketch:

Proof of (2.18) (sketch).

Let FiF() for i=1,2. With the help of (I.3.9) and (I.3.10) we have a map prF1-1prF1:K()K(F){AK()|F1A}. This induces a map :GalGalF1, which respects gallery equivalence. (This map corresponds in a sense with the map r in prop. (2.10)). Now let G(G2). Then F1=(G1)(G2)=G.

In fact this is the way (2.18) and hence its consequence (3.11) was discovered. We have given the other approach, because the geometrical background seems more clear.

§3. An open cover of the universal covering space

After these preliminaries we are going to construct an open cover 𝒰^ of the universal covering space of Y(𝕍,H,,p), where (𝕍,H,,p) is a hyperplane system as before.

(3.1) Definition. We call a facet FF() minimal if it is so for the natural order relation i.e. if there is no facet EF contained in the closure F.

(3.2) Definition. We call a hyperplane system central if H=𝕍 and MM. We call a system non central if there doesn't exist a unique minimal facet.

(3.3) Remark. As the terminology suggests, we can suppose that a system is either central or non central. For, suppose there exists a unique minimal facet Fmin. Then since each facet contains at least one minimal facet, we have for all FF() that FminF. Hence St(Fmin)F. Hence St(Fmin)=H. The only hyperplanes, which meet St(Fmin) are the elements of Fmin, so =Fmin. By (2.11) we have YY(𝕍,𝕍,,p). Since we are only interested in the homotopy type of Y, we see that indeed we can suppose that the system is central in the sense of def. (3.2), if there exists a unique minimal facet. Note that we have:

(3.4) Fmin=MM

(3.5) Definition. We agree that Fmin in the non central case (this fits in with (3.4)!). We define the set of proper facets by:

F*() F()-{Fmin}

(hence F*()=F() in the non central case). A proper facet is called properly minimal if its closure doesn't contain a different proper facet. The set of properly minimal facets is denoted by Fmin*().

(3.6) Definition. Let p:YY be the universal covering. For FF() define: 𝒰(F) the collection of components of p-1(Y(F)) and:

𝒰FF*() 𝒰(F),𝒰min FFmin*() 𝒰(F)

If the system is central, then is finite, hence pMpM is locally finite on 𝕍. For AK(p) we define: 𝒱(A) the collection of components of p-1(Re-1(A)) and 𝒱AK(p)𝒱(A) and we agree that 𝒱 if the system is non central. Furthermore we define:

𝒰^𝒰𝒱, 𝒰^min 𝒰min𝒱 Σnerf(𝒰), Σmin nerf(𝒰min) Σ^ nerf(𝒰^), Σ^min nerf(𝒰^min).

(3.7) Definition. We call a hyperplane system locally-K(π,1) if (𝕍,𝕍,F,p|F) is a K(π,1)-system for each FF*(). This means that YF and hence (by (2.11)) Y(F) are K(π,1)'s.

(3.8) Example. A simple, simplicial system is locally-K(π,1) (central or non central). This follows from the theorem of Deligne (1.2) and the observation that the local system (𝕍,𝕍,F,p|F) is simplicial if the original system is so, and this local system is clearly central.

(3.9) Example. If it is true that a simple, simplicial system is K(π,1) (conjecture (1.3)) at least in the case of an affine linear Coxeter group then the hyperplane systems associated to the extended Coxeter groups (III.1.6) are locally-K(π,1).

(3.10) Example. (The "triangle"). Let 𝕍=H be a plane and let ={l1,l2,l3} consist of three different lines with lilj, but l1l2l3= (so a non central system!). See Figure 24. Consider the simple hyperplane system of this configuration. The local systems are easily seen to be simplicial, hence K(π,1). So the system is locally-K(π,1). But it is not K(π,1)! This is well known and can be seen by the method of §4 (see (4.17)).

1 1 2 2 3 3 P1 P2 P3 C D Figure 24.

(3.11) Theorem. Let (𝕍,H,,p) be a hyperplane system which is locally-K(π,1). Then 𝒰^ as defined in (3.6) is a simple covering of the universal covering space of Y(𝕍,H,,p). In particular this universal covering space is homotopy equivalent to the geometric realisation |Σ^| of the abstract simplicial complex Σ^, the nerf of the cover 𝒰^.

Proof.

First we prove the following remark: If Ui𝒰(Fi) for i=1,2 and UU1U2, then FF1F2 exists and U𝒰(F), i.e. 𝒰 is closed under intersections. Now St(F1)St(F2), because U1U2 so it equals St(F) (theorem (2.8)). Let h=Imp. Then we have

(3.12) h-1(St(F)) =h-1(St(F1)) h-1(St(F2))

The right hand side is a union of intersections of elements in 𝒰(F1) and 𝒰(F2) and each intersection is in turn a union of components. So, for instance U1U2 is a union of elements in 𝒰(F) and we have to show that this union consists of just one element, that is we have to see that U is connected.

For this purpose suppose a,bU and let γi be a curve in Ui from a to b (each Ui is connected). We have γ1γ2 in Y, because Y is simply connected. So γ1γ2 in Y, where γipγi. Now γi is in Y(Fi) so by corollary (2.15) we know that γiγ where γ is a curve in Y(F). Let γ be the unique lift of γ to Y with γ(C)=a, then γγi in Y, in particular γ(1)=b. Furthermore γ is inside U so we have connected a and b by a curve in U, hence U is connected. This proves the first remark i.e. that 𝒰 is closed under intersections.

It can be verified that for U𝒰(F), p|U:UY(F) is a universal covering. For instance the simply connectedness of U follows from the fact that Y(F)Y induces an injection of fundamental groups (2.12). The locally-K(π,1) assumption ensures now that Y(F) is a K(π,1) if FF*(). Hence U is contractible then. Together with the first remark it now follows that all intersections of elements of 𝒰 are contractible. In the non central case the elements of 𝒰 cover Y and so this is then indeed a simple cover.

In the central case we still have to cover the points of Y with Imp(x)Fmin. This can be done with the elements of 𝒱. Since these elements are mutually disjoint, we still have only to prove that UV is contractible (if nonempty) for U𝒰(F) and V𝒱(A), where FF*() and AK(p). We have:

(3.13) p-1(Re-1(A)) p-1(Y(F))= p-1(A+-1St(F))

Argueing as after (3.12) we can conclude from (3.13) that UV is a union of components of the right hand side of (3.13). We will see that this union consist of just one element i.e. That UV is connected. The only thing we have to do is to prove the following analog of corollary (2.15): Let γ1 (respectively γ2) be a curve in Re-1(A) (respectively in Y(F)=Im-1(St(F)), where we consider Im:YH) and suppose γ1γ2 in Y. Then there is a curve γ in Re-1(A) Im-1(St(F))= A+-1St(F) such that γγ1γ2 in Y. To prove this we define a curve γ by γ(t) Re(γ1(t))+ -1Im(γ2(t)). Let G be a homotopy from γ1 to γ2 then (s,t)(Re(γ1(t)),Im(Gs(t))) gives a homotopy from γ1 to γ. So γγ2γ1, as we wanted.

