Chapter IV. The $K(\pi ,1)\text{-Problem}$

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 $(𝕍,H,ℳ,p)$ be a hyperplane system. In this chapter we consider the question under which circumstances one can prove that the space $Y≔Y(𝕍,H,ℳ,p)$ is a $K(\pi ,1)\text{.}$ If this is the case we call the system $(𝕍,H,ℳ,p)$ a $K(\pi ,1)\text{-system.}$

(1.1) Definition. A hyperplane system is called simplicial, if the cone on a chamber is a simplicial cone, i.e. of the form $\bigcap _{i=1}^k{\varphi }_i^{-1}\left((0,\infty )\right)$ for some independent set $\{{\varphi }_1,\dots ,{\varphi }_k\}\subset {(𝕍\times ℝ)}^{*}\text{.}$ The cone on a set $A\subset 𝕍$ is defined to be $\{(v,t)\in 𝕍\times ℝ\hspace{0.17em}|\hspace{0.17em}t>0,\frac{v}{t}\in A\}\text{.}$ 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 $K(\pi ,1)\text{.}$

(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 $\widehat{𝒰}$ of the universal covering space $\stackrel{\sim }{Y}$ of $Y\text{.}$ To ensure that the abstract simplicial complex $\hat{\Sigma }≔\text{nerf}\left(\widehat{𝒰}\right)$ has simplices with a finite number of vertices we suppose from now on that the configuration satisfies:

(1.4) Assumption. For each facet $F\in F\left(ℳ\right)$ the number of facets contained in $\bar{F}$ is finite.

Assuming that the system is $\text{locally-}K(\pi ,1)$ (3.7), by which we mean that the star subsystems (2.9) are $K(\pi ,1)\text{'s,}$ we prove that this cover is simple i.e. that intersections are contractible. Then the geometric realisation $\left|\hat{\Sigma }\right|$ of $\hat{\Sigma }$ has the same homotopy type as $\stackrel{\sim }{Y}$ (Weil [Wei1952]). So the $K(\pi ,1)\text{-problem}$ is reduced to a contractibility problem for a simplicial complex, if one knows that the system is locally $K(\pi ,1)\text{.}$ But this $\text{"locally-}K(\pi ,1)\text{"}$ 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 $𝕍\text{.}$

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 $K(\pi ,1)\text{-system.}$ In the central case the simpliciality takes care for a nice triangulation of a sphere $S$ around the origin. The simplicial complex ${\hat{\Sigma }}_{\text{min}}$ (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 $\hat{Z}(𝕍,H,ℳ,p)$ homotopic to $\left|\hat{\Sigma }\right|\text{.}$ Besides this the space $\hat{Z}$ 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 $K(\pi ,1)$ 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 $(𝕍,H,ℳ,p)$ be a hyperplane system. We need some more notions and properties; first of the configuration $(𝕍,H,ℳ)\text{.}$

(2.1) Definition. Let $F\in F\left(ℳ\right),$ then the star of $F,$ $\text{St}\left(F\right)$ is defined by $\text{St}\left(F\right)≔\bigcup _{\bar{G}\supset F}G\text{.}$

(2.2) Proposition. We have $\text{St}\left(F\right)\underset{\text{(i)}}{=}\text{pr}(ℳ-{ℳ}_F\subset ℳ)\left(F\right)\underset{\text{(ii)}}{\in }K(ℳ-{ℳ}_F)\text{.}$

		Proof.
	(ii) Since $F\not\subset M$ for all $M\in ℳ-{ℳ}_F,$ $F$ is contained in a chamber with respect to $ℳ-{ℳ}_F\text{.}$ (i) By (ii) The right hand side of (i) equals $\bigcap _{M\in ℳ-{ℳ}_F}{D}_M\left(F\right)\text{.}$ Now let $F\subset \bar{G},$ then by criterion (I.3.5) for each $M\in ℳ-{ℳ}_F$ we have $G\subset D_M\left(F\right),$ so $G$ is contained in the right hand side. And conversely. $\square$

(2.3) Corollary. $\text{St}\left(F\right)$ is open in $𝕍\text{.}$

(2.4) Lemma. Let $F_1,F_2\in F\left(ℳ\right)\text{.}$ Then the following properties are equivalent:

(i)	$F_1\subset {\bar{F}}_2$
(ii)	$\text{St}\left(F_2\right)\subset \text{St}\left(F_1\right)$
(iii)	For each $A\in K\left(ℳ\right)$ we have $A\subset \text{St}\left(F_2\right)\Rightarrow A\subset \text{St}\left(F_1\right)\text{.}$

		Proof.
	(i) $\Rightarrow$ (ii) Let $E\in F\left(ℳ\right)$ and $E\subset \text{St}\left(F_2\right)\text{.}$ So $F_2\subset \bar{E}\text{.}$ Then $F_1\subset {\bar{F}}_2\subset \bar{E},$ so $E\subset \text{St}\left(F_2\right)\text{.}$ (ii) $\Rightarrow$ (iii) is trivial. (iii) $\Rightarrow$(i) Let $M\in ℳ-{ℳ}_{F_1},$ By criterion (I.3.5) we have to show that $F_2\subset D_M\left(F_1\right)\text{.}$ Suppose not. Then $F_2\subset {\bar{D}}_M,$ where $D_M$ is the other halfspace. So $\text{St}\left(F_2\right)\cap D_M$ 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) $A\subset \text{St}\left(F_1\right),$ so $A\subset D_M\left(F_1\right),$ contradiction with $A\subset D_M\text{.}$ $\square$

(2.5) Lemma. Let $F\subset \bar{G},$ $F,G\in F\left(ℳ\right),$ then:

(i)	$L_F\subset L_G,$ so $\text{dim}\left(F\right)\le \text{dim}\left(G\right)\text{.}$
(ii)	$L_F=L_G\Rightarrow F=G\text{.}$
(iii)	$F\ne G\Rightarrow \text{dim}\left(F\right)<\text{dim}\left(G\right)\text{.}$

		Proof.
	(i) If $G\subset M$ then $F\subset M,$ so $L_F=\bigcap _{F\subset M}M\subset \bigcap _{G\subset M}M=L_G\text{.}$ (ii) If $L_F=L_G,$ then $F\subset M\iff G\subset M$ for all $M\in ℳ\text{.}$ And with (I.3.5): $F\not\subset M\iff G\not\subset M\iff G\subset D_M\left(F\right)\text{.}$ It follows by definition of facet that $F=G\text{.}$ (iii) follows from (i) and (ii). $\square$

(2.6) Proposition. Let $F_1,F_2\in F\left(ℳ\right)$ and ${\bar{F}}_1\cap {\bar{F}}_2\ne \varnothing \text{.}$ Then there is a facet $F\in F\left(ℳ\right)$ such that ${\bar{F}}_1\cap {\bar{F}}_2=\bar{F}\text{.}$

		Proof.
	${\bar{F}}_1\cap {\bar{F}}_2$ is a union of facets (since ${\bar{F}}_i$ is so (I.3.4)). Take $F\subset {\bar{F}}_1\cap {\bar{F}}_2$ with maximal dimension. Suppose $a\in {\bar{F}}_1\cap {\bar{F}}_2\text{.}$ To prove: $a\in \bar{F}\text{.}$ Choose $f\in F\text{.}$ If $a=f,$ then finished. If $a\ne f,$ then consider the line $l$ through $a$ and $f\text{.}$ Case 1: $l\cap L_F$ contains only $f\text{.}$ Since $\text{St}\left(F\right)$ is open (2.3), $\text{St}\left(F\right)\cap l$ is open in $l,$ so there is a $e\in [a,f)\cap \text{St}\left(F\right)\text{.}$ Let $E$ be the facet containing the point $e\text{.}$ Then $F\subset \bar{E},$ but $F\ne E,$ for $e\notin F\text{.}$ So by (2.5)(iii) $\text{dim}\left(E\right)>\text{dim}\left(F\right)\text{.}$ But since ${\bar{F}}_1\cap {\bar{F}}_2$ is convex, we have $e\in {\bar{F}}_1\cap {\bar{F}}_2,$ so $E\subset {\bar{F}}_1\cap {\bar{F}}_2\text{.}$ Hence $\text{dim}\left(E\right)\le \text{dim}\left(F\right),$ a contradiction. Case 2: $l\subset L_F\text{.}$ Since $F$ is open in $L_F,$ $l\cap F$ is an open interval on $l$ containing $f\text{.}$ If $(a,f]\subset F,$ then $a\in \bar{F}$ and we are ready. If $(x,f]\subset F$ and $x\notin F$ for some $x\in (a,f),$ then there is a $M\in ℳ,$ such that $x\in M$ and $f\notin M\text{.}$ So $a$ and $f$ are at different sides of $M,$ but ${\bar{F}}_1\cap {\bar{F}}_2$ is an intersection of hyperplanes in $ℳ$ and closed halfspaces. $\overline{D_M\left(f\right)}$ is one of them, so then $a\notin {\bar{F}}_1\cap {\bar{F}}_2,$ contradiction. $\square$

(2.7) Definition. Let $F_1,F_2\in F\left(ℳ\right)\text{.}$ If ${\bar{F}}_1\cap {\bar{F}}_2\ne \varnothing ,$ then we denote the unique facet $F\in F\left(ℳ\right)$ such that $\bar{F}={\bar{F}}_1\cap {\bar{F}}_2$ as exists by (2.6) by $F_1\wedge F_2\text{.}$ If $F_1$ and $F_2$ are such that there exists a $G\in F\left(ℳ\right)$ such that $\bar{G}\supset F_1\cup F_2,$ then by locally finiteness there are only finitely may such facets $G\text{.}$ Then by (2.6) thre is a facet $F\in F\left(ℳ\right)$ such that $\bar{F}=\bigcap _{\bar{G}\supset F_1\cup F_2}\bar{G}\text{.}$ We denote this facet by $F_1\vee F_2\text{.}$

Note that for $i=1,2,$ $F_1\wedge F_2\subset {\bar{F}}_i$ (respectively $F_i\subset \overline{F_1\vee F_2}\text{)}$ and this "meet" (respectively "join") is maximal (respectively minimal) with this property, if it exists.

(2.8) Theorem. If $F_1,F_2\in F\left(ℳ\right),$ then $\text{St}\left(F_1\right)\cap \text{St}\left(F_2\right)$ is nonempty, if and only if $F_1\vee F_2$ is defined and in this case $\text{St}\left(F_1\right)\cap \text{St}\left(F_2\right)=\text{St}(F_1\vee F_2)\text{.}$

		Proof.
	$\text{St}\left(F_1\right)\cap \text{St}\left(F_2\right)=\left(\bigcup _{\bar{G}\supset F_1}G\right)\cap \left(\bigcup _{\bar{G}\supset F_2}G\right)=\bigcup _{\bar{G}\supset F_1\cup F_2}G\underset{\text{(\$)}}{=}\bigcup _{\bar{G}\supset F_1\vee F_2}G≕\text{St}(F_1\vee F_2)\text{.}$ The equality ($) holds because $F_1\cup F_2\subset \bar{G}\iff \overline{F_1\vee F_2}\subset \bar{G}\iff F_1\vee F_2\subset \bar{G}\text{.}$ $\square$

(2.9) Definition. (cf. def. (II.4.8)) Let again $F\in F\left(ℳ\right)$ and define

$$\begin{array}{ccc}{Y}_F& ≔& Y(𝕍,𝕍,{ℳ}_F,p_{|{ℳ}_F})\text{and}\\ Y\left(F\right)& ≔& Y(𝕍,\text{St}\left(F\right),{ℳ}_F,p_{|{ℳ}_F})\end{array}$$

Note that $Y\left(F\right)=Y\cap {\text{Im}}^{-1}\left(\text{St}\left(F\right)\right)$ (or simply ${\text{Im}}^{-1}\left(\text{St}\left(F\right)\right)$ if $\text{Im}$ is considered as a map $Y\to H\text{).}$ Furthermore note that $Y\left(F\right)\subset Y\subset Y_F\text{.}$

(2.10) Proposition. (cf. prop. (II.4.9)). Let $F\in F\left(ℳ\right)\text{.}$ Then there exists a map $r:Y_F\to Y\left(F\right)$ with the properties:

(i)	The map $r$ is a homotopy inverse of the inclusion $i:Y\left(F\right)\subset Y_F\text{.}$
(ii)	For another $F\prime \in F\left(ℳ\right)$ such that $F\vee F\prime$ exists we have $r\left(Y\left(F\prime \right)\right)\subset Y\left(F\prime \right)\text{.}$

		Proof.
	Choose $f\in F$ and let $B$ be a closed ball around $f$ inside $\text{St}\left(F\right)$ in some metric $𝕍$ $\text{(}\text{St}\left(F\right)$ is open!). For each $y\in 𝕍$ there is a unique $r\left(y\right)\in B,$ such that $[f,r\left(y\right)]=[f,y]\cap B\text{.}$ Now $r$ is a continuous map $𝕍\to \text{St}\left(F\right)\text{.}$ 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 $f\in F\subset \overline{F\vee F\prime }\subset \overline{\text{St}\left(F\prime \right)}\text{.}$ $$\begin{array}{c} y f F B r(y) St(F) \\ \text{Figure 23.}\end{array}$$ $\square$

(2.11) Corollary. The spaces $Y\left(F\right)$ and $Y_F$ are homotopy equivalent.