So UV is indeed connected. p|UV: UVA+ -1St(F) is again a universal covering. The base space is convex, hence contractible (and p|UV is even a homeomorphism). So 𝒰^ is indeed a simple open cover.

As said before, the second statement of the theorem follows from Weil [Wei1952].

(3.14) Remark. Note that we did not suppose simpliciality nor centrality. This in contrast with Deligne. His ingredient which is comparable with corollary (2.15) is [Del1972] 1.31. And there he uses the existence of a gallery Δ, which depends on the fact that the situation is central.

(3.15) Remark. We have the same result for the open cover 𝒰^min and the abstract simplicial complex Σ^min, as easily can be seen. The space |Σ^| looks like a "thickening" of the space |Σ^min|.

§4. The Simplicial Complex in terms of Galleries

In this paragraph we study the simplicial complex defined in the previous paragraph with the help of the galleries of the hyperplane system. We fix a chamber CK(), a point cC and a point cY above -1cY, where p:YY is the universal covering.

(4.1) Definition. For FF() define:

GalCF AK(),FA GalCA

(Note that FAASt(F)) and recall (2.16):

GalF { C0U1 C1Uk CkGal| FCi }

(4.2) Definition. Let GiGalCFi for i=1,2. We write (G1,F1)(G2,F2) if

(i) F1F2
(ii) There is a GGalF1 such that G2G1G (gallery equivalence (I.4.7))

Note that St(F2)St(F1) by (i), so GalCF2GalCF1, so (ii) make sense. Note also that this relation is transitive.

G G1 G2 St(F1) St(F2) Figure 25.

(4.3) Definition. If G1,G2GalCF then we call G1 and G2 F-equivalent, notation: G1FG2, if (G1,F)(G2,F) (then automatically (G2,F)(G1,F)) i.e. if there is a GGalF such that G2G1G. The F-equivalence classes are denoted by galCF. If ggalCF, then end(g)F. Note that for all AK() galCA=GalCA/, where is gallery equivalence. Furthermore galFF*()galCF. (note that this is a disjoint union).

(4.4) Definition. Let gigalCFi for i=1,2. Then g1g2 means that (G1,F1)(G2,F2), where Gi is in the Fi-equivalence class gi. It is easily checked that this indeed does not depend on the choice of the representatives.

(4.5) Proposition

(i) The relation is transitive.
(ii) Suppose g2galCF2 and F1F2, then there is a unique g1galCF1, such that g1g2.
(iii) If g1,g2gal and there exists a ggal such that gig, then there exists a unique g1g2 such that gig1g2, which is the smallest, i.e. if gig then g1g2g.
(iv) If g1,g2gal and there exists a ggal such that ggi then there exists a unique g1g2 such that g1g2gi, which is the biggest, i.e. if ggi then gg1g2.
(v) If g1,g2gal then g1g2 if and only if g1g2 as sets.
(vi) If g1,g2gal then g1g2 if and only if there is a ggal such that gig.

Proof.

(i) is trivial.

(ii) Let G2g2. Then G2GalCF2GalCF1, so G2 also determines a g1g2, g1galCF1. If also g1g2 and G1g1, then G2G1G for some GGalCF1, by the definition of g1g2. But this says at the same time that G1F1G2, so g1=g1.

(iii) Suppose gigalCFi, ggalCF and gig. Then FiF, so F1F2F is defined. By (ii) there is a unique g1g2galC,F1F2 such that g1g2g. FiF1F2, so by (ii) there is a unique gigalFi, such that gig1g2. By (i) gig and again by (ii) we have gi=gi. We still have to prove that this construction of g1g2 does not depend on the choice of g, so suppose hgalC,F1F2 has also the property that gih. Choose representatives G1,G2,G and H of g1,g2,g and h respectively and let Gi (respectively Gi) be an element of GalFi, such that GGi·Gi (respectively HGi·Gi), as exists, since gig1g2 (respectively g1h). Then G·(Gi)-1H·(Gi)-1, so we have g-1·G(Gi)-1·GiGalFi. Then by corollary (2.18) we have (Gi)-1·GiG for some gGalF1F2. Then H-1·GG, hence GF1F2H, so g1g2=h.

(iv) If ggal there are only finitely many hgal such that hg, as follows from (ii) and (1.4). Therefore (iv) follows from (iii).

(v) Suppose gigalCFi and g1g2. So F1F2. Let G2g2, then G2GalCF2GalCF1. If G1g1, then since g1g2, there is a GGalF1 such that G2G1G, so G1F1G2. Hence G2g1. So g2g1. Now suppose g2g1. First we prove F1F2. For each chamber ASt(F2) there is a G2g2 such that A=end(G2). Since g2g1, we also have ASt(F2). By lemma (2.4) it follows that F1F2. Now let Gigi, then G2g1, so G2F1G1. Hence there is a GGalF1 such that G2G1G. So g1g2.

(vi) If there is a ggal such that gig, then ggi by (v), so g1g2. If Gg1g2, then end(G)St(F1)St(F2). So F1F2 exists and G determines a ggalC,F1F2 such that gig.

(4.6) Remark. g1 and g2 is not necessarily an element of galC,end(g1)end(g2).

(4.7) Definition. We define a mapping u: { (G,F)|F F*(),G GalCF } 𝒰 as follows: Let γ be a curve in Y such that φ(G)={γ}. Then γ(1)Y(F). Let γ be the unique lift with γ(0)=c. Then γ(1) is determined by G and is contained in a unique u(G,F)𝒰(F). In the central case we define also a mapping v:GalCC× K(p)𝒱 as follows: Let (G,A)GalCC×K(p) and suppose γ is a loop in Y such that φ(G)={γ}. Then the unique lift of γ[-1c,-1c+a], where aA, that starts in c, has his endpoint in a uniquely determined v(G,A)𝒱(A).

(4.8) Lemma. Let Uiu(Gi,Fi) for i=1,2. Then U1U2 if and only if (G2,F2)(G1,F1).

Proof.

Let U1U2, then clearly St(F1)St(F2), so F2F1. Let γi be as in definition (4.7). Let γ be a curve in U2 from γ1(1) to γ2(1) and γpγ. Then γ is a curve in Y(F2), so by lemma (2.17) we have φ(G)={γ} for some GGalF. Now γ2=γ1γ so by lemma (2.17) we have φ(G)={γ} for some GGalF. Now γ2=γ1γ so by (I.4.10) G2G1·G. So indeed (G2,F2)(G1,F1). The reverse implication can also easily be seen.

(4.9) Theorem

(i) The mapping u induces a bijection between gal and 𝒰, the vertices of Σ.
(ii) The mapping v induces a bijection between galCC×K(p) and 𝒱, the vertices of Σ^, which are not vertices of Σ.

Proof.

(i) Let U𝒰(F), AK(), ASt(F), aA and a a point in U above -1a, γ a curve from c to a and γ=pγ. According to (I.4.10) there is a GGalCAGalCF, such that φ(G)={γ}. Then u(G,F)=U, so u is surjective. From lemma (4.8) it follows that u(G1,F1)=u(G2,F2) if and only if F1=F2F and G1FG2. Hence u can be defined on gal and is then injective.

(ii) Let V𝒱(A), AK(p). Let γ1 be a curve from c to a point in V. Now γ1pγ1 is a curve in Y from -1c to a point γ1(1)Re-1(A). We may assume that γ1(1)=a+-1c for some aA. Then γ1[a+-1c,-1c] is a loop in Y on -1c, so by (I.4.10) there is a G1GalCC such that φ(G1)={γ1[a+-1c,-1c]}, hence the surjectivity. It is also straightforward to see that v(G1,A1)=v(G2,A2) if and only if A1=A2 and G1G2. Hence we have a bijection v:galCC×K(p)𝒱.

So from now on we can identify the vertices of Σ with the set gal. We proceed by describing the simplices of Σ:

(4.10) Theorem. A set {g1,,gk}, gigal constitutes a simplex in Σ if and only if there is a ggal such that gig for i=1,,k.

Proof.

Let GiGalCFi for i=1,,k and Uiu(Gi,Fi). Then by definition {U1,,Uk} constitutes a simplex if and only if UU1Uk. Then U𝒰(F) for some FF*(), hence U=u(G,F) for some GGalCF. Since UUi, it follows from lemma (4.8) that (Gi,Fi)(G,F).

(4.11) Corollary. The simplicial complex Σ is isomorphic to nerf(gal).

Proof.

This follows from (4.10) with the help of proposition (4.5)(vi).

(4.12) Remark. If we define galminFFmin*galCF, then it can be seen also, that the simplicial complex Σmin is isomorphic to nerf(galmin).

(4.13) Definition. An element ggal (or a representative Gg) as in theorem (4.10) is said to point to that simplex {g1,,gk}. Let ggalCF and {F1,,Fk} a set of facets in F. Then by proposition (4.5)(ii) there is a uniquely determined gigalFi with gig and they constitute a simplex. In particular, if {F1,,Fk} is the set of all the facets in F, then this simplex is denoted by σ(g). The set of all simplices to which g points is exactly the set of all subsimplices of σ(g). The maximal simplices of Σ are the simplices of the form σ(g) for ggalCA for some AK().

(4.14) Definition. Let the system be central. For a vertex V of Σ^, which is not a vertex of Σ (i.e. an element of 𝒱) we denote by Σ(V) the subcomplex of Σ consisting of all simplices that form with V a simplex of Σ^. The corresponding subcomplex of Σ^ is denoted by Σ^(V). For (g,A)galCC×K(p) we write Σ(g,A) (respectively Σ^(g,A)) instead of Σ(v(g,A)) (resp. Σ^(v(g,A))). We define Σmin(V) and Σ^min(V) analogously.

In remark (6.21) we will see that Σ(g,A) (resp. Σ(g,A)) has the homotopy type of a sphere (respectively cell). Now we still have to finish our description of Σ^ in the central case:

(4.15) Theorem. Suppose that the hyperplane system is central and let (g,A)galCC×K(p) and Gg. Then the galleries that point to simplices of Σ(g,A) are equivalent to galleries of the form G·Gd where Gd is a direct replacing gallery of CED with DK() and Epr(p(C,D)p)(A).

Proof.

Let G1G and let the curve γ1 and the point aA be as in the proof of (4.9)(ii). Let G2GalCF point to a simplex {U1,,Uk} of Σ (definition (4.13)); suppose that this simplex forms a simplex in Σ^ with V=v(G1,A). This means that if Uu(G2,F) then UV. For we may suppose that, if Ui𝒰(Fi) then F=F1Fk, hence U=i=1kUi. Now φ(G2)={γ2}, where γ2 is a curve in Y from -1c to -1d, with dDK(), DSt(F). Let γ be a curve in Y from c to a point in UV above a+-1d and put γpγ. Now define the "straight" curves:

λ1 [ -1c, a+-1c ] λ2 [ a+-1c, a+-1d ] λ3 [ a+-1d, -1d ]

Then ?? γλ2-1 and γ2γλ3. Hence φ(G1)={γλ2-1λ1-1}, φ(G2)={γλ3}. Now φ(G1-1G2)={λ1λ2λ3} and by the definition of the functor φ (I.1.3) the latter equals φ(CED), where Epr(p(C,D)p)(A). So we have G2G1·CED as has to be proven.

λ1 λ2 λ3 γ γ1 γ2 -1 -1c -1d a+-1c a+-1d Re-1(A) ?(F)=Im-1(St(F)) Figure 26.

If moreover the hyperplane system is simple, then we have still another description of the extension ΣΣ^:

(4.16) Theorem. Suppose we have a simple central hyperplane system. Then there is a bijection between galCC×K(p) and AK()galCA, the set of equivalence classes of galleries starting in C. So the subcomplexes Σ(g,A) can also be labelled by the latter. If ggalCA and Gg, then the galleries that point to simplices of Σ(g) are equivalent with galleries of the form G·P where P is a direct positive gallery.

Proof.

Let us have the situation of theorem (4.15) and suppose moreover that the system is simple. Then we have K(p)=K(), so AK(). Let C,C1,,Cn=A and A=D0,D1,,Dk=D be direct Tits galleries and E1pr((C,A))(A) and E2pr((A,D))(A) Then AEE1E2, so GdR CEDR CE1AE2D R C-C1- Cn · D0+D1+ Dk.

(where R denotes pregallery equivalence, see (I.1.4)). By proposition (I.4.8) the first and the last gallery are also gallery equivalent, so G·GdG·P, where P is the positive gallery D0+D1+Dk and GG·C-C1--Cn. Now each gallery is equivalent to a gallery of this form and this induces the desired bijection between galCC×K(p) and AK()galCA.

(4.17) Example. As an illustration let us determine the complex Σ^min of the hyperplane system "the triangle", example (3.10) (equals Σmin in this non central case). It is easily seen that the group galCC (which is in general the fundamental group of Y !) is isomorphic to 3, by the isomorphism (n1,n2,n3)g1n1g2n2g3n3, where gi is the equivalence class of the gallery Giii (in the notation used in this kind of examples, see §(I.5)). For instance the commutation relation G1·G2G2·G1 is illustrated by flip relations around the plinth P3:

1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1122 1212 1221 2121 2211 Figure 27.