(2.12) Corollary. Let $j$ be the inclusion $Y\left(F\right)\subset Y$ and $a\in Y\left(F\right),$ then the homomorphism $j_{*}:\pi (Y\left(F\right),a)\to \pi (Y,a)$ is injective.

(2.13) Lemma. Let $F_1,F_2\in F\left(ℳ\right)$ such that $F_1\vee F_2$ exists and let $a\in Y(F_1\vee F_2)\text{.}$ Then

$$\begin{array}{cc}\text{(2.14)}& \pi (Y\left(F_1\right),a)\cap \pi (Y\left(F_2\right),a)=\pi (Y(F_1\vee F_2),a)\end{array}$$

where these fundamental groups are seen as subgroups of $\pi (Y,a),$ 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). $\square$

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 $F_i\in F\left(ℳ\right)$ and ${\gamma }_i$ a curve in $Y\left(F_i\right)\text{.}$ Suppose ${\gamma }_1$ is homotopic to ${\gamma }_2$ in $Y\text{.}$ Then $F_1\vee F_2$ exists and the curves ${\gamma }_i$ are homotopic in $Y$ to a curve $\gamma$ in $Y(F_1\vee F_2)\text{.}$

		Proof.
	It is easy to see that we may restrict ourselves to the case that ${\gamma }_1$ and ${\gamma }_2$ are loops, say from a point $a\in Y(F_1\vee F_2)\text{.}$ From ${\gamma }_1{\simeq }_Y{\gamma }_2$ it follows that $\left\{{\gamma }_1\right\}=\left\{{\gamma }_2\right\}$ 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 $\left\{\gamma \right\}$ where $\gamma$ is a curve in $Y(F_1\vee F_2)\text{.}$ $\square$

(2.16) Definition. Let $F\in F\left(ℳ\right)\text{.}$ Define:

$$\begin{array}{ccc}{\text{Gal}}^F& ≔& \{{C_0}^{U_1}{C}_1\dots {}^{U_k}{C}_k\in \text{Gal}\hspace{0.17em}|\hspace{0.17em}F\subset {\bar{C}}_i\}\text{and}\\ {\Pi }^F& ≔& \{x\in \Pi \hspace{0.17em}|\hspace{0.17em}x\in \left\{\gamma \right\}\text{for some curve}\hspace{0.17em}\gamma \hspace{0.17em}\text{with}\hspace{0.17em}\gamma \left(t\right)\subset Y\left(F\right)\hspace{0.17em}\text{for}\hspace{0.17em}t\in [0,1]\}\end{array}$$

(2.17) Lemma. We have $\phi \left({\text{Gal}}^F\right)={\Pi }^F\text{.}$

		Proof.
	This is a simple inspection of the proof of (I.1.5). $\square$

(2.18) Corollary of (2.15). For $i=1,2$ let $F_i\in F\left(ℳ\right)\text{.}$ If $G_i\in {\text{Gal}}^{F_i}$ and $G_{\prime }\sim G_2,$ then $F_1\vee F_2$ exists and $G_i\sim G$ for some gallery $G\in {\text{Gal}}^{F_1\vee F_2}\text{.}$

		Proof.
	Let ${\gamma }_i$ be a curve in $Y\left(F_i\right)$ such that $\phi \left(G_i\right)=\left\{{\gamma }_i\right\}$ as exists by (2.17). From $G_1\sim G_2$ it follows that $\phi \left(G_1\right)=\phi \left(G_2\right),$ hence ${\gamma }_1\simeq {\gamma }_2\text{.}$ So by (2.15) there is a curve $\gamma$ in $Y(F_1\vee F_2)$ such that $\gamma \simeq {\gamma }_i\text{.}$ Then there is by (2.17) a $G\in {\text{Gal}}^{F_1\vee F_2}$ such that $\phi \left(G\right)=\left\{\gamma \right\}\text{.}$ Then $\phi \left(G\right)=\phi \left(G_i\right),$ hence $G\sim G_i\text{.}$ $\square$

(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 $F_i\in F\left(ℳ\right)$ for $i=1,2\text{.}$ With the help of (I.3.9) and (I.3.10) we have a map ${\text{pr}}_{F_1}^{-1}\circ {\text{pr}}_{F_1}:K\left(ℳ\right)\to K\left({ℳ}_F\right)\to \{A\in K\left(ℳ\right)\hspace{0.17em}|\hspace{0.17em}{F}_1\subset \bar{A}\}\text{.}$ This induces a map $ℛ:\text{Gal}\to {\text{Gal}}^{F_1},$ which respects gallery equivalence. (This map corresponds in a sense with the map $r$ in prop. (2.10)). Now let $G≔ℛ\left(G_2\right)\text{.}$ Then $F_1=ℛ\left(G_1\right)\sim ℛ\left(G_2\right)=G\text{.}$ $\square$

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 $\widehat{𝒰}$ of the universal covering space of $Y(𝕍,H,ℳ,p),$ where $(𝕍,H,ℳ,p)$ is a hyperplane system as before.

(3.1) Definition. We call a facet $F\in F\left(ℳ\right)$ minimal if it is so for the natural order relation i.e. if there is no facet $E\ne F$ contained in the closure $\bar{F}\text{.}$

(3.2) Definition. We call a hyperplane system central if $H=𝕍$ and $\bigcap _{M\in ℳ}M\ne \varnothing \text{.}$ 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 $F_{\text{min}}\text{.}$ Then since each facet contains at least one minimal facet, we have for all $F\in F\left(ℳ\right)$ that $F_{\text{min}}\subset \bar{F}\text{.}$ Hence $\text{St}\left(F_{\text{min}}\right)\subset \bar{F}\text{.}$ Hence $\text{St}\left(F_{\text{min}}\right)=H\text{.}$ The only hyperplanes, which meet $\text{St}\left(F_{\text{min}}\right)$ are the elements of ${ℳ}_{F_{\text{min}}},$ so $ℳ={ℳ}_{F_{\text{min}}}\text{.}$ By (2.11) we have $Y\simeq Y(𝕍,𝕍,ℳ,p)\text{.}$ 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:

$$\begin{array}{cc}\text{(3.4)}& F_{\text{min}}=\bigcap _{M\in ℳ}M\end{array}$$

(3.5) Definition. We agree that $F_{\text{min}}≔\varnothing$ in the non central case (this fits in with (3.4)!). We define the set of proper facets by:

$$F^{*}\left(ℳ\right)≔F\left(ℳ\right)-\left\{F_{\text{min}}\right\}$$

(hence $F^{*}\left(ℳ\right)=F\left(ℳ\right)$ 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 $F_{\text{min}}^{*}\left(ℳ\right)\text{.}$

(3.6) Definition. Let $p:\stackrel{\sim }{Y}\to Y$ be the universal covering. For $F\in F\left(ℳ\right)$ define: $𝒰\left(F\right)≔$ the collection of components of $p^{-1}\left(Y\left(F\right)\right)$ and:

$$𝒰≔\bigcup _{F\in F^{*}\left(ℳ\right)}𝒰\left(F\right),{𝒰}_{\text{min}}≔\bigcup _{F\in F_{\text{min}}^{*}\left(ℳ\right)}𝒰\left(F\right)$$

If the system is central, then $ℳ$ is finite, hence $p_{ℳ}≔\bigcup _{M\in ℳ}{p}_M$ is locally finite on $𝕍\text{.}$ For $A\in K\left(p_{ℳ}\right)$ we define: $𝒱\left(A\right)≔$ the collection of components of $p^{-1}\left({\text{Re}}^{-1}\left(A\right)\right)$ and $𝒱≔\bigcup _{A\in K\left(p_{ℳ}\right)}𝒱\left(A\right)$ and we agree that $𝒱≔\varnothing$ if the system is non central. Furthermore we define:

$$\begin{array}{cc}\widehat{𝒰}≔𝒰\cup 𝒱,& {\widehat{𝒰}}_{\text{min}}≔{𝒰}_{\text{min}}\cup 𝒱\\ \Sigma ≔\text{nerf}\left(𝒰\right),& {\Sigma }_{\text{min}}≔\text{nerf}\left({𝒰}_{\text{min}}\right)\\ \hat{\Sigma }≔\text{nerf}\left(\widehat{𝒰}\right),& {\hat{\Sigma }}_{\text{min}}≔\text{nerf}\left({\widehat{𝒰}}_{\text{min}}\right)\text{.}\end{array}$$

(3.7) Definition. We call a hyperplane system $\text{locally-}K(\pi ,1)$ if $(𝕍,𝕍,{ℳ}_F,p_{|{ℳ}_F})$ is a $K(\pi ,1)\text{-system}$ for each $F\in F^{*}\left(ℳ\right)\text{.}$ This means that $Y_F$ and hence (by (2.11)) $Y\left(F\right)$ are $K(\pi ,1)\text{'s.}$

(3.8) Example. A simple, simplicial system is $\text{locally-}K(\pi ,1)$ (central or non central). This follows from the theorem of Deligne (1.2) and the observation that the local system $(𝕍,𝕍,{ℳ}_F,p_{|{ℳ}_F})$ 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 $K(\pi ,1)$ (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 $\text{locally-}K(\pi ,1)\text{.}$

(3.10) Example. (The "triangle"). Let $𝕍=H$ be a plane and let $ℳ=\{l_1,l_2,l_3\}$ consist of three different lines with $l_i\cap l_j\ne \varnothing ,$ but $l_1\cap l_2\cap l_3=\varnothing$ (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 $K(\pi ,1)\text{.}$ So the system is $\text{locally-}K(\pi ,1)\text{.}$ But it is not $K(\pi ,1)!$ This is well known and can be seen by the method of §4 (see (4.17)).

$$\begin{array}{c} 1 1 2 2 3 3 P1 P2 P3 C D \\ \text{Figure 24.}\end{array}$$

(3.11) Theorem. Let $(𝕍,H,ℳ,p)$ be a hyperplane system which is $\text{locally-}K(\pi ,1)\text{.}$ Then $\widehat{𝒰}$ as defined in (3.6) is a simple covering of the universal covering space of $Y(𝕍,H,ℳ,p)\text{.}$ In particular this universal covering space is homotopy equivalent to the geometric realisation $\left|\hat{\Sigma }\right|$ of the abstract simplicial complex $\hat{\Sigma },$ the nerf of the cover $\widehat{𝒰}\text{.}$

		Proof.
	First we prove the following remark: If $U_i\in 𝒰\left(F_i\right)$ for $i=1,2$ and $U≔U_1\cap U_2\ne \varnothing ,$ then $F≔F_1\vee F_2$ exists and $U\in 𝒰\left(F\right),$ i.e. $𝒰$ is closed under intersections. Now $\text{St}\left(F_1\right)\cap \text{St}\left(F_2\right)\ne \varnothing ,$ because $U_1\cap U_2\ne \varnothing$ so it equals $\text{St}\left(F\right)$ (theorem (2.8)). Let $h=\text{Im}\circ p\text{.}$ Then we have $$\begin{array}{cc}\text{(3.12)}& h^{-1}\left(\text{St}\left(F\right)\right)=h^{-1}\left(\text{St}\left(F_1\right)\right)\cap h^{-1}\left(\text{St}\left(F_2\right)\right)\end{array}$$ The right hand side is a union of intersections of elements in $𝒰\left(F_1\right)$ and $𝒰\left(F_2\right)$ and each intersection is in turn a union of components. So, for instance $U_1\cap U_2$ is a union of elements in $𝒰\left(F\right)$ 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 $a,b\in U$ and let ${\stackrel{\sim }{\gamma }}_i$ be a curve in $U_i$ from $a$ to $b$ (each $U_i$ is connected). We have ${\stackrel{\sim }{\gamma }}_1\simeq {\stackrel{\sim }{\gamma }}_2$ in $\stackrel{\sim }{Y},$ because $\stackrel{\sim }{Y}$ is simply connected. So ${\gamma }_1\simeq {\gamma }_2$ in $Y,$ where ${\gamma }_i≔p\circ {\stackrel{\sim }{\gamma }}_i\text{.}$ Now ${\gamma }_i$ is in $Y\left(F_i\right)$ so by corollary (2.15) we know that ${\gamma }_i\simeq \gamma$ where $\gamma$ is a curve in $Y\left(F\right)\text{.}$ Let $\stackrel{\sim }{\gamma }$ be the unique lift of $\gamma$ to $\stackrel{\sim }{Y}$ with $\stackrel{\sim }{\gamma }\left(C\right)=a,$ then $\stackrel{\sim }{\gamma }\simeq {\stackrel{\sim }{\gamma }}_i$ in $Y,$ in particular $\stackrel{\sim }{\gamma }\left(1\right)=b\text{.}$ Furthermore $\stackrel{\sim }{\gamma }$ 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 $U\in 𝒰\left(F\right),$ $p_{|U}:U\to Y\left(F\right)$ is a universal covering. For instance the simply connectedness of $U$ follows from the fact that $Y\left(F\right)\subset Y$ induces an injection of fundamental groups (2.12). The $\text{locally-}K(\pi ,1)$ assumption ensures now that $Y\left(F\right)$ is a $K(\pi ,1)$ if $F\in F^{*}\left(ℳ\right)\text{.}$ 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 $\stackrel{\sim }{Y}$ and so this is then indeed a simple cover. In the central case we still have to cover the points of $\stackrel{\sim }{Y}$ with $\text{Im}\circ p\left(x\right)\in F_{\text{min}}\text{.}$ This can be done with the elements of $𝒱\text{.}$ Since these elements are mutually disjoint, we still have only to prove that $U\cap V$ is contractible (if nonempty) for $U\in 𝒰\left(F\right)$ and $V\in 𝒱\left(A\right),$ where $F\in F^{*}\left(ℳ\right)$ and $A\in K\left(p_{ℳ}\right)\text{.}$ We have: $$\begin{array}{cc}\text{(3.13)}& p^{-1}\left({\text{Re}}^{-1}\left(A\right)\right)\cap p^{-1}\left(Y\left(F\right)\right)=p^{-1}(A+\sqrt{-1}\text{St}\left(F\right))\end{array}$$ Argueing as after (3.12) we can conclude from (3.13) that $U\cap V$ 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 $U\cap V$ is connected. The only thing we have to do is to prove the following analog of corollary (2.15): Let ${\gamma }_1$ (respectively ${\gamma }_2\text{)}$ be a curve in ${\text{Re}}^{-1}\left(A\right)$ (respectively in $Y\left(F\right)={\text{Im}}^{-1}\left(\text{St}\left(F\right)\right),$ where we consider $\text{Im}:Y\to H\text{)}$ and suppose ${\gamma }_1\simeq {\gamma }_2$ in $Y\text{.}$ Then there is a curve $\gamma$ in ${\text{Re}}^{-1}\left(A\right)\cap {\text{Im}}^{-1}\left(\text{St}\left(F\right)\right)=A+\sqrt{-1}\text{St}\left(F\right)$ such that $\gamma \simeq {\gamma }_1\simeq {\gamma }_2$ in $Y\text{.}$ To prove this we define a curve $\gamma$ by $\gamma \left(t\right)≔\text{Re}\left({\gamma }_1\left(t\right)\right)+\sqrt{-1}\text{Im}\left({\gamma }_2\left(t\right)\right)\text{.}$ Let $G$ be a homotopy from ${\gamma }_1$ to ${\gamma }_2$ then $(s,t)\to (\text{Re}\left({\gamma }_1\left(t\right)\right),\text{Im}\left(G_s\left(t\right)\right))$ gives a homotopy from ${\gamma }_1$ to $\gamma \text{.}$ So $\gamma \simeq {\gamma }_2\simeq {\gamma }_1,$ as we wanted. So $U\cap V$ is indeed connected. $p_{|U\cap V}:U\cap V\to A+\sqrt{-1}\text{St}\left(F\right)$ is again a universal covering. The base space is convex, hence contractible (and $p_{|U\cap V}$ is even a homeomorphism). So $\widehat{𝒰}$ is indeed a simple open cover. As said before, the second statement of the theorem follows from Weil [Wei1952]. $\square$