The minimal facets are the points: Fmin*()={P1,P2,P3}. For each ggalCPi there is a unique representative Ging. So the vertices of Σmin:galmin=i=13galCPi can be labelled by a disjoint union of three copies of :×{1,2,3}. The two-simplices {(n1,1),(n2,2),(n3,3)} are the maximal simplices of Σmin. The system is locally-K(π,1), so by theorem (3.11) or in fact remark (3.15) we have |Σmin|Y. Now we can see why the system is not K(π,1): Take a subcomplex ΣΣmin consisting of the simplices {(n1,1),(n2,2),(n3,3)} with ni{a,b}, where a and b are two different integers. Then |Σ| is a 2-sphere (the surface of the platonic solid in figure 28). Since the simplicial complex |Σmin| is two dimensional |Σ| generates a nontrivial element in π2(Y).

(a,1) (a,2) (a,3) (b,1) (b,2) (b,3) Figure 28.

§5. The Artin Complex and the Extended Artin Complex

In the case that the hyperplane system is associated to a linear Coxeter group or an extended Coxeter group (i.e. a generalised root system) the simplicial complex constructed in §3 can be described in terms of the groups involved. The construction turns out to be of the following general type:

(5.1) Definition. Let G be a group and a finite collection of subgroups. We define an abstract simplicial complex, called the cosetnerf of the group G and the collection as follows:

cnerf(G,) nerf(HG/H).

(5.2) Remark. Since {1}HH, the collection itself is a (maximal) simplex of cnerf(G,). This simplex serves as a fundamental domain for the natural action of G on cnerf(G,).

(5.3) Lemma. the map Gx{xH|H} induces a bijection between G/(HH) and the set of maximal simplices of cnerf(G,).

Proof.

Straightforward.

(5.4) Lemma. If GG, then the space |cnerf(G,)| is homeomorphic to |cnerf(G,)|×G/G, where we give G/G the discrete topology. Hence |cnerf(G,)| is a disjoint union of spaces homeomorphic to |cnerf(G,)|.

Proof.

Let H. Then each left coset of G in G is a disjoint union of left cosets of H. So for each element of G/G we have a subcomplex of |cnerf(G,)|. The space |cnerf(G,)| is a disjoint union of this kind of subcomplexes and a point in one such a subcomplex cannot be connected to a point in another. Furthermore they are all isomorphic to |cnerf(G,)|, as can be seen from the action of G on |cnerf(G,)|.

(5.5) Lemma. The space |cnerf(G,)| is connected if and only if G is generated by HH.

Proof.

If |cnerf(G,)| is connected, then by (5.3) the index of the subgroup generated by the elements of is one. Conversely, let G be generated by HH. Then an arbitrary vertex of nerf(G,) is of the form xH,xH and H. Write x=a1a2ak, aiHi for i=1,,k and let xiaiai. Then, since aiHi=xixiHi=x1xi-1Hi=ai-1Hi the sequence |{xH,xkHk}|, |{xk-1Hk,xk-1Hk-1}|, , |{x1H2,x1H1}| is a sequence of 1-simplices connecting the vertex xH to a vertex of the fundamental simplex.

(5.6) Remark. It is convenient to restrict ourselves to collections , which have the following two properties:

(i) HH ={1}.
(ii) G is generated by HH.

By lemma (5.4) we do not lose much generality by supposing (ii). If (ii) is supposed we will suppress G in the notation: cnerf().

Property (i) ensures with the help of lemma (5.3) that we have a bijection

(5.7) σ:G{maximal simplices of cnerf()}.

In all the cases we will consider, the properties (i) and (ii) will be satisfied.

(5.8) Example. Let (W,{σ1,,σ}) be a finite, affine or compact hyperbolic system i.e. the subgroups WiW{1,,}-{i} for i=1,, are finite. Then the collection {Wi|i{1,,}} satisfies (5.6) (i) and (ii). Now cnerf() is the well known Coxeter complex. Note that if W is finite, then |cnerf()| is homeomorphic to the -1-sphere.

(5.9) Theorem. Suppose a simple hyperplane system of a linear Coxeter group W is given. Let (AW,{s1,,s}) be the associated Artin system. Then:

(i) The simplicial complex Σ of the hyperplane system as defined in (3.6) is isomorphic to the cosetnerf of the collection of subgroups of AW of the form AWJ, where J{1,,} and WJ is finite. This collection satisfies (5.6) (i) and (ii).
(ii) The system is central if and only if W is finite and in this case there is a bijection m ("midpoint") from AW to the vertices of Σ^, which are not vertices of Σ. If xAW, then the maximal simplices of the subcomplex Σ(m(x)) of Σ are the simplices of the form: σ(xw),wW, where W has to be interpreted as a subset of AW by remark (II.3.19).

Proof.

The fact that the collection of subgroups of AW as in (i) satisfies (5.61)(i) follows from theorem (II.4.13)(ii). We have a surjective map f:GalAW as the composite GalGal/Wφ/Wπ/WAW, where the first is canonical projection and the two others are discussed in the proof of (II.3.8)(see (II.3.161). Now we claim that galgf(g)AW induces a bijection between gal and the cosets of the subgroups of AW mentioned in (i). For, if G1g, ggalCF, then the elements of g are equivalent to galleries of the form G1G2 with G2GalF. After dividing out the action of W, we can suppose FC, then f(GalF)=AWJ, where J{1,,} is of the type F, i.e. J={j|ϕj(F)={0}}. WJ is finite because FI and J is a proper subset, because FF*(). Now f(g) is the coset f(G1)AWJ. Conversely, if WJ is finite, but J{1,,} and xAW, it is easy to see that f-1(xAWJ)galCF, for some proper facet F determined by π(x)W and WJ. (i) now follows from corollary (4.11). (ii) follows from (4.16).

(5.10) Definition. Let AW be an extended Artin group of rank as in definition (III.2.13). For J{1,,} we define the following subgroups:

AWJ the group generated bysj,tj, jJ. π-1(W)J the group generated bysj(n), jJandn.

(5.11) Remarks. (i) We have π-1(W)Jπ-1(WJ), but the other inclusion need not hold (take for instance J=). What we do have is:

π-1(W)J= (π|AWJ)-1 (WJ)=AWJ π-1(WJ).

(ii) Analogous to theorem (II.4.13) it can be proved that the flip relations form a complete set of relations for π-1(W)J and also:

(5.12) π-1(W)J1 π-1(W)J2 =π-1(W)J1J2 .

(iii) From (ii) it can be seen that (as the notation already suggests) (AWJ,{sj,tj}jJ) forms an extended Artin system and we have AWJ1 AWJ2= AWJ1J2.