(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 $\Delta ,$ which depends on the fact that the situation is central.

(3.15) Remark. We have the same result for the open cover ${\widehat{𝒰}}_{\text{min}}$ and the abstract simplicial complex ${\hat{\Sigma }}_{\text{min}},$ as easily can be seen. The space $\left|\hat{\Sigma }\right|$ looks like a "thickening" of the space $\left|{\hat{\Sigma }}_{\text{min}}\right|\text{.}$

§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 $C\in K\left(ℳ\right),$ a point $c\in C$ and a point $\stackrel{\sim }{c}\in \stackrel{\sim }{Y}$ above $\sqrt{-1}c\in Y,$ where $p:\stackrel{\sim }{Y}\to Y$ is the universal covering.

(4.1) Definition. For $F\in F\left(ℳ\right)$ define:

$${\text{Gal}}_{CF}≔\bigcup _{A\in K\left(ℳ\right),F\subset \bar{A}}{\text{Gal}}_{CA}$$

(Note that $F\subset \bar{A}\iff A\subset \text{St}\left(F\right)\text{)}$ and recall (2.16):

$${\text{Gal}}^F≔\{{C_0}^{U_1}{C}_1\dots {}^{U_k}{C}_k\in \text{Gal}\hspace{0.17em}|\hspace{0.17em}F\subset {\bar{C}}_i\}$$

(4.2) Definition. Let $G_i\in {\text{Gal}}_{C{F}_i}$ for $i=1,2\text{.}$ We write $(G_1,F_1)\subseteq (G_2,F_2)$ if

(i)	$F_1\subset {\bar{F}}_2$
(ii)	There is a $G\in {\text{Gal}}^{F_1}$ such that $G_2\sim G_{1}G$ (gallery equivalence (I.4.7))

Note that $\text{St}\left(F_2\right)\subset \text{St}\left(F_1\right)$ by (i), so ${\text{Gal}}_{C{F}_2}\subset {\text{Gal}}_{C{F}_1},$ so (ii) make sense. Note also that this relation $\subseteq$ is transitive.

$$\begin{array}{c} G G1 G2 St(F1) St(F2) \\ \text{Figure 25.}\end{array}$$

(4.3) Definition. If $G_1,G_2\in {\text{Gal}}_{CF}$ then we call $G_1$ and $G_2$ $F\text{-equivalent,}$ notation: $G_1{\sim }_F{G}_2,$ if $(G_1,F)\subseteq (G_2,F)$ (then automatically $(G_2,F)\subseteq (G_1,F)\text{)}$ i.e. if there is a $G\in {\text{Gal}}^F$ such that $G_2\sim G_{1}G\text{.}$ The $F\text{-equivalence}$ classes are denoted by ${\text{gal}}_{CF}\text{.}$ If $g\in {\text{gal}}_{CF},$ then $\text{end}\left(g\right)≔F\text{.}$ Note that for all $A\in K\left(ℳ\right)$ ${\text{gal}}_{CA}={\text{Gal}}_{CA}/\sim ,$ where $\sim$ is gallery equivalence. Furthermore $\text{gal}≔\bigcup _{F\in F^{*}\left(ℳ\right)}{\text{gal}}_{CF}\text{.}$ (note that this is a disjoint union).

(4.4) Definition. Let $g_i\in {\text{gal}}_{C{F}_i}$ for $i=1,2\text{.}$ Then $g_1\le g_2$ means that $(G_1,F_1)\subseteq (G_2,F_2),$ where $G_i$ is in the $F_i\text{-equivalence}$ class $g_i\text{.}$ It is easily checked that this indeed does not depend on the choice of the representatives.

(4.5) Proposition

(i)	The relation $\le$ is transitive.
(ii)	Suppose $g_2\in {\text{gal}}_{C{F}_2}$ and $F_1\subset {\bar{F}}_2,$ then there is a unique $g_1\in {\text{gal}}_{C{F}_1},$ such that $g_1\le g_2\text{.}$
(iii)	If $g_1,g_2\in \text{gal}$ and there exists a $g\in \text{gal}$ such that $g_i\le g,$ then there exists a unique $g_1\vee g_2$ such that $g_i\le g_1\vee g_2,$ which is the smallest, i.e. if $g_i\le g$ then $g_1\vee g_2\le g\text{.}$
(iv)	If $g_1,g_2\in \text{gal}$ and there exists a $g\in \text{gal}$ such that $g\le g_i$ then there exists a unique $g_1\wedge g_2$ such that $g_1\wedge g_2\le g_i,$ which is the biggest, i.e. if $g\le g_i$ then $g\le g_1\wedge g_2\text{.}$
(v)	If $g_1,g_2\in \text{gal}$ then $g_1\le g_2$ if and only if $g_1\supset g_2$ as sets.
(vi)	If $g_1,g_2\in \text{gal}$ then $g_1\cap g_2\ne \varnothing$ if and only if there is a $g\in \text{gal}$ such that $g_i\le $ g\text{.}$$

		Proof.
	(i) is trivial. (ii) Let $G_2\in g_2\text{.}$ Then $G_2\in {\text{Gal}}_{C{F}_2}\subset {\text{Gal}}_{C{F}_1},$ so $G_2$ also determines a $g_1\le g_2,$ $g_1\in {\text{gal}}_{C{F}_1}\text{.}$ If also $g_1^{\prime }\le g_2$ and $G_1\in g_1^{\prime },$ then $G_2\sim G_{1}G$ for some $G\in {\text{Gal}}_{C{F}_1},$ by the definition of $g_1^{\prime }\le g_2\text{.}$ But this says at the same time that $G_1{\sim }_{F_1}{G}_2,$ so $g_1=g_1^{\prime }\text{.}$ (iii) Suppose $g_i\in {\text{gal}}_{C{F}_i},$ $g\in {\text{gal}}_{CF}$ and $g_i\le g\text{.}$ Then $F_i\subset \bar{F},$ so $F_1\vee F_2\subset \bar{F}$ is defined. By (ii) there is a unique $g_1\vee g_2\in {\text{gal}}_{C,F_1\vee F_2}$ such that $g_1\vee g_2\le g\text{.}$ $F_i\subset \overline{F_1\vee F_2},$ so by (ii) there is a unique $g_i^{\prime }\in {\text{gal}}_{F_i},$ such that $g_i^{\prime }\le g_1\vee g_2\text{.}$ By (i) $g_i^{\prime }\le g$ and again by (ii) we have $g_i^{\prime }=g_i\text{.}$ We still have to prove that this construction of $g_1\vee g_2$ does not depend on the choice of $g,$ so suppose $h\in {\text{gal}}_{C,F_1\vee F_2}$ has also the property that $g_i\le h\text{.}$ Choose representatives $G_1,G_2,G$ and $H$ of $g_1,g_2,g$ and $h$ respectively and let $G_i^{\prime }$ (respectively $G_i^{\prime \prime }\text{)}$ be an element of ${\text{Gal}}^{F_i},$ such that $G\sim G_i·G_i^{\prime }$ (respectively $H\sim G_i·G_i^{\prime \prime }\text{),}$ as exists, since $g_i\le g_1\vee g_2$ (respectively $g_1\le h\text{).}$ Then $G·{\left(G_i^{\prime }\right)}^{-1}\sim H·{\left(G_i^{\prime \prime }\right)}^{-1},$ so we have $g^{-1}·G\sim {\left(G_i^{\prime \prime }\right)}^{-1}·G_i^{\prime }\in {\text{Gal}}^{F_i}\text{.}$ Then by corollary (2.18) we have ${\left(G_i^{\prime \prime }\right)}^{-1}·G_i^{\prime }\sim G\prime$ for some $g\prime \in {\text{Gal}}^{F_1\vee F_2}\text{.}$ Then $H^{-1}·G\sim G\prime ,$ hence $G{\sim }_{F_1\vee F_2}H,$ so $g_1\vee g_2=h\text{.}$ (iv) If $g\in \text{gal}$ there are only finitely many $h\in \text{gal}$ such that $h\le g,$ as follows from (ii) and (1.4). Therefore (iv) follows from (iii). (v) Suppose $g_i\in {\text{gal}}_{C{F}_i}$ and $g_1\le g_2\text{.}$ So $F_1\subset {\bar{F}}_2\text{.}$ Let $G_2\in g_2,$ then $G_2\in {\text{Gal}}_{C{F}_2}\subset {\text{Gal}}_{C{F}_1}\text{.}$ If $G_1\in g_1,$ then since $g_1\le g_2,$ there is a $G\in {\text{Gal}}^{F_1}$ such that $G_2\sim G_{1}G,$ so $G_1{\sim }_{F_1}{G}_2\text{.}$ Hence $G_2\in g_1\text{.}$ So $g_2\subset g_1\text{.}$ Now suppose $g_2\subset g_1\text{.}$ First we prove $F_1\subset {\bar{F}}_2\text{.}$ For each chamber $A\subset \text{St}\left(F_2\right)$ there is a $G_2\in g_2$ such that $A=\text{end}\left(G_2\right)\text{.}$ Since $g_2\subset g_1,$ we also have $A\subset \text{St}\left(F_2\right)\text{.}$ By lemma (2.4) it follows that $F_1\subset {\bar{F}}_2\text{.}$ Now let $G_i\in g_i,$ then $G_2\in g_1,$ so $G_2{\sim }_{F_1}{G}_1\text{.}$ Hence there is a $G\in {\text{Gal}}^{F_1}$ such that $G_2\sim G_{1}G\text{.}$ So $g_1\le g_2\text{.}$ (vi) If there is a $g\in \text{gal}$ such that $g_i\le g,$ then $g\subset g_i$ by (v), so $g_1\cap g_2\ne \varnothing \text{.}$ If $G\in g_1\cap g_2,$ then $\text{end}\left(G\right)\subset \text{St}\left(F_1\right)\cap \text{St}\left(F_2\right)\text{.}$ So $F_1\vee F_2$ exists and $G$ determines a $g\in {\text{gal}}_{C,F_1\vee F_2}$ such that $g_i\le g\text{.}$ $\square$

(4.6) Remark. $g_1$ and $g_2$ is not necessarily an element of ${\text{gal}}_{C,\text{end}\left(g_1\right)\wedge \text{end}\left(g_2\right)}\text{.}$

(4.7) Definition. We define a mapping $u:\left\{(G,F)\hspace{0.17em}\right|\hspace{0.17em}F\in F^{*}\left(ℳ\right),G\in {\text{Gal}}_{CF}\}\to 𝒰$ as follows: Let $\gamma$ be a curve in $Y$ such that $\phi \left(G\right)=\left\{\gamma \right\}\text{.}$ Then $\gamma \left(1\right)\in Y\left(F\right)\text{.}$ Let $\stackrel{\sim }{\gamma }$ be the unique lift with $\stackrel{\sim }{\gamma }\left(0\right)=\stackrel{\sim }{c}\text{.}$ Then $\stackrel{\sim }{\gamma }\left(1\right)$ is determined by $G$ and is contained in a unique $u(G,F)\in 𝒰\left(F\right)\text{.}$ In the central case we define also a mapping $v:{\text{Gal}}_{CC}\times K\left(p_{ℳ}\right)\to 𝒱$ as follows: Let $(G,A)\in {\text{Gal}}_{CC}\times K\left(p_{ℳ}\right)$ and suppose $\gamma$ is a loop in $Y$ such that $\phi \left(G\right)=\left\{\gamma \right\}\text{.}$ Then the unique lift of $\gamma [\sqrt{-1}c,\sqrt{-1}c+a],$ where $a\in A,$ that starts in $\stackrel{\sim }{c},$ has his endpoint in a uniquely determined $v(G,A)\in 𝒱\left(A\right)\text{.}$