(5.13) Theorem. Suppose a hyperplane system of an extended Coxeter group of rank is given. Let AW be the associated extended Artin group of . Then:

(i) The simplicial complex Σ of the hyperplane system is isomorphic to the cosetnerf of the collection subgroups of π-1(W) of the form π-1(W)J where J{1,,} and WJ is finite. This collection satisfies (5.6)(i) and (ii).
(ii) The system is central if and only if W is finite and in this case there is a bijection m ("midpoints") from AW to the vertices of Σ^ which are not vertices of Σ. If x·eqAW is the unique way to write an element of AW as product of xπ-1(W) and qQ, then the maximal simplices of the subcomplex Σ(m(xeq)) are the simplices of the form σ(xλq(w)) ,wW where W has to be interpreted as a subset of π-1(W), which can be done by remark (II.3.19) and lemma (III.4.1) and λ is the pseudo conjugation the action of Q on the set π-1(W) as defined in (III.4.14).

Proof.

(i) is proved in a similar way as the corresponding statements of (5.9). Note that (5.6)(i) is satisfied by (5.12). It is in principle possible to give a bijection between AW and galCC×K(p) and prove (ii) with the help of (4.15). But let us give a more direct and instructive proof: Let V𝒱(A) be a vertex of Σ^, which is not a vertex of Σ. Here A=w(q+U0)K(p) for unique wW, qQ and U0 as in the proof of (III.3.5)(iii).

λ1 λ2 λ3 λ4 γ -1c -1d a+-1d w(q+-1c) w(q)+-1d Re-1(A) Im-1(D) Figure 29.

A maximal simplex of Σ is determined by a U𝒰(n), D=w(C)K() for a unique wW. This simplex forms with V a simplex in Σ^ (i.e. is a simplex of Σ(V)) iff UV. We may assume, that there is a point in this inter¬¨section, which projects (by YY) on a+-1dRe-1(A)Im-1(D)Y, where aw(q+c)w(q+U0)=A and dw(c)D (recall that cU0C). Let γ be a curve in Y from c to this point in UV. Then γpγ is a curve in Y from -1c to a+-1d. Define the "straight" curves λ1,,λ4 as follows

λ1 [ w(q+-1c), w(q+c) ] = [ w(q+-1c) ,a ] , λ2 [ w(q+c),w(q+c) +-1w(c) ] =[a,a+-1d], λ3 [ w(q+c)+-1w (c),w(q)+ -1w(c) ] = [ a+-1d,w(q) +-1d ] , λ4 [ w(q)+-1w (c) ] = [ w(q)+-1d, -1d ] .

Note that λ2Re-1(A) and λ3,λ4Im-1(D). Now the curve γλ2-1λ1-1 projects on a loop in X representing an element in AW=π(X,*) of the form x·eq, xπ-1(W), qQ as above. Conversely each element in AW determines a curve in Y from -1c to a point of the form w(q+-1c). Composing this curve with λ1 and lifting to Y the endpoint lies in a V𝒱. This way we get the desired bijection m:AW𝒱.

The curve γλ3λ4 projects on a loop representing an element y of π-1(W), such that π(y)=w. This describes the bijection between π-1(W) and the maximal simplices of Σ given by (i).

Since γλ3λ4 γλ2-1 λ1-1λ1 λ2λ3λ4 we have

($) y=x·eq·z,

for some zAW, represented by the projection of λ1λ2λ3λ4. To analyse this element z, we note that λ1λ2λ3 projects on the same loop in X as λ1λ2λ3, where

λ1 [-1c,c], λ2 [ c,c+-1w-1 w(c) ] , λ3 [ c+-1w-1w (c),-1w-1 w(c) ] ,

the transforms of λi under e-q·w-1W. Inspection of these formulas gives that λ1λ2λ3 projects on a loop representing the element w-1wWπ-1(W).

The curve λ4 projects on the same loop as its transform under (w)-1· e-w(q) W: [ -1c, (w)-1 (-w(q))+ -1c ] , so it projects on a loop representing (w)-1(w(-q)) QAW. we conclude that z= w-1w· e(w)-1w(-q) .

The reasoning has shown that if σ(y) is a simplex of Σ(m(x·eq)), then y=x·eq·z=x·eq·w·e(w)-1(-q)x·λq(w), where ww-1w. Conversely, it can be checked that σ(x·λq(w)) is a simplex of Σ(m(x·eq)) for all xπ-1(W), qQ and wW.

(5.14) Remark. In the situation of a linear Coxeter group as well the situation of an extended Coxeter group the simplicial complex Σ^min can be described by adding to the conditions on J in (5.9)(i) (respectively (5.13)(i), namely that J{1,,} and WJ finite, that J is maximal with respect to these properties.

(5.15) Remark. In the case that W is finite the sets J{1,,} with the properties mentioned in (5.14) are precisely the sets J with |J|=-1, hence of the form {1,,}-{j} for a j{1,,}. Hence for a given xAW (respectively x·eqAW) we have a bijection between the maximal simplices of the Coxeter complex of W (example (5.8)) and the maximal simplices of Σmin(m(x)) (respectively Σmin(m(x·eq))). This bijection induces a homeomorphism between the -1-sphere and |Σmin(m(x))| (respectively |Σmin(m(x·eq))|).

§6. The Contractibility Problem

If the space |Σ^| is contractible and the system is locally-K(π,1), then the system is K(π,1), as follows from theorem (3.11). To investigate the question of contractibility of |Σ^| we construct a space Z^=Z^(𝕍,H,,p) in terms of facets and galleries, which turns out to be homotopy equivalent to |Σ^|, as follows:

(6.1) Definition. Let SH-Fmin. Note that in the non central case S=H and in the central case S=𝕍-MM, which has the homotopy type of a sphere, hence the use of the symbol S.

(6.2) Definition. For FF*() we define F^F-Fmin (hence, by (3.5) F in the non central case).

(6.3) Definition. We first define the sets:

Z { (x,g)S×gal |xend(g)^ } and forggal: Zg end(g)^ ×{g}.

We give Zg the topology of end(g)^𝕍 and we give Z the topology of the disjoint union of the spaces Zg (Note that the set Z is a disjoint union of the sets Zg, ggal). Furthermore for ggal: Zgend(g)×{g}. Note that this fits in with the notation, because Zg is indeed the closure of Zg in Z. Now we come to the definition of the space Z:

(6.4) Definition. We define an equivalence relation on Z as follows: If (xi,gi)Z for i=1,2, then (x1,g1)(x2,g2) if x1=x2 and if Fend(gi) is the facet containing xi and gi the unique element of galCF (see (4.5)(ii)) such that gigi, then g1=g2.

Let Z denotes the quotient space Z/ and b:ZZ the projection.

(6.5) Remark. From the definition it follows that each equivalence class contains a unique element (x,g) with the property that xend(g). So Z=ggalZg, where Zgb(Zg). This is a disjoint union. The sets Zg are the facets of the space Z.

(6.6) Remark. To clarify the picture of the space Z (see also fig. 30) we note the following: Since each point in S is contained in A for some chamber A we could have restricted ourselves to elements g such that end(g)K() (and g is then nothing but an equivalence class of galleries). I.e. Z can also be considered as a quotient space ggalCA, AK() Zg.

0 1 q Z H=𝕍=and={0,1} Figure 30.

(6.7) Definition. Let ggal. We define the star of the facet Zg (and denote it by Str to avoid confusion with the notations St and Star) by: Str(g)ghZh

(6.8) Proposition. Str(g) is open.

Proof.

Let kgal. It suffices to show that

(6.9) Zk b-1 (Str(g))= { (x,k) Zk |xSt(end(g)) }

if the l.h.s. is nonempty, because the r.h.s. of (6.9) is open in Zk. Let xend(k)^ and b(x,k)Str(g). By the definition of Str(g) (6.7) we have b(x,k)Zh for some hgal with gh. Then (x,k)(x,h) and xend(h), which entails hk. So first of all we see that, if the l.h.s. of (6.9) is nonempty, then gk. Furthermore from gh it follows that end(g)end(h), so xSt(end(g)), which proves that the l.h.s. is contained in the r.h.s.

Now we assume that the l.h.s. is nonempty (hence as we saw above we know that gk) and prove that the r.h.s. is contained in the l.h.s.: Suppose (x,k)Zk and xSt(end(g)). Then xF for some facet F with end(g)F. Furthermore Fend(k), so there is a unique hgalCF, such that hk. end(g)F, so there is a unique element ggalC,end(g) such that gh. So gk, but we also know that gk, so g=g and hence gh. From b(x,k)=b(x,h)Zh and gh, we conclude that b(x,k)Str(g). Hence (x,k) is element of the r.h.s. of (6.9).

(6.10) Proposition. Let ggal. Then:

(i) b(Zg) =hgZh
(ii) b(Zg) is closed in Z, hence Zg=b(Zg).

Proof.

(i) is clear from def. (6.4) and remark (6.5).

(ii) Suppose kgal is cuh that Zkb(Zg)=, then by (i): kg. We show that Str(k)b(Zg)=, which proves (ii): Suppose ZkStr(k)b(Zg), then kh and hg, hence kg, contradiction.

(6.11) Remark. We have Z=ggalZg and a map defined on Z is continuous, if and only if the restrictions to Zg, ggal are continuous. If we wish we even can restrict ourselves to ggalCA with AK().

(6.12) Proposition. Let g1,g2gal. Then Str(g1)Str(g2) is nonempty if and only if g1g2 exists and then it equals Str(g1g2).

Proof.

If nonempty, then there is a Zg contained in the intersection and then gig i=1,2. So then g1g2 exists. And conversely. Str(g1) Str(g2)= gihZh= g1g2hZh Str(g1g2).

(6.13) Proposition. Let ggal. Then Str(g) is contractible.

Proof.

Choose a point x0end(g). Then b(x0,g)ZgStr(g) and we are going to contract Str(g) on b(x0,g). Let kgal and suppose, that the left hand side of (6.9) is nonempty (i.e. gk). By the equality (6.9) we see that we can define a continuous map ϕk:A×[0,1]A by ϕk((x,k),t)(tx0+(1-t)x,k), where A is the set in (6.9). It is not difficult to check, that (x1,k1)(x2,k2) then ϕk1((x1,k1),t) (ϕk((x2,k2),t)). So we have a continuous map ϕ:Str(g)×[0,1]Str(g), which is the desired contraction on b(x0,g).

(6.14) Theorem. The space Z is homotopy equivalent to the space |Σ|.

Proof.

It follows from (6.8) and (6.12) that {Str(g)}ggal is an open cover of Z, whose nerf is isomorphic to Σ. This cover is simple by (6.13) and (6.12), so the theorem follows.

(6.15) Discussion. We have a map q:ZS, sending the class of (x,g) to x. Suppose we have a central system. For AK(p) and ggalCC we define a section s=s(g,A):SZ of q as follows:

If xS, then xD^ for some DK(). Now let kgalCD be the class of G·Gd, where Gg and Gd is a direct replacing gallery of the direct pregallery CED with Epr(p(C,D)p)(A). Then s(x)b(x,k), We have to check that this does not depend on the choices involved. We sketch this check for the choice of another DK(), such that xD^. Let FDD be the facet such that xF. Then, as we saw in (I.3.9) and (I.3.10) the map prF restricted to {BK()|FB} is a bijection on K(F). So we can take BprF-1(prF(C)). Then a direct Tits gallery from C to B, followed by a direct Tits gallery from B to D, is a direct Tits gallery from C to D. So Gd can be chosen of the form G1G2,G2GalF and Gd=G1G2,G2GalF. Then Gd and Gd are F-equivalent.

B C D D Figure 31.

The map s is continuous, because it is so when one restricts it to a set D^ and S is a finite union of such in S closed sets. Furthermore it is clear that qs=idS and sq|S(g,A)=idS(g,A), where S(g,A)s(g,A)(S).

So we conclude from this discussion, that we have a collection of spheres in Z in the central case, labelled by galCC×K(p). Now the hope is that we can prove in some cases that Z is just a bouquet of these spheres. I.e. if one attach cells to these spheres one gets a contractible space. Indeed, constructing the space Z^, we are going to attach for each (g,A)galCC×K(p) a cell B(g,A). Now, if it could happen that S(g1,A1)=S(g2,A2), with (g1,A2)(g2,A2), then the contractibility of Z^ would be doubtful. But we have:

(6.16) Lemma. If S(g1,A1)=S(g2,A2), then (g1,A1)=(g2,A2).

Proof (sketch).

Let xC, then s(g1,A1)(x)=s(g2,A2)(x), so (x,g1)(x,g2), which implies g1=g2. From s(g1,A1)(-x)=s(g2,A2)(-x) it follows that G·CA1(-C)RG·CA2(-C), where Gg1=g2. So CA1(-C)RCA2(-C), from which it can be proved that A1=A2, by a simple application of an invariant technique, which we do not want to develop here.

(6.17) Definition. We define a space Z^=Z^(𝕍,H,,p) as follows: If the system is non central then Z^Z. If the system is central, then we attach a cell B(g,A) to each sphere S(g,A) of Z. Precisely: Z^ is a quotient of Z(𝕍×(galCC×K(p))), where points of S(g,A) are identified with points of 𝕍×{(g,A)} by S(g,A)qS𝕍𝕍×{(g,A)}

Note that we can extend q:Z^𝕍 in the central case, so we have a projection q:Z^H in both cases.