(4.8) Lemma. Let $U_i≔u(G_i,F_i)$ for $i=1,2\text{.}$ Then $U_1\subset U_2$ if and only if $(G_2,F_2)\subseteq (G_1,F_1)\text{.}$

		Proof.
	Let $U_1\subset U_2,$ then clearly $\text{St}\left(F_1\right)\subset \text{St}\left(F_2\right),$ so $F_2\subset {\bar{F}}_1\text{.}$ Let ${\stackrel{\sim }{\gamma }}_i$ be as in definition (4.7). Let $\stackrel{\sim }{\gamma }$ be a curve in $U_2$ from ${\stackrel{\sim }{\gamma }}_1\left(1\right)$ to ${\stackrel{\sim }{\gamma }}_2\left(1\right)$ and $\gamma ≔p\circ \stackrel{\sim }{\gamma }\text{.}$ Then $\gamma$ is a curve in $Y\left(F_2\right),$ so by lemma (2.17) we have $\phi \left(G\right)=\left\{\gamma \right\}$ for some $G\in {\text{Gal}}^F\text{.}$ Now ${\gamma }_2={\gamma }_1\gamma$ so by lemma (2.17) we have $\phi \left(G\right)=\left\{\gamma \right\}$ for some $G\in {\text{Gal}}^F\text{.}$ Now ${\gamma }_2={\gamma }_1\gamma$ so by (I.4.10) $G_2\sim G_1·G\text{.}$ So indeed $(G_2,F_2)\subseteq (G_1,F_1)\text{.}$ The reverse implication can also easily be seen. $\square$

(4.9) Theorem

(i)	The mapping $u$ induces a bijection between $\text{gal}$ and $𝒰,$ the vertices of $\Sigma \text{.}$
(ii)	The mapping $v$ induces a bijection between ${\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ and $𝒱,$ the vertices of $\hat{\Sigma },$ which are not vertices of $\Sigma \text{.}$

		Proof.
	(i) Let $U\in 𝒰\left(F\right),$ $A\in K\left(ℳ\right),$ $A\subset \text{St}\left(F\right),$ $a\in A$ and $\stackrel{\sim }{a}$ a point in $U$ above $\sqrt{-1}a,$ $\stackrel{\sim }{\gamma }$ a curve from $\stackrel{\sim }{c}$ to $\stackrel{\sim }{a}$ and $\gamma =p\circ \stackrel{\sim }{\gamma }\text{.}$ According to (I.4.10) there is a $G\in {\text{Gal}}_{CA}\subset {\text{Gal}}_{CF},$ such that $\phi \left(G\right)=\left\{\gamma \right\}\text{.}$ Then $u(G,F)=U,$ so $u$ is surjective. From lemma (4.8) it follows that $u(G_1,F_1)=u(G_2,F_2)$ if and only if $F_1=F_2≕F$ and $G_1{\sim }_F{G}_2\text{.}$ Hence $u$ can be defined on $\text{gal}$ and is then injective. (ii) Let $V\in 𝒱\left(A\right),$ $A\in K\left(p_{ℳ}\right)\text{.}$ Let ${\stackrel{\sim }{\gamma }}_1$ be a curve from $\stackrel{\sim }{c}$ to a point in $V\text{.}$ Now ${\gamma }_1≔p\circ {\stackrel{\sim }{\gamma }}_1$ is a curve in $Y$ from $\sqrt{-1}c$ to a point ${\gamma }_1\left(1\right)\in {\text{Re}}^{-1}\left(A\right)\text{.}$ We may assume that ${\gamma }_1\left(1\right)=a+\sqrt{-1}c$ for some $a\in A\text{.}$ Then ${\gamma }_1[a+\sqrt{-1}c,\sqrt{-1}c]$ is a loop in $Y$ on $\sqrt{-1}c,$ so by (I.4.10) there is a $G_1\in {\text{Gal}}_{CC}$ such that $\phi \left(G_1\right)=\left\{{\gamma }_1[a+\sqrt{-1}c,\sqrt{-1}c]\right\},$ hence the surjectivity. It is also straightforward to see that $v(G_1,A_1)=v(G_2,A_2)$ if and only if $A_1=A_2$ and $G_1\sim G_2\text{.}$ Hence we have a bijection $v:{\text{gal}}_{CC}\times K\left(p_{ℳ}\right)\to 𝒱\text{.}$ $\square$

So from now on we can identify the vertices of $\Sigma$ with the set $\text{gal}\text{.}$ We proceed by describing the simplices of $\Sigma :$

(4.10) Theorem. A set $\{g_1,\dots ,g_k\},$ $g_i\in \text{gal}$ constitutes a simplex in $\Sigma$ if and only if there is a $g\in \text{gal}$ such that $g_i\le g$ for $i=1,\dots ,k\text{.}$

		Proof.
	Let $G_i\in {\text{Gal}}_{C{F}_i}$ for $i=1,\dots ,k$ and $U_i≔u(G_i,F_i)\text{.}$ Then by definition $\{U_1,\dots ,U_k\}$ constitutes a simplex if and only if $U≔U_1\cap \dots \cap U_k\ne \varnothing \text{.}$ Then $U\in 𝒰\left(F\right)$ for some $F\in F^{*}\left(ℳ\right),$ hence $U=u(G,F)$ for some $G\in {\text{Gal}}_{CF}\text{.}$ Since $U\subset U_i,$ it follows from lemma (4.8) that $(G_i,F_i)\subseteq (G,F)\text{.}$ $\square$

(4.11) Corollary. The simplicial complex $\Sigma$ is isomorphic to $\text{nerf}\left(\text{gal}\right)\text{.}$

		Proof.
	This follows from (4.10) with the help of proposition (4.5)(vi). $\square$

(4.12) Remark. If we define ${\text{gal}}_{\text{min}}≔\bigcup _{F\in F_{\text{min}}^{*}}{\text{gal}}_{CF},$ then it can be seen also, that the simplicial complex ${\Sigma }_{\text{min}}$ is isomorphic to $\text{nerf}\left({\text{gal}}_{\text{min}}\right)\text{.}$

(4.13) Definition. An element $g\in \text{gal}$ (or a representative $G\in g\text{)}$ as in theorem (4.10) is said to point to that simplex $\{g_1,\dots ,g_k\}\text{.}$ Let $g\in {\text{gal}}_{CF}$ and $\{F_1,\dots ,F_k\}$ a set of facets in $\bar{F}\text{.}$ Then by proposition (4.5)(ii) there is a uniquely determined $g_i\in {\text{gal}}_{F_i}$ with $g_i\le g$ and they constitute a simplex. In particular, if $\{F_1,\dots ,F_k\}$ is the set of all the facets in $\bar{F},$ then this simplex is denoted by $\sigma \left(g\right)\text{.}$ The set of all simplices to which $g$ points is exactly the set of all subsimplices of $\sigma \left(g\right)\text{.}$ The maximal simplices of $\Sigma$ are the simplices of the form $\sigma \left(g\right)$ for $g\in {\text{gal}}_{CA}$ for some $A\in K\left(ℳ\right)\text{.}$

(4.14) Definition. Let the system be central. For a vertex $V$ of $\hat{\Sigma },$ which is not a vertex of $\Sigma$ (i.e. an element of $𝒱\text{)}$ we denote by $\Sigma \left(V\right)$ the subcomplex of $\Sigma$ consisting of all simplices that form with $V$ a simplex of $\hat{\Sigma }\text{.}$ The corresponding subcomplex of $\hat{\Sigma }$ is denoted by $\hat{\Sigma }\left(V\right)\text{.}$ For $(g,A)\in {\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ we write $\Sigma (g,A)$ (respectively $\hat{\Sigma }(g,A)\text{)}$ instead of $\Sigma \left(v(g,A)\right)$ (resp. $\hat{\Sigma }\left(v(g,A)\right)\text{).}$ We define ${\Sigma }_{\text{min}}\left(V\right)$ and ${\hat{\Sigma }}_{\text{min}}\left(V\right)$ analogously.

In remark (6.21) we will see that $\Sigma (g,A)$ (resp. $\Sigma (g,A)\text{)}$ has the homotopy type of a sphere (respectively cell). Now we still have to finish our description of $\hat{\Sigma }$ in the central case:

(4.15) Theorem. Suppose that the hyperplane system is central and let $(g,A)\in {\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ and $G\in g\text{.}$ Then the galleries that point to simplices of $\Sigma (g,A)$ are equivalent to galleries of the form $G·G_d$ where $G_d$ is a direct replacing gallery of $C^{E}D$ with $D\in K\left(ℳ\right)$ and $E≔\text{pr}(p(C,D)\subset p_{ℳ})\left(A\right)\text{.}$

		Proof.
	Let $G_1≔G$ and let the curve ${\gamma }_1$ and the point $a\in A$ be as in the proof of (4.9)(ii). Let $G_2\in {\text{Gal}}_{CF}$ point to a simplex $\{U_1,\dots ,U_k\}$ of $\Sigma$ (definition (4.13)); suppose that this simplex forms a simplex in $\hat{\Sigma }$ with $V=v(G_1,A)\text{.}$ This means that if $U≔u(G_2,F)$ then $U\cap V\ne \varnothing \text{.}$ For we may suppose that, if $U_i\in 𝒰\left(F_i\right)$ then $F=F_1\vee \dots \vee F_k,$ hence $U=\bigcap _{i=1}^k{U}_i\text{.}$ Now $\phi \left(G_2\right)=\left\{{\gamma }_2\right\},$ where ${\gamma }_2$ is a curve in $Y$ from $\sqrt{-1}c$ to $\sqrt{-1}d,$ with $d\in D\in K\left(ℳ\right),$ $D\subset \text{St}\left(F\right)\text{.}$ Let $\stackrel{\sim }{\gamma }$ be a curve in $Y$ from $\stackrel{\sim }{c}$ to a point in $U\cap V$ above $a+\sqrt{-1}d$ and put $\gamma ≔p\circ \stackrel{\sim }{\gamma }\text{.}$ Now define the "straight" curves: $$\begin{array}{ccc}{\lambda }_1& ≔& [\sqrt{-1}c,a+\sqrt{-1}c]\\ {\lambda }_2& ≔& [a+\sqrt{-1}c,a+\sqrt{-1}d]\\ {\lambda }_3& ≔& [a+\sqrt{-1}d,\sqrt{-1}d]\end{array}$$ Then ?? $\simeq \gamma {\lambda }_2^{-1}$ and ${\gamma }_2\simeq \gamma {\lambda }_3\text{.}$ Hence $\phi \left(G_1\right)=\left\{\gamma {\lambda }_2^{-1}{\lambda }_1^{-1}\right\},$ $\phi \left(G_2\right)=\left\{\gamma {\lambda }_3\right\}\text{.}$ Now $\phi \left(G_1^{-1}{G}_2\right)=\left\{{\lambda }_1{\lambda }_2{\lambda }_3\right\}$ and by the definition of the functor $\phi$ (I.1.3) the latter equals $\phi \left(C^{E}D\right),$ where $E≔\text{pr}(p(C,D)\subset p_{ℳ})\left(A\right)\text{.}$ So we have $G_2\sim G_1·C^{E}D$ as has to be proven. $$\begin{array}{c} λ1 λ2 λ3 γ γ1 γ2 -1 -1c -1d a+-1c a+-1d Re-1(A) ?(F)=Im-1(St(F)) \\ \text{Figure 26.}\end{array}$$ $\square$

If moreover the hyperplane system is simple, then we have still another description of the extension $\Sigma \subset \hat{\Sigma }:$

(4.16) Theorem. Suppose we have a simple central hyperplane system. Then there is a bijection between ${\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ and ${\bigcup }_{A\in K\left(ℳ\right)}{\text{gal}}_{CA},$ the set of equivalence classes of galleries starting in $C\text{.}$ So the subcomplexes $\Sigma (g,A)$ can also be labelled by the latter. If $g\in {\text{gal}}_{CA}$ and $G\in g,$ then the galleries that point to simplices of $\Sigma \left(g\right)$ 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 $K\left(p_{ℳ}\right)=K\left(ℳ\right),$ so $A\subset K\left(ℳ\right)\text{.}$ Let $C,C_1,\dots ,C_n=A$ and $A=D_0,D_1,\dots ,D_k=D$ be direct Tits galleries and $E_1≔\text{pr}(ℳ(C,A)\subset ℳ)\left(A\right)$ and $E_2≔\text{pr}(ℳ(A,D)\subset ℳ)\left(A\right)$ Then $\varnothing \ne A\subset E\cap E_1\cap E_2,$ so $G_d{\sim }_R{C}^{E}D{\sim }_R{C}^{E_1}{A}^{E_2}D{\sim }_R{C}^{-}{C}_1^{-}\dots C_n·{D_0}^{+}{D_1}^{+}\dots D_k\text{.}$ (where ${\sim }_R$ denotes pregallery equivalence, see (I.1.4)). By proposition (I.4.8) the first and the last gallery are also gallery equivalent, so $G·G_d\sim G\prime ·P,$ where $P$ is the positive gallery ${D_0}^{+}{D}_1\dots {}^{+}{D}_k$ and $G\prime ≔G·C^{-}{C_1}^{-}\dots {}^{-}{C}_n\text{.}$ Now each gallery is equivalent to a gallery of this form and this induces the desired bijection between ${\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ and $\bigcup _{A\in K\left(ℳ\right)}{\text{gal}}_{CA}\text{.}$ $\square$