(6.18) Theorem. The space Z^ is homotopy equivalent to the space |Σ^|.

Proof.

For the non central case this is theorem (6.14). So suppose central. Then: The in Z open sets Str(g) for ggal induce open sets in Z^ This follows from the fact that q(Str(g))=St(end(g)) is open in 𝕍. These open sets still do not cover Z^. But for each (g,A)galCC×K(p) 𝕍×{(g,A)} induces an open set in Z^ and together with the first sets this gives an open cover of Z^. It can be verified, that this cover is simple and that its nerf is isomorphic to Σ^.

(6.19) Corollary. If the system is locally-K(π,1), then the universal covering space Y is homotopy equivalent to Z^.

(6.20) Remark. It can be proved that the homotopy equivalence of (6.19) can be realised by the following map ψ:YZ^, which makes the diagram shown commutative. Let yY and set xIm(p(y))H. If xFmin, then yV𝒱 and ψ(y) is the point in Z^ determined by x and v-1(V)galCC×K(p). If xS, then xF for an FF*(). Then yU𝒰(F) and ψ(y) is the point in Z^ determined by b(x,u-1(U))Z. (Recall the maps u and v from (4.9)).

(6.21) Remark. The spaces |Σ(g,A)| are homotopy equivalent to a sphere, since they can be seen to correspond to the spheres S(g,A) by the homotopy equivalence of theorem (6.14).

Y ψ Z^ p Y q Im H

(6.22) Remark. If the system is simplicial and if MM={0}, then the hyperplanes cut out a triangulation of a sphere around the origin. Deligne uses this triangulation to define his complex (which coincides with |Σ^min|). But the use of the simpliciality condition for this purpose is not the essential use as we can see from the constructions of this chapter. The geometric simplicial complex on the sphere is exactly |Σmin(g,A)| (cf. (5.14) and (5.15)).

Now we are going to define some conditions on the galleries, which seem to be satisfied in many cases and which are sufficient to prove that Z is contractible in the non central case (for a simple system it follows then for a central system too).

(6.23) Definition. Let GGalCF. Recall (I.4.1) that |G| is the length of G. The gallery G is called F-minimal, if |G||G| for all G in the F-equivalence class of G. A gallery G is called minimal, if it is end(G)-minimal, i.e. if its length is minimal for the gallery equivalence class to which it belongs. Note that F-minimality implies minimality, but in general not conversely.

If ggalCF, we define the length of g, notation also |g| to be the length of an F-minimal representative of g.

(6.24) Definition. Let G,GGal. G is said to be left (resp. right) contained in G, if GG·G (resp. GG·G) for some GGal. Recall that denotes flip equivalence (I.4.5).

(6.25) Definition. Let g be an equivalence class of galleries. Then a gallery G is called a direct starting (resp. ending) piece of g, if G is direct and left (resp. right) contained in some minimal representative of g.

(6.26) Example. The galleries 1 2 and 3_ 2 are direct starting pieces of the gallery class that is depicted in figure 12 (example (I.5.5)). The galleries 2 1_ and 2 3 are the direct ending pieces.

(6.27) Example. Let us consider the configuration of figure 32. The galleries G1,,G6, starting in C, where

G1 1_255 2_1 G2 1_3_ 5531 G3 3_1_ 5531 G4 1_3_ 5513 G5 3_1_ 5513 G6 3_455 4_3

(Example is due to the computer) are all the minimal representatives of a gallery class g.

1 2 3 4 5 5 A B C Figure 32.

We see that the (maximal) direct starting nieces of g are: 1_ 2 5, 1_ 3_ 5 (3_1_5) and 3_ 4 5. The (maximal) direct ending pieces: 5 2_ 1, 5 3 1 (513) and 5 4_ 3, ending in chamber C. This example is interesting because this equivalence class splits up in three flip equivalence classes: {G1}, {G2,G3,G4,G5} and {G6}.

(6.28) Definition. A gallery class g (with Abeg(g)) is said to satisfy the direct starting pieces property (DSPP), if there exists a direct gallery Gd, such that:

(i) Each direct starting piece of g is left contained in Gd.
(ii) For each MA(A,end(Gd)) there is a minimal representative of g, which starts through M, i.e. which Deligne gallery is of the form A, spMA(A), (definition of spF: (I.3.11)).

The direct ending pieces property (DEPP) is defined analogously. It is easily seen that all the gallery classes of a hyperplane system satisfy DSPP if and only if all the classes satisfy DEPP and in this case the hyperplane system is said to satisfy the direct pieces property (DPP).

(6.29) Example. For the examples (6.26) and (6.27) the classes mentioned have both properties: Gd=123_3_21 for (6.26) and for example (6.27): Gd=1_253_4 for the direct starting pieces (note that Gd is direct, because the underlying Tits gallery is so and chamber A prescribes the signs). And Gd=4_352_1 for the direct ending pieces (now chamber B prescribes the signs of Gd).

(6.30) Definition. A hyperplane system is said to satisfy the facet minimality property, if for each FF() and G1GalCF, G1 is F-minimal and G2GalF, G2 is minimal, then G1G2 is minimal (if defined).

(6.31) Theorem. Let (𝕍,H,,p) be a non central hyperplane system, which satisfies the direct pieces property and the facet minimality property. Then Z(𝕍,H,,p) is contractible.

For the proof we need a simple lemma, which tells us how we can project the closure of a chamber on its boundary in the direction of another chamber:

(6.32) Lemma. Let A,BK(), AB. Set

E MA(A,B) MA

Then EA is a strong deformation retract of A.

Proof.

Choose bB. for xA, A[b,x] is of the form [d(x),x] for some unique d(x)A. We prove that d(A)=E. First remark that,

xd(A) d(x)=x[b,x] A={x}. x b E d(x) A Figure 33.

Let xA. If there is a MA(A,B) with xM, then [b,x]DM(A)={x}, for bDM(A,B). So then xd(A). Conversely, if there is no such M, then for all MA we have [b,x]DM(A){x}. So then [b,x]A=MA([b,x]DM(A)){x}. This shows that d(A)=E. The continuity of d is now also clear. By the convexity of A, we see that A×[0,1](x,t)(1-t)x+t·d(x)A gives a deformation retraction of A on E.

Proof of theorem (6.31).

We construct a filtration of Z: For n+ we define zn union of all Zg, where ggalCA, AK() and |g|n. Then CZ0Z1Z2ZnZn+1Z and Z=n+Zn, so it suffices to show that Zn is a (strong) deformation retract of Zn+1. Therefore we define a continuous map κt:Zn+1Zn+1, t[0,1], which is a (strong) deformation retraction of Zn+1 on Zn. Let us define this first on Zg for a ggalCA, AK() and |g|n+1. If |g|n, then we define κt to be the identity on Zg.

Now let |g|=n+1. Let Gd be the direct gallery, such that all the direct ending pieces of g are right contained in Gd (so in particular end(Gd)=A). Let Bbeg(Gd). Then AB, for |g|>0. Since the system is non central, we have A=A^, so Zg=A×{g}A. So we can define κt on Zg with the help of the deformation retraction of lemma (6.32). Let us check that κt is now well defined on Zn+1 by proving that the definitions coincide on ZgZh.

There is only something to prove in the case that at least one, say g, of the gallery classes has length n+1. So |g|=n+1 and we use the same notations for g and its attributes as above. Suppose hgalCD, DK(). ZgZh implies that AD, so this intersection equals F, where FAD (2.7). Now there is a unique kgalCF such that kg and kh. Choose an F-minimal representative G1 of k and let G2GalF be a minimal gallery such that G1G2 is a representative of g. Let also G3GalF be minimal, such that G1G3 is a representative of h. It follows from the facet minimality property, that G1G2 and G1G3 are minimal, in particular |G1G2|=n+1. Now we may suppose that |G2|1, for otherwise |G1|=n+1, hence |G3|=0, so then A=D=F and g=h and there is nothing to prove, go we can write G2=G2G2 with |G2|=1 and let G2 pass through the wall MA. Since G2 is a direct ending piece of g, we have MA(A,B) by (DEPP). Now G2GalF, so G2 also and hence MF. So FE, where E is as in lemma (6.32). It follows that κt restricted to the intersection ZgZh is the identity for each t[0,1], in particular coincidence of the definition on Zg and on Zh So κt is veil defined and it is continuous by remark (6.11).

We still have to prove κ1(Zn+1)Zn. Let g and its attributes be again as above and suppose xA. We have to show, that κ1(b(x,g)), which equals b(d(x),g) by definition, is contained in Zn, where d:AA is the map constructed in the proof of lemma (6.32). By this lemma d(x)M for some MA(A,B), so by (DEPP)(ii), we know that g has a minimal representative of the form G1G2, where |G2|=1 and G2 passes through M. So d(x) is element of the closure F of a common face F of A and end(G1). The pair (G1,F) determines an element hgalCF and G1 and G1G2 are F-equivalent, so (d(x),h)(d(x),g) and since |h|=n, it follows that b(d(x),h), hence also b(d(x),g) is contained in Zn.

(6.33) Corollary. Let we have a system as in theorem (6.31). Suppose the system is also locally-K(π,1). Then the system is K(π,1).

In the simple case there is no need to restrict ourselves to non central systems:

(6.34) Theorem. Let a simple hyperplane system be given, which satisfies the direct pieces property and the facet minimality property. If the system is locally-K(π,1), then it is K(π,1).

Proof.

In the noncentral case this is corollary (6.33). So suppose the system is central. Choose a hyperplane N and a form ϕ𝕍*, such that N=ϕ-1(0). Let Hϕ-1((0,)). Consider ϕ as an element of (𝕍)*. Let Yϕ-1(1)Y. Then it is easily seen that Y is a strong deformation retract of YY(,H,-{N}). It is also not difficult to see, that *×Y(t,z)tzY defines a homeomorphism. So the system (𝕍,𝕍,) is K(π,1) if and only if (𝕍,H,-{N}) is K(π,1). And the latter is K(π,1) by (6.33).

§7. The Direct Piece Property; Conjectures

(7.1) Conjecture. A simplicial system has the direct pieces property and the facet minimality property.

If this conjecture is true, at least for simple systems one can prove conjecture (1.3). This proof is independent of the result of Deligne (1.2), because to prove the locally-K(π,1)-property one can apply induction on the codimension of MM in 𝕍.

In this paragraph we want to give a sketch of a possible proof of the conjectured implication: simpliciality direct pieces property. First we remark that it is only necessary to prove (6.28)(i), because in the simplicial case it can be proved that (ii) is satisfied if one takes the length of Gd of (i) minimal. Now we need a definition:

(7.2) Definition. We say that a gallery G reduces to a gallery G, notation GG, if we can obtain G from G by length reducing cancel operations. If we can take these cancel operations as follows: (Gdm·G1·AUBUA·G2,Gdm·G1·G2), where Gdm is the maximal direct starting piece of the second gallery, we say that G reduces to G respecting the direct starting pieces, notation G!G.

If gfl is a flip class of galleries, then Ggfl (resp. G!gfl) means that there is a Ggfl, such that GG (resp. G!G).

Now by very general reasoning it can be seen that if {G1,,Gk} is a finite collection of representatives of a gallery class g, then there exists a flip class gflg such that Gigfl for i=1,,k. In particular there exists such a class for all the minimal representatives of g. Now (6.28) would be proved if we have:

(7.3) Conjecture. By the minimality of the representatives it follows that we can choose gfl such that we even have Gi!gfl for all the minimal representatives Gi of g.

and:

(7.4) Proposition. (6.28) is true for flip equivalence.

Proof.

This can be proved analogously as the key lemma of Deligne [Del1972] 1.19.

Here is the essential use of the simpliciality.

It is interesting to ask if a somewhat stronger form of the direct pieces property is true, namely if we replace the word "minimal" by "irreducible", where:

(7.5) Definition. We call a gallery irreducible, if it is not flip equivalent to a gallery, that contains a cancellation, i.e. that is of the form G1·AUBUA·G2.

Note that minimal implies irreducible, but not conversely:

(7.6) Example. Consider G1221_1_2_ in figure 12, page 40. Then no flip operations nor length reducing cancel operations are possible, so G is irreducible. But G 3_2231_2_ 3_222_1_3 3_21_3, so G is not minimal.

If we not only replace "minimal" by "irreducible", but require an additional condition (which is satisfied in all the examples we have considered so far) we get the following:

(7.7) Definition. A hyperplane system as said to satisfy the strong direct pieces property, if for each gallery class g, there exists a direct gallery Gd, such that direct starting pieces of irreducible representatives of g are left contained in Gd and moreover if gig for i=1,2 is a flip class of irreducible galleries and Gi is the maximal direct starting piece of gi, then |G1|+|G2|>|Gd|.

(7.8) Theorem. If the hyperplane system satisfies the strong direct pieces property, then we have the following solution to the word problem of the galleries: Let GGal and beg(A)A. If GA, then G is reducible.

Proof.

If G is irreducible, then G=G1·G2-1 for two irreducible and equivalent galleries G1 and G2. Using the second condition in def.(7.7), we get a contradiction. The details are left to the reader.

If simplicial systems would satisfy the strong direct pieces property, then theorem (7.8) would give a very simple solution to the word problem for Artin groups. Note that for braid groups it would mean the following: "If a braid is trivial, then this can be shown without increasing the number of crossings".

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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