(4.17) Example. As an illustration let us determine the complex ${\hat{\Sigma }}_{\text{min}}$ of the hyperplane system "the triangle", example (3.10) (equals ${\Sigma }_{\text{min}}$ in this non central case). It is easily seen that the group ${\text{gal}}_{CC}$ (which is in general the fundamental group of $Y$ !) is isomorphic to ${\mathbb{Z}}^3,$ by the isomorphism $(n_1,n_2,n_3)\to {g_1}^{n_1}{g_2}^{n_2}{g_3}^{n_3},$ where $g_i$ is the equivalence class of the gallery $G_i≔i\hspace{0.17em}i$ (in the notation used in this kind of examples, see §(I.5)). For instance the commutation relation $G_1·G_2\sim G_2·G_1$ is illustrated by flip relations around the plinth $P_3:$

$$\begin{array}{ccccc} 1 1 2 2 & 1 1 2 2 & 1 1 2 2 & 1 1 2 2 & 1 1 2 2 \\ 1\hspace{0.17em}1\hspace{0.17em}2\hspace{0.17em}2& 1\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}2& 1\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}1& 2\hspace{0.17em}1\hspace{0.17em}2\hspace{0.17em}1& 2\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}1\\ \multicolumn{5}{c}{\text{Figure 27.}}\end{array}$$

The minimal facets are the points: $F_{\text{min}}^{*}\left(ℳ\right)=\{P_1,P_2,P_3\}\text{.}$ For each $g\in {\text{gal}}_{C{P}_i}$ there is a unique representative $G_i^n\in g\text{.}$ So the vertices of ${\Sigma }_{\text{min}}:{\text{gal}}_{\text{min}}=\bigcup _{i=1}^3{\text{gal}}_{C{P}_i}$ can be labelled by a disjoint union of three copies of $\mathbb{Z}:\mathbb{Z}\times \{1,2,3\}\text{.}$ The two-simplices $\{(n_1,1),(n_2,2),(n_3,3)\}$ are the maximal simplices of ${\Sigma }_{\text{min}}\text{.}$ The system is $\text{locally-}K(\pi ,1),$ so by theorem (3.11) or in fact remark (3.15) we have $\left|{\Sigma }_{\text{min}}\right|\simeq \stackrel{\sim }{Y}\text{.}$ Now we can see why the system is not $K(\pi ,1):$ Take a subcomplex $\Sigma \prime \subset {\Sigma }_{\text{min}}$ consisting of the simplices $\{(n_1,1),(n_2,2),(n_3,3)\}$ with $n_i\in \{a,b\},$ where $a$ and $b$ are two different integers. Then $\left|\Sigma \prime \right|$ is a 2-sphere (the surface of the platonic solid in figure 28). Since the simplicial complex $\left|{\Sigma }_{\text{min}}\right|$ is two dimensional $\left|\Sigma \prime \right|$ generates a nontrivial element in ${\pi }_2\left(\stackrel{\sim }{Y}\right)\text{.}$

$$\begin{array}{c} (a,1) (a,2) (a,3) (b,1) (b,2) (b,3) \\ \text{Figure 28.}\end{array}$$

§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 $\mathscr{H}$ a finite collection of subgroups. We define an abstract simplicial complex, called the cosetnerf of the group $G$ and the collection $\mathscr{H}$ as follows:

$$\text{cnerf}(G,\mathscr{H})≔\text{nerf}(\bigcup _{H\in \mathscr{H}}G/H)\text{.}$$

(5.2) Remark. Since $\varnothing \ne \left\{1\right\}\subset \bigcap _{H\in \mathscr{H}}H,$ the collection $\mathscr{H}$ itself is a (maximal) simplex of $\text{cnerf}(G,\mathscr{H})\text{.}$ This simplex serves as a fundamental domain for the natural action of $G$ on $\text{cnerf}(G,\mathscr{H})\text{.}$

(5.3) Lemma. the map $G\ni x\to \left\{xH\hspace{0.17em}\right|\hspace{0.17em}H\in \mathscr{H}\}$ induces a bijection between $G/\left(\bigcap _{H\in \mathscr{H}}H\right)$ and the set of maximal simplices of $\text{cnerf}(G,\mathscr{H})\text{.}$

		Proof.
	Straightforward. $\square$

(5.4) Lemma. If $G\subset G\prime ,$ then the space $\left|\text{cnerf}(G\prime ,\mathscr{H})\right|$ is homeomorphic to $\left|\text{cnerf}(G,\mathscr{H})\right|\times G\prime /G,$ where we give $G\prime /G$ the discrete topology. Hence $\left|\text{cnerf}(G\prime ,\mathscr{H})\right|$ is a disjoint union of spaces homeomorphic to $\left|\text{cnerf}(G,\mathscr{H})\right|\text{.}$

		Proof.
	Let $H\in \mathscr{H}\text{.}$ Then each left coset of $G$ in $G\prime$ is a disjoint union of left cosets of $H\text{.}$ So for each element of $G\prime /G$ we have a subcomplex of $\left|\text{cnerf}(G\prime ,\mathscr{H})\right|\text{.}$ The space $\left|\text{cnerf}(G\prime ,\mathscr{H})\right|$ 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 $\left|\text{cnerf}(G,\mathscr{H})\right|,$ as can be seen from the action of $G\prime$ on $\left|\text{cnerf}(G\prime ,\mathscr{H})\right|\text{.}$ $\square$

(5.5) Lemma. The space $\left|\text{cnerf}(G,\mathscr{H})\right|$ is connected if and only if $G$ is generated by $\bigcup _{H\in \mathscr{H}}H\text{.}$

		Proof.
	If $\left|\text{cnerf}(G,\mathscr{H})\right|$ is connected, then by (5.3) the index of the subgroup generated by the elements of $\mathscr{H}$ is one. Conversely, let $G$ be generated by $\bigcup _{H\in \mathscr{H}}H\text{.}$ Then an arbitrary vertex of $\text{nerf}(G,\mathscr{H})$ is of the form $xH,x\in H$ and $H\in \mathscr{H}\text{.}$ Write $x=a_1{a}_2\dots a_k,$ $a_i\in H_i\in \mathscr{H}$ for $i=1,\dots ,k$ and let $x_i≔a_i\dots a_i\text{.}$ Then, since $a_i{H}_i=x_i\dots x_i{H}_i=x_1\dots x_{i-1}{H}_i=a_{i-1}{H}_i$ the sequence $\left|\{xH,x_k{H}_k\}\right|,\left|\{x_{k-1}{H}_k,x_{k-1}{H}_{k-1}\}\right|,\dots ,\left|\{x_1{H}_2,x_1{H}_1\}\right|$ is a sequence of 1-simplices connecting the vertex $xH$ to a vertex of the fundamental simplex. $\square$

(5.6) Remark. It is convenient to restrict ourselves to collections $\mathscr{H},$ which have the following two properties:

(i)	$\bigcap _{H\in \mathscr{H}}H=\left\{1\right\}\text{.}$
(ii)	$G$ is generated by $\bigcup _{H\in \mathscr{H}}H\text{.}$

By lemma (5.4) we do not lose much generality by supposing (ii). If (ii) is supposed we will suppress $G$ in the notation: $\text{cnerf}\left(\mathscr{H}\right)\text{.}$

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

$$\begin{array}{cc}\text{(5.7)}& \sigma :G\to \left\{\text{maximal simplices of cnerf}\left(\mathscr{H}\right)\right\}\text{.}\end{array}$$

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

(5.8) Example. Let $(W,\{{\sigma }_1,\dots ,{\sigma }_{\ell }\})$ be a finite, affine or compact hyperbolic system i.e. the subgroups $W_i≔W_{\{1,\dots ,\ell \}-\left\{i\right\}}$ for $i=1,\dots ,\ell$ are finite. Then the collection $\mathscr{H}≔\left\{W_i\hspace{0.17em}\right|\hspace{0.17em}i\in \{1,\dots ,\ell \}\}$ satisfies (5.6) (i) and (ii). Now $\text{cnerf}\left(\mathscr{H}\right)$ is the well known Coxeter complex. Note that if $W$ is finite, then $\left|\text{cnerf}\left(\mathscr{H}\right)\right|$ is homeomorphic to the $\ell -1\text{-sphere.}$

(5.9) Theorem. Suppose a simple hyperplane system of a linear Coxeter group $W$ is given. Let $(A_W,\{s_1,\dots ,s_{\ell }\})$ be the associated Artin system. Then:

(i)

The simplicial complex $\Sigma$ of the hyperplane system as defined in (3.6) is isomorphic to the cosetnerf of the collection of subgroups of $A_W$ of the form $A_{W_J},$ where $J⫋\{1,\dots ,\ell \}$ and $W_J$ 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 $A_W$ to the vertices of $\hat{\Sigma },$ which are not vertices of $\Sigma \text{.}$ If $x\in A_W,$ then the maximal simplices of the subcomplex $\Sigma \left(m\left(x\right)\right)$ of $\Sigma$ are the simplices of the form: $$\sigma \left(xw\right),\hspace{0.17em}w\in W,$$ where $W$ has to be interpreted as a subset of $A_W$ by remark (II.3.19).

		Proof.
	The fact that the collection of subgroups of $A_W$ as in (i) satisfies (5.61)(i) follows from theorem (II.4.13)(ii). We have a surjective map $f:\text{Gal}\to A_W$ as the composite $\text{Gal}\to \text{Gal}/W\underset{\phi /W}{\to }\pi /W\underset{\cong }{\to }{A}_W,$ 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 $\text{gal}\ni g\to f\left(g\right)\subset A_W$ induces a bijection between $\text{gal}$ and the cosets of the subgroups of $A_W$ mentioned in (i). For, if $G_1\in g,$ $g\in {\text{gal}}_{CF},$ then the elements of $g$ are equivalent to galleries of the form $G_1{G}_2$ with $G_2\in {\text{Gal}}^F\text{.}$ After dividing out the action of $W,$ we can suppose $F\subset \bar{C},$ then $f\left({\text{Gal}}^F\right)=A_{W_J},$ where $J⫋\{1,\dots ,\ell \}$ is of the type $F,$ i.e. $J=\left\{j\hspace{0.17em}\right|\hspace{0.17em}{\varphi }_j\left(F\right)=\left\{0\right\}\}\text{.}$ $W_J$ is finite because $F\subset \stackrel{\circ }{I}$ and $J$ is a proper subset, because $F\in F^{*}\left(ℳ\right)\text{.}$ Now $f\left(g\right)$ is the coset $f\left(G_1\right)A_{W_J}\text{.}$ Conversely, if $W_J$ is finite, but $J\ne \{1,\dots ,\ell \}$ and $x\in A_W,$ it is easy to see that $f^{-1}\left(x{A}_{W_J}\right)\in {\text{gal}}_{CF},$ for some proper facet $F$ determined by $\pi \left(x\right)\in W$ and $W_J\text{.}$ (i) now follows from corollary (4.11). (ii) follows from (4.16). $\square$

(5.10) Definition. Let $A_{\stackrel{\sim }{W}}$ be an extended Artin group of rank $\ell$ as in definition (III.2.13). For $J\subset \{1,\dots ,\ell \}$ we define the following subgroups:

$$\begin{array}{ccc}{A}_{{\stackrel{\sim }{W}}_J}& ≔& \text{the group generated by}\hspace{0.17em}{s}_j,t_j,\hspace{0.17em}j\in J\text{.}\\ {\pi }^{-1}{\left(W\right)}_J& ≔& \text{the group generated by}\hspace{0.17em}{s}_j^{\left(n\right)},\hspace{0.17em}j\in J\hspace{0.17em}\text{and}\hspace{0.17em}n\in \mathbb{Z}\text{.}\end{array}$$

(5.11) Remarks. (i) We have ${\pi }^{-1}{\left(W\right)}_J\subset {\pi }^{-1}\left(W_J\right),$ but the other inclusion need not hold (take for instance $J=\varnothing \text{).}$ What we do have is:

$${\pi }^{-1}{\left(W\right)}_J={\left({\pi }_{|A_{{\stackrel{\sim }{W}}_J}}\right)}^{-1}\left(W_J\right)=A_{{\stackrel{\sim }{W}}_J}{\pi }^{-1}\left(W_J\right)\text{.}$$

(ii) Analogous to theorem (II.4.13) it can be proved that the flip relations form a complete set of relations for ${\pi }^{-1}{\left(W\right)}_J$ and also:

$$\begin{array}{cc}\text{(5.12)}& {\pi }^{-1}{\left(W\right)}_{J_1}\cap {\pi }^{-1}{\left(W\right)}_{J_2}={\pi }^{-1}{\left(W\right)}_{J_1\cap J_2}\text{.}\end{array}$$

(iii) From (ii) it can be seen that (as the notation already suggests) $(A_{{\stackrel{\sim }{W}}_J},{\{s_j,t_j\}}_{j\in J})$ forms an extended Artin system and we have $A_{{\stackrel{\sim }{W}}_{J_1}}\cap A_{{\stackrel{\sim }{W}}_{J_2}}=A_{{\stackrel{\sim }{W}}_{J_1\cap J_2}}\text{.}$

(5.13) Theorem. Suppose a hyperplane system of an extended Coxeter group of rank $\ell$ is given. Let $A_{\stackrel{\sim }{W}}$ be the associated extended Artin group of $\ell \text{.}$ Then:

(i)

The simplicial complex $\Sigma$ of the hyperplane system is isomorphic to the cosetnerf of the collection subgroups of ${\pi }^{-1}\left(W\right)$ of the form ${\pi }^{-1}{\left(W\right)}_J$ where $J⫋\{1,\dots ,\ell \}$ and $W_J$ 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 $A_{\stackrel{\sim }{W}}$ to the vertices of $\hat{\Sigma }$ which are not vertices of $\Sigma \text{.}$ If $x·e^q\in A_{\stackrel{\sim }{W}}$ is the unique way to write an element of $A_{\stackrel{\sim }{W}}$ as product of $x\in {\pi }^{-1}\left(W\right)$ and $q\in Q,$ then the maximal simplices of the subcomplex $\Sigma \left(m\left(x{e}^q\right)\right)$ are the simplices of the form $$\sigma \left(x{\lambda }_q\left(w\right)\right),\hspace{0.17em}w\in W$$ where $W$ has to be interpreted as a subset of ${\pi }^{-1}\left(W\right),$ which can be done by remark (II.3.19) and lemma (III.4.1) and $\lambda$ is the pseudo conjugation the action of $Q$ on the set ${\pi }^{-1}\left(W\right)$ 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 $A_{\stackrel{\sim }{W}}$ and ${\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ and prove (ii) with the help of (4.15). But let us give a more direct and instructive proof: Let $V\in 𝒱\left(A\right)$ be a vertex of $\hat{\Sigma },$ which is not a vertex of $\Sigma \text{.}$ Here $A=w(q+U_0)\in K\left(p_{ℳ}\right)$ for unique $w\in W,$ $q\in Q$ and $U_0$ as in the proof of (III.3.5)(iii). $$\begin{array}{c} λ1 λ2 λ3 λ4 γ -1c -1d a+-1d w(q+-1c) w(q)+-1d Re-1(A) Im-1(D) \\ \text{Figure 29.}\end{array}$$ A maximal simplex of $\Sigma$ is determined by a $U\in 𝒰\left(n\right),$ $D=w\prime \left(C\right)\in K\left(ℳ\right)$ for a unique $w\prime \in W\text{.}$ This simplex forms with $V$ a simplex in $\hat{\Sigma }$ (i.e. is a simplex of $\Sigma \left(V\right)\text{)}$ iff $U\cap V\ne \varnothing \text{.}$ We may assume, that there is a point in this inter¬section, which projects (by $\stackrel{\sim }{Y}\to Y\text{)}$ on $a+\sqrt{-1}d\in {\text{Re}}^{-1}\left(A\right)\cap {\text{Im}}^{-1}\left(D\right)\subset Y,$ where $a≔w(q+c)\in w(q+U_0)=A$ and $d≔w\prime \left(c\right)\in D$ (recall that $c\in U_0\subset C\text{).}$ Let $\stackrel{\sim }{\gamma }$ be a curve in $\stackrel{\sim }{Y}$ from $\stackrel{\sim }{c}$ to this point in $U\cap V\text{.}$ Then $\gamma ≔p\circ \stackrel{\sim }{\gamma }$ is a curve in $Y$ from $\sqrt{-1}c$ to $a+\sqrt{-1}d\text{.}$ Define the "straight" curves ${\lambda }_1,\dots ,{\lambda }_4$ as follows $$\begin{array}{ccc}{\lambda }_1& ≔& [w\left(q+\sqrt{-1}c\right),w(q+c)]=[w(q+\sqrt{-1}c),a],\\ {\lambda }_2& ≔& [w(q+c),w(q+c)+\sqrt{-1}w\prime \left(c\right)]=[a,a+\sqrt{-1}d],\\ {\lambda }_3& ≔& [w(q+c)+\sqrt{-1}w\prime \left(c\right),w\left(q\right)+\sqrt{-1}w\prime \left(c\right)]=[a+\sqrt{-1}d,w\left(q\right)+\sqrt{-1}d],\\ {\lambda }_4& ≔& [w\left(q\right)+\sqrt{-1}w\prime \left(c\right)]=[w\left(q\right)+\sqrt{-1}d,\sqrt{-1}d]\text{.}\end{array}$$ Note that ${\lambda }_2\subset {\text{Re}}^{-1}\left(A\right)$ and ${\lambda }_3,{\lambda }_4\subset {\text{Im}}^{-1}\left(D\right)\text{.}$ Now the curve $\gamma {\lambda }_2^{-1}{\lambda }_1^{-1}$ projects on a loop in $X$ representing an element in $A_{\stackrel{\sim }{W}}=\pi (X,*)$ of the form $x·e^q,$ $x\in {\pi }^{-1}\left(W\right),$ $q\in Q$ as above. Conversely each element in $A_{\stackrel{\sim }{W}}$ determines a curve in $Y$ from $\sqrt{-1}c$ to a point of the form $w(q+\sqrt{-1}c)\text{.}$ Composing this curve with ${\lambda }_1$ and lifting to $\stackrel{\sim }{Y}$ the endpoint lies in a $V\in 𝒱\text{.}$ This way we get the desired bijection $m:A_{\stackrel{\sim }{W}}\to 𝒱\text{.}$ The curve $\gamma {\lambda }_3{\lambda }_4$ projects on a loop representing an element $y$ of ${\pi }^{-1}\left(W\right),$ such that $\pi \left(y\right)=w\prime \text{.}$ This describes the bijection between ${\pi }^{-1}\left(W\right)$ and the maximal simplices of $\Sigma$ given by (i). Since $\gamma {\lambda }_3{\lambda }_4\simeq \gamma {\lambda }_2^{-1}{\lambda }_1^{-1}{\lambda }_1{\lambda }_2{\lambda }_3{\lambda }_4$ we have $$\begin{array}{cc}\text{(\$)}& y=x·e^q·z,\end{array}$$ for some $z\in A_{\stackrel{\sim }{W}},$ represented by the projection of ${\lambda }_1{\lambda }_2{\lambda }_3{\lambda }_4\text{.}$ To analyse this element $z,$ we note that ${\lambda }_1{\lambda }_2{\lambda }_3$ projects on the same loop in $X$ as ${\lambda }_1^{\prime }{\lambda }_2^{\prime }{\lambda }_3^{\prime },$ where $$\begin{array}{ccc}{\lambda }_1^{\prime }& ≔& [\sqrt{-1}c,c],\\ {\lambda }_2^{\prime }& ≔& [c,c+\sqrt{-1}{w}^{-1}w\prime \left(c\right)],\\ {\lambda }_3^{\prime }& ≔& [c+\sqrt{-1}{w}^{-1}w\prime \left(c\right),\sqrt{-1}{w}^{-1}w\prime \left(c\right)],\end{array}$$ the transforms of ${\lambda }_i$ under $e^{-q}·w^{-1}\in \stackrel{\sim }{W}\text{.}$ Inspection of these formulas gives that ${\lambda }_1^{\prime }{\lambda }_2^{\prime }{\lambda }_3^{\prime }$ projects on a loop representing the element $w^{-1}w\prime \in W\subset {\pi }^{-1}\left(W\right)\text{.}$ The curve ${\lambda }_4$ projects on the same loop as its transform under ${\left(w\prime \right)}^{-1}·e^{-w\left(q\right)}\in \stackrel{\sim }{W}:[\sqrt{-1}c,{\left(w\prime \right)}^{-1}(-w\left(q\right))+\sqrt{-1}c],$ so it projects on a loop representing ${\left(w\prime \right)}^{-1}\left(w(-q)\right)\in Q\subset A_{\stackrel{\sim }{W}}\text{.}$ we conclude that $z=w^{-1}w\prime ·e^{{\left(w\prime \right)}^{-1}w(-q)}\text{.}$ The reasoning has shown that if $\sigma \left(y\right)$ is a simplex of $\Sigma \left(m(x·e^q)\right),$ then $y=x·e^q·z=x·e^q·w^{\prime \prime }·e^{{\left(w^{\prime \prime }\right)}^{-1}(-q)}≕x·{\lambda }_q\left(w^{\prime \prime }\right),$ where $w^{\prime \prime }≔w^{-1}w\prime \text{.}$ Conversely, it can be checked that $\sigma (x·{\lambda }_q\left(w\right))$ is a simplex of $\Sigma \left(m(x·e^q)\right)$ for all $x\in {\pi }^{-1}\left(W\right),$ $q\in Q$ and $w\in W\text{.}$ $\square$

(5.14) Remark. In the situation of a linear Coxeter group as well the situation of an extended Coxeter group the simplicial complex ${\hat{\Sigma }}_{\text{min}}$ can be described by adding to the conditions on $J$ in (5.9)(i) (respectively (5.13)(i), namely that $J⫋\{1,\dots ,\ell \}$ and $W_J$ finite, that $J$ is maximal with respect to these properties.

(5.15) Remark. In the case that $W$ is finite the sets $J\subset \{1,\dots ,\ell \}$ with the properties mentioned in (5.14) are precisely the sets $J$ with $\left|J\right|=\ell -1,$ hence of the form $\{1,\dots ,\ell \}-\left\{j\right\}$ for a $j\in \{1,\dots ,\ell \}\text{.}$ Hence for a given $x\in A_W$ (respectively $x·e^q\in A_{\stackrel{\sim }{W}}\text{)}$ we have a bijection between the maximal simplices of the Coxeter complex of $W$ (example (5.8)) and the maximal simplices of ${\Sigma }_{\text{min}}\left(m\left(x\right)\right)$ (respectively ${\Sigma }_{\text{min}}\left(m(x·e^q)\right)\text{).}$ This bijection induces a homeomorphism between the $\ell -1\text{-sphere}$ and $\left|{\Sigma }_{\text{min}}\left(m\left(x\right)\right)\right|$ (respectively $\left|{\Sigma }_{\text{min}}\left(m(x·e^q)\right)\right|\text{).}$

§6. The Contractibility Problem

If the space $\left|\hat{\Sigma }\right|$ is contractible and the system is $\text{locally-}K(\pi ,1),$ then the system is $K(\pi ,1),$ as follows from theorem (3.11). To investigate the question of contractibility of $\left|\hat{\Sigma }\right|$ we construct a space $\hat{Z}=\hat{Z}(𝕍,H,ℳ,p)$ in terms of facets and galleries, which turns out to be homotopy equivalent to $\left|\hat{\Sigma }\right|,$ as follows:

(6.1) Definition. Let $S≔H-F_{\text{min}}\text{.}$ Note that in the non central case $S=H$ and in the central case $S=𝕍-\bigcap _{M\in ℳ}M,$ which has the homotopy type of a sphere, hence the use of the symbol $S\text{.}$

(6.2) Definition. For $F\in F^{*}\left(ℳ\right)$ we define $\hat{\bar{F}}≔\bar{F}-F_{\text{min}}$ (hence, by (3.5) $≔\bar{F}$ in the non central case).

(6.3) Definition. We first define the sets:

$$\begin{array}{ccc}Z\prime & ≔& \{(x,g)\in S\times \text{gal}\hspace{0.17em}|\hspace{0.17em}x\in \widehat{\bar{\text{end}\left(g\right)}}\}\text{and for}g\in \text{gal}:\\ {\bar{Z}}_g^{\prime }& ≔& \widehat{\bar{\text{end}\left(g\right)}}\times \left\{g\right\}\text{.}\end{array}$$

We give ${\bar{Z}}_g^{\prime }$ the topology of $\widehat{\bar{\text{end}\left(g\right)}}\subset 𝕍$ and we give $Z\prime$ the topology of the disjoint union of the spaces ${\bar{Z}}_g^{\prime }$ (Note that the set $Z\prime$ is a disjoint union of the sets ${\bar{Z}}_g^{\prime },$ $g\in \text{gal}\text{).}$ Furthermore for $g\in \text{gal}:$ $Z_g^{\prime }≔\text{end}\left(g\right)\times \left\{g\right\}\text{.}$ Note that this fits in with the notation, because ${\bar{Z}}_g^{\prime }$ is indeed the closure of $Z_g^{\prime }$ in $Z\prime \text{.}$ Now we come to the definition of the space $Z:$

(6.4) Definition. We define an equivalence relation on $Z\prime$ as follows: If $(x_i,g_i)\in Z\prime$ for $i=1,2,$ then $(x_1,g_1)\sim (x_2,g_2)$ if $x_1=x_2$ and if $F\subset \overline{\text{end}\left(g_i\right)}$ is the facet containing $x_i$ and $g_i^{\prime }$ the unique element of ${\text{gal}}_{CF}$ (see (4.5)(ii)) such that $g_i^{\prime }\le g_i,$ then $g_1^{\prime }=g_2^{\prime }\text{.}$

Let $Z$ denotes the quotient space $Z\prime /\sim$ and $b:Z\prime \to Z$ the projection.

(6.5) Remark. From the definition it follows that each equivalence class contains a unique element $(x,g)$ with the property that $x\in \text{end}\left(g\right)\text{.}$ So $Z=\bigcup _{g\in \text{gal}}{Z}_g,$ where $Z_g≔b\left(Z_g^{\prime }\right)\text{.}$ This is a disjoint union. The sets $Z_g$ are the facets of the space $Z\text{.}$

(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 $\bar{A}$ for some chamber $A$ we could have restricted ourselves to elements $g$ such that $\text{end}\left(g\right)\in K\left(ℳ\right)$ (and $g$ is then nothing but an equivalence class of galleries). I.e. $Z$ can also be considered as a quotient space $\bigcup _{g\in {\text{gal}}_{CA}},$ $A\in K\left(ℳ\right)$ ${\bar{Z}}_g^{\prime }\text{.}$

$$\begin{array}{c} 0 1 q ℝ Z \\ H=𝕍=ℝ\hspace{0.17em}\text{and}\hspace{0.17em}ℳ=\{0,1\}\\ \text{Figure 30.}\end{array}$$

(6.7) Definition. Let $g\in \text{gal}\text{.}$ We define the star of the facet $Z_g$ (and denote it by Str to avoid confusion with the notations St and Star) by: $\text{Str}\left(g\right)≔\bigcup _{g\le h}{Z}_h$

(6.8) Proposition. $\text{Str}\left(g\right)$ is open.

		Proof.
	Let $k\in \text{gal}\text{.}$ It suffices to show that $$\begin{array}{cc}\text{(6.9)}& {\bar{Z}}_k^{\prime }\cap b^{-1}\left(\text{Str}\left(g\right)\right)=\{(x,k)\in {\bar{Z}}_k^{\prime }\hspace{0.17em}|\hspace{0.17em}x\in \text{St}\left(\text{end}\left(g\right)\right)\}\end{array}$$ if the l.h.s. is nonempty, because the r.h.s. of (6.9) is open in ${\bar{Z}}_k^{\prime }\text{.}$ Let $x\in \widehat{\bar{\text{end}\left(k\right)}}$ and $b(x,k)\in \text{Str}\left(g\right)\text{.}$ By the definition of $\text{Str}\left(g\right)$ (6.7) we have $b(x,k)\in Z_h$ for some $h\in \text{gal}$ with $g\le h\text{.}$ Then $(x,k)\sim (x,h)$ and $x\in \text{end}\left(h\right),$ which entails $h\le k\text{.}$ So first of all we see that, if the l.h.s. of (6.9) is nonempty, then $g\le k\text{.}$ Furthermore from $g\le h$ it follows that $\text{end}\left(g\right)\subset \overline{\text{end}\left(h\right)},$ so $x\in \text{St}\left(\text{end}\left(g\right)\right),$ 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 $g\le k\text{)}$ and prove that the r.h.s. is contained in the l.h.s.: Suppose $(x,k)\in {\bar{Z}}_k^{\prime }$ and $x\in \text{St}\left(\text{end}\left(g\right)\right)\text{.}$ Then $x\in F$ for some facet $F$ with $\text{end}\left(g\right)\subset \bar{F}\text{.}$ Furthermore $F\subset \overline{\text{end}\left(k\right)},$ so there is a unique $h\in {\text{gal}}_{CF},$ such that $h\le k\text{.}$ $\text{end}\left(g\right)\subset \bar{F},$ so there is a unique element $g\prime \in {\text{gal}}_{C,\text{end}\left(g\right)}$ such that $g\prime \le h\text{.}$ So $g\prime \le k,$ but we also know that $g\le k,$ so $g=g\prime$ and hence $g\le h\text{.}$ From $b(x,k)=b(x,h)\in Z_h$ and $g\le h,$ we conclude that $b(x,k)\in \text{Str}\left(g\right)\text{.}$ Hence $(x,k)$ is element of the r.h.s. of (6.9). $\square$

(6.10) Proposition. Let $g\in \text{gal}\text{.}$ Then:

(i)	$b\left({\bar{Z}}_g^{\prime }\right)=\bigcup _{h\le g}{Z}_h$
(ii)	$b\left({\bar{Z}}_g^{\prime }\right)$ is closed in $Z,$ hence ${\bar{Z}}_g=b\left({\bar{Z}}_g^{\prime }\right)\text{.}$

		Proof.
	(i) is clear from def. (6.4) and remark (6.5). (ii) Suppose $k\in \text{gal}$ is cuh that $Z_k\cap b\left({\bar{Z}}_g^{\prime }\right)=\varnothing ,$ then by (i): $k\nleqq g\text{.}$ We show that $\text{Str}\left(k\right)\cap b\left({\bar{Z}}_g^{\prime }\right)=\varnothing ,$ which proves (ii): Suppose $Z_k\subset \text{Str}\left(k\right)\cap b\left({\bar{Z}}_g^{\prime }\right),$ then $k\le h$ and $h\le g,$ hence $k\le g,$ contradiction. $\square$

(6.11) Remark. We have $Z=\bigcup _{g\in \text{gal}}{\bar{Z}}_g$ and a map defined on $Z$ is continuous, if and only if the restrictions to ${\bar{Z}}_g,$ $g\in \text{gal}$ are continuous. If we wish we even can restrict ourselves to $g\in {\text{gal}}_{CA}$ with $A\in K\left(ℳ\right)\text{.}$

(6.12) Proposition. Let $g_1,g_2\in \text{gal}\text{.}$ Then $\text{Str}\left(g_1\right)\cap \text{Str}\left(g_2\right)$ is nonempty if and only if $g_1\vee g_2$ exists and then it equals $\text{Str}(g_1\vee g_2)\text{.}$

		Proof.
	If nonempty, then there is a $Z_g$ contained in the intersection and then $g_i\le g$ $i=1,2\text{.}$ So then $g_1\vee g_2$ exists. And conversely. $\text{Str}\left(g_1\right)\cap \text{Str}\left(g_2\right)=\bigcup _{g_i\le h}{Z}_h=\bigcup _{g_1\vee g_2\le h}{Z}_h≕\text{Str}(g_1\vee g_2)\text{.}$ $\square$

(6.13) Proposition. Let $g\in \text{gal}\text{.}$ Then $\text{Str}\left(g\right)$ is contractible.

		Proof.
	Choose a point $x_0\in \text{end}\left(g\right)\text{.}$ Then $b(x_0,g)\in Z_g\subset \text{Str}\left(g\right)$ and we are going to contract $\text{Str}\left(g\right)$ on $b(x_0,g)\text{.}$ Let $k\in \text{gal}$ and suppose, that the left hand side of (6.9) is nonempty (i.e. $g\le k\text{).}$ By the equality (6.9) we see that we can define a continuous map ${\varphi }_k:A\times [0,1]\to A$ by ${\varphi }_k((x,k),t)≔(t{x}_0+(1-t)x,k),$ where $A$ is the set in (6.9). It is not difficult to check, that $(x_1,k_1)\sim (x_2,k_2)$ then ${\varphi }_{k_1}((x_1,k_1),t)\sim \left({\varphi }_k((x_2,k_2),t)\right)\text{.}$ So we have a continuous map $\varphi :\text{Str}\left(g\right)\times [0,1]\to \text{Str}\left(g\right),$ which is the desired contraction on $b(x_0,g)\text{.}$ $\square$

(6.14) Theorem. The space $Z$ is homotopy equivalent to the space $\left|\Sigma \right|\text{.}$

		Proof.
	It follows from (6.8) and (6.12) that ${\left\{\text{Str}\left(g\right)\right\}}_{g\in \text{gal}}$ is an open cover of $Z,$ whose nerf is isomorphic to $\Sigma \text{.}$ This cover is simple by (6.13) and (6.12), so the theorem follows. $\square$

(6.15) Discussion. We have a map $q:Z\to S,$ sending the class of $(x,g)$ to $x\text{.}$ Suppose we have a central system. For $A\in K\left(p_{ℳ}\right)$ and $g\in {\text{gal}}_{CC}$ we define a section $s=s(g,A):S\to Z$ of $q$ as follows:

If $x\in S,$ then $x\in \hat{\bar{D}}$ for some $D\in K\left(ℳ\right)\text{.}$ Now let $k\in {\text{gal}}_{CD}$ be the class of $G·G_d,$ where $G\in g$ and $G_d$ is a direct replacing gallery of the direct pregallery $C^{E}D$ with $E≔\text{pr}(p(C,D)\subset p_{ℳ})\left(A\right)\text{.}$ Then $s\left(x\right)≔b(x,k),$ We have to check that this does not depend on the choices involved. We sketch this check for the choice of another $D\prime \in K\left(ℳ\right),$ such that $x\in \hat{\bar{D}}\prime \text{.}$ Let $F\subset \bar{D}\cap \bar{D}\prime$ be the facet such that $x\in F\text{.}$ Then, as we saw in (I.3.9) and (I.3.10) the map ${\text{pr}}_F$ restricted to $\{B\in K\left(ℳ\right)\hspace{0.17em}|\hspace{0.17em}F\subset \bar{B}\}$ is a bijection on $K\left({ℳ}_F\right)\text{.}$ So we can take $B≔{\text{pr}}_F^{-1}\left({\text{pr}}_F\left(C\right)\right)\text{.}$ 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 $D\text{.}$ So $G_d$ can be chosen of the form $G_1{G}_2,G_2\in {\text{Gal}}^F$ and $G_d^{\prime }=G_1{G}_2^{\prime },G_2^{\prime }\in {\text{Gal}}^F\text{.}$ Then $G_d$ and $G_d^{\prime }$ are $F\text{-equivalent.}$

$$\begin{array}{c} B C D D′ \\ \text{Figure 31.}\end{array}$$

The map $s$ is continuous, because it is so when one restricts it to a set $\hat{\bar{D}}$ and $S$ is a finite union of such in $S$ closed sets. Furthermore it is clear that $q\circ s={\text{id}}_S$ and $s\circ q_{|S(g,A)}={\text{id}}_{S(g,A)},$ where $S(g,A)≔s(g,A)\left(S\right)\text{.}$

So we conclude from this discussion, that we have a collection of spheres in $Z$ in the central case, labelled by ${\text{gal}}_{CC}\times K\left(p_{ℳ}\right)\text{.}$ 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 $\hat{Z},$ we are going to attach for each $(g,A)\in {\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ a cell $B(g,A)\text{.}$ Now, if it could happen that $S(g_1,A_1)=S(g_2,A_2),$ with $(g_1,A_2)\ne (g_2,A_2),$ then the contractibility of $\hat{Z}$ would be doubtful. But we have:

(6.16) Lemma. If $S(g_1,A_1)=S(g_2,A_2),$ then $(g_1,A_1)=(g_2,A_2)\text{.}$

		Proof (sketch).
	Let $x\in C,$ then $s(g_1,A_1)\left(x\right)=s(g_2,A_2)\left(x\right),$ so $(x,g_1)\sim (x,g_2),$ which implies $g_1=g_2\text{.}$ From $s(g_1,A_1)(-x)=s(g_2,A_2)(-x)$ it follows that $G·C^{A_1}(-C){\sim }_{R}G·C^{A_2}(-C),$ where $G\in g_1=g_2\text{.}$ So $C^{A_1}(-C){\sim }_R{C}^{A_2}(-C),$ from which it can be proved that $A_1=A_2,$ by a simple application of an invariant technique, which we do not want to develop here. $\square$

(6.17) Definition. We define a space $\hat{Z}=\hat{Z}(𝕍,H,ℳ,p)$ as follows: If the system is non central then $\hat{Z}≔Z\text{.}$ If the system is central, then we attach a cell $B(g,A)$ to each sphere $S(g,A)$ of $Z\text{.}$ Precisely: $\hat{Z}$ is a quotient of $Z\cup (𝕍\times ({\text{gal}}_{CC}\times K\left(p_{ℳ}\right))),$ where points of $S(g,A)$ are identified with points of $𝕍\times \left\{(g,A)\right\}$ by $S(g,A)\underset{q}{\overset{\cong }{\to }}S\subset 𝕍\cong 𝕍\times \left\{(g,A)\right\}$

Note that we can extend $q:\hat{Z}\to 𝕍$ in the central case, so we have a projection $q:\hat{Z}\to H$ in both cases.

(6.18) Theorem. The space $\hat{Z}$ is homotopy equivalent to the space $\left|\hat{\Sigma }\right|\text{.}$

		Proof.
	For the non central case this is theorem (6.14). So suppose central. Then: The in $Z$ open sets $\text{Str}\left(g\right)$ for $g\in \text{gal}$ induce open sets in $\hat{Z}$ This follows from the fact that $q\left(\text{Str}\left(g\right)\right)=\text{St}\left(\text{end}\left(g\right)\right)$ is open in $𝕍\text{.}$ These open sets still do not cover $\hat{Z}\text{.}$ But for each $(g,A)\in {\text{gal}}_{CC}\times K\left(p_{ℳ}\right)$ $𝕍\times \left\{(g,A)\right\}$ induces an open set in $\hat{Z}$ and together with the first sets this gives an open cover of $\hat{Z}\text{.}$ It can be verified, that this cover is simple and that its nerf is isomorphic to $\hat{\Sigma }\text{.}$ $\square$

(6.19) Corollary. If the system is $\text{locally-}K(\pi ,1),$ then the universal covering space $\stackrel{\sim }{Y}$ is homotopy equivalent to $\hat{Z}\text{.}$

(6.20) Remark. It can be proved that the homotopy equivalence of (6.19) can be realised by the following map $\psi :\stackrel{\sim }{Y}\to \hat{Z},$ which makes the diagram shown commutative. Let $y\in \stackrel{\sim }{Y}$ and set $x≔\text{Im}\left(p\left(y\right)\right)\in H\text{.}$ If $x\in F_{\text{min}},$ then $y\in V\in 𝒱$ and $\psi \left(y\right)$ is the point in $\hat{Z}$ determined by $x$ and $v^{-1}\left(V\right)\in {\text{gal}}_{CC}\times K\left(p_{ℳ}\right)\text{.}$ If $x\in S,$ then $x\in F$ for an $F\in F^{*}\left(ℳ\right)\text{.}$ Then $y\in U\in 𝒰\left(F\right)$ and $\psi \left(y\right)$ is the point in $\hat{Z}$ determined by $b(x,u^{-1}\left(U\right))\in Z\text{.}$ (Recall the maps $u$ and $v$ from (4.9)).

(6.21) Remark. The spaces $\left|\Sigma (g,A)\right|$ 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).

$$\begin{array}{cccc}& \stackrel{\sim }{Y}& \stackrel{\psi }{⟶}& \hat{Z}\\ p& \downarrow \\ & Y& \swarrow q\\ \text{Im}& \downarrow \\ & H\end{array}$$

(6.22) Remark. If the system is simplicial and if $\bigcap _{M\in ℳ}M=\left\{0\right\},$ then the hyperplanes cut out a triangulation of a sphere around the origin. Deligne uses this triangulation to define his complex (which coincides with $\left|{\hat{\Sigma }}_{\text{min}}\right|\text{).}$ 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 $\left|{\Sigma }_{\text{min}}(g,A)\right|$ (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 $G\in {\text{Gal}}_{CF}\text{.}$ Recall (I.4.1) that $\left|G\right|$ is the length of $G\text{.}$ The gallery $G$ is called $F\text{-minimal,}$ if $\left|G\right|\le \left|G\prime \right|$ for all $G\prime$ in the $F\text{-equivalence}$ class of $G\text{.}$ A gallery $G$ is called minimal, if it is $\text{end}\left(G\right)\text{-minimal,}$ i.e. if its length is minimal for the gallery equivalence class to which it belongs. Note that $F\text{-minimality}$ implies minimality, but in general not conversely.

If $g\in {\text{gal}}_{CF},$ we define the length of $g,$ notation also $\left|g\right|$ to be the length of an $F\text{-minimal}$ representative of $g\text{.}$

(6.24) Definition. Let $G,G\prime \in \text{Gal}\text{.}$ $G$ is said to be left (resp. right) contained in $G\prime ,$ if $G\prime \approx G·G^{\prime \prime }$ (resp. $G\prime \approx G^{\prime \prime }·G\text{)}$ for some $G^{\prime \prime }\in \text{Gal}\text{.}$ Recall that $\approx$ 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 $g\text{.}$

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

(6.27) Example. Let us consider the configuration of figure 32. The galleries $G_1,\dots ,G_6,$ starting in $C,$ where

$$\begin{array}{ccc}{G}_1& ≔& \underset{\_}{1}\hspace{0.17em}2\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}\underset{\_}{2}\hspace{0.17em}1\\ G_2& ≔& \underset{\_}{1}\hspace{0.17em}\underset{\_}{3}\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}3\hspace{0.17em}1\\ G_3& ≔& \underset{\_}{3}\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}3\hspace{0.17em}1\\ G_4& ≔& \underset{\_}{1}\hspace{0.17em}\underset{\_}{3}\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}1\hspace{0.17em}3\\ G_5& ≔& \underset{\_}{3}\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}1\hspace{0.17em}3\\ G_6& ≔& \underset{\_}{3}\hspace{0.17em}4\hspace{0.17em}5\hspace{0.17em}5\hspace{0.17em}\underset{\_}{4}\hspace{0.17em}3\end{array}$$

(Example is due to the computer) are all the minimal representatives of a gallery class $g\text{.}$

$$\begin{array}{c} 1 2 3 4 5 5 A B C \\ \text{Figure 32.}\end{array}$$

We see that the (maximal) direct starting nieces of $g$ are: $\underset{\_}{1}$ 2 5, $\underset{\_}{1}$ $\underset{\_}{3}$ 5 $(\approx \hspace{0.17em}\underset{\_}{3}\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}5)$ and $\underset{\_}{3}$ 4 5. The (maximal) direct ending pieces: 5 $\underset{\_}{2}$ 1, 5 3 1 $(\approx \hspace{0.17em}5\hspace{0.17em}1\hspace{0.17em}3)$ and 5 $\underset{\_}{4}$ 3, ending in chamber $C\text{.}$ This example is interesting because this equivalence class splits up in three flip equivalence classes: $\left\{G_1\right\},$ $\{G_2,G_3,G_4,G_5\}$ and $\left\{G_6\right\}\text{.}$

(6.28) Definition. A gallery class $g$ (with $A≔\text{beg}\left(g\right)\text{)}$ is said to satisfy the direct starting pieces property (DSPP), if there exists a direct gallery $G_d,$ such that:

(i)	Each direct starting piece of $g$ is left contained in $G_d\text{.}$
(ii)	For each $M\in {ℳ}^A\cap ℳ(A,\text{end}\left(G_d\right))$ there is a minimal representative of $g,$ which starts through $M,$ i.e. which Deligne gallery is of the form $A,$ ${\text{sp}}_{M\cap \bar{A}}\left(A\right),$ $\dots$ (definition of ${\text{sp}}_F:$ (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: $G_d=1\hspace{0.17em}2\hspace{0.17em}\underset{\_}{3}\approx \underset{\_}{3}\hspace{0.17em}2\hspace{0.17em}1$ for (6.26) and for example (6.27): $G_d=\underset{\_}{1}\hspace{0.17em}2\hspace{0.17em}5\hspace{0.17em}\underset{\_}{3}\hspace{0.17em}4$ for the direct starting pieces (note that $G_d$ is direct, because the underlying Tits gallery is so and chamber $A$ prescribes the signs). And $G_d=\underset{\_}{4}\hspace{0.17em}3\hspace{0.17em}5\hspace{0.17em}\underset{\_}{2}\hspace{0.17em}1$ for the direct ending pieces (now chamber $B$ prescribes the signs of $G_d\text{).}$

(6.30) Definition. A hyperplane system is said to satisfy the facet minimality property, if for each $F\in F\left(ℳ\right)$ and $G_1\in {\text{Gal}}_{CF},$ $G_1$ is $F\text{-minimal}$ and $G_2\in {\text{Gal}}^F,$ $G_2$ is minimal, then $G_1{G}_2$ is minimal (if defined).

(6.31) Theorem. Let $(𝕍,H,ℳ,p)$ be a non central hyperplane system, which satisfies the direct pieces property and the facet minimality property. Then $Z(𝕍,H,ℳ,p)$ 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 $A,B\in K\left(ℳ\right),$ $A\ne B\text{.}$ Set

$$E≔\bigcup _{M\in {ℳ}^A\cap ℳ(A,B)}M\cap \bar{A}$$

Then $E\subset \partial A$ is a strong deformation retract of $\bar{A}\text{.}$

		Proof.
	Choose $b\in B\text{.}$ for $x\in \bar{A},$ $\bar{A}\cap [b,x]$ is of the form $[d\left(x\right),x]$ for some unique $d\left(x\right)\in \bar{A}\text{.}$ We prove that $d\left(\bar{A}\right)=E\text{.}$ First remark that, $$x\in d\left(\bar{A}\right)\iff d\left(x\right)=x\iff [b,x]\cap \bar{A}=\left\{x\right\}\text{.}$$ $$\begin{array}{c} x b E d(x) A‾ \\ \text{Figure 33.}\end{array}$$ Let $x\in \bar{A}\text{.}$ If there is a $M\in {ℳ}^A\cap ℳ(A,B)$ with $x\in M,$ then $[b,x]\cap \overline{D_M\left(A\right)}=\left\{x\right\},$ for $b\notin \overline{D_M(A,B)}\text{.}$ So then $x\in d\left(\bar{A}\right)\text{.}$ Conversely, if there is no such $M,$ then for all $M\in {ℳ}^A$ we have $[b,x]\cap \overline{D_M\left(A\right)}\ne \left\{x\right\}\text{.}$ So then $[b,x]\cap \bar{A}=\bigcap _{M\in {ℳ}^A}([b,x]\cap \overline{D_M\left(A\right)})\ne \left\{x\right\}\text{.}$ This shows that $d\left(\bar{A}\right)=E\text{.}$ The continuity of $d$ is now also clear. By the convexity of $\bar{A},$ we see that $\bar{A}\times [0,1]\ni (x,t)\to (1-t)x+t·d\left(x\right)\in \bar{A}$ gives a deformation retraction of $\bar{A}$ on $E\text{.}$ $\square$

		Proof of theorem (6.31).
	We construct a filtration of $Z:$ For $n\in {\mathbb{Z}}^{+}$ we define $z^n≔$ union of all ${\bar{Z}}_g,$ where $g\in {\text{gal}}_{CA},$ $A\in K\left(ℳ\right)$ and $\left|g\right|\le n\text{.}$ Then $\bar{C}\cong Z^0\subset Z^1\subset Z^2\subset \dots \subset Z^n\subset Z^{n+1}\subset \dots \subset Z$ and $Z=\bigcup _{n\in {\mathbb{Z}}^{+}}{Z}^n,$ so it suffices to show that $Z^n$ is a (strong) deformation retract of $Z^{n+1}\text{.}$ Therefore we define a continuous map ${\kappa }_t:Z^{n+1}\to Z^{n+1},$ $t\in [0,1],$ which is a (strong) deformation retraction of $Z^{n+1}$ on $Z^n\text{.}$ Let us define this first on ${\bar{Z}}_g$ for a $g\in {\text{gal}}_{CA},$ $A\in K\left(ℳ\right)$ and $\left|g\right|\le n+1\text{.}$ If $\left|g\right|\le n,$ then we define ${\kappa }_t$ to be the identity on ${\bar{Z}}_g\text{.}$ Now let $\left|g\right|=n+1\text{.}$ Let $G_d$ be the direct gallery, such that all the direct ending pieces of $g$ are right contained in $G_d$ (so in particular $\text{end}\left(G_d\right)=A\text{).}$ Let $B≔\text{beg}\left(G_d\right)\text{.}$ Then $A\ne B,$ for $\left|g\right|>0\text{.}$ Since the system is non central, we have $\bar{A}=\hat{\bar{A}},$ so ${\bar{Z}}_g^{\prime }=\bar{A}\times \left\{g\right\}\cong \bar{A}\text{.}$ So we can define ${\kappa }_t$ on ${\bar{Z}}_g$ with the help of the deformation retraction of lemma (6.32). Let us check that ${\kappa }_t$ is now well defined on $Z^{n+1}$ by proving that the definitions coincide on ${\bar{Z}}_g\cap {\bar{Z}}_h\text{.}$ There is only something to prove in the case that at least one, say $g,$ of the gallery classes has length $n+1\text{.}$ So $\left|g\right|=n+1$ and we use the same notations for $g$ and its attributes as above. Suppose $h\in {\text{gal}}_{CD},$ $D\in K\left(ℳ\right)\text{.}$ ${\bar{Z}}_g\cap {\bar{Z}}_h\ne \varnothing$ implies that $\bar{A}\cap \bar{D}\ne \varnothing ,$ so this intersection equals $\bar{F},$ where $F≔A\wedge D$ (2.7). Now there is a unique $k\in {\text{gal}}_{CF}$ such that $k\le g$ and $k\le h\text{.}$ Choose an $F\text{-minimal}$ representative $G_1$ of $k$ and let $G_2\in {\text{Gal}}^F$ be a minimal gallery such that $G_1{G}_2$ is a representative of $g\text{.}$ Let also $G_3\in {\text{Gal}}^F$ be minimal, such that $G_1{G}_3$ is a representative of $h\text{.}$ It follows from the facet minimality property, that $G_1{G}_2$ and $G_1{G}_3$ are minimal, in particular $\left|G_1{G}_2\right|=n+1\text{.}$ Now we may suppose that $\left|G_2\right|\ge 1,$ for otherwise $\left|G_1\right|=n+1,$ hence $\left|G_3\right|=0,$ so then $A=D=F$ and $g=h$ and there is nothing to prove, go we can write $G_2=G_2^{\prime }{G}_2^{\prime \prime }$ with $\left|G_2^{\prime \prime }\right|=1$ and let $G_2^{\prime \prime }$ pass through the wall $M\in {ℳ}^A\text{.}$ Since $G_2^{\prime \prime }$ is a direct ending piece of $g,$ we have $M\in {ℳ}^A\cap ℳ(A,B)$ by (DEPP). Now $G_2\in {\text{Gal}}^F,$ so $G_2^{\prime \prime }$ also and hence $M\in {ℳ}_F\text{.}$ So $F\subset E,$ where $E$ is as in lemma (6.32). It follows that ${\kappa }_t$ restricted to the intersection ${\bar{Z}}_g\cap {\bar{Z}}_h$ is the identity for each $t\in [0,1],$ in particular coincidence of the definition on ${\bar{Z}}_g$ and on ${\bar{Z}}_h$ So ${\kappa }_t$ is veil defined and it is continuous by remark (6.11). We still have to prove ${\kappa }_1\left(Z^{n+1}\right)\subset Z^n\text{.}$ Let $g$ and its attributes be again as above and suppose $x\in \bar{A}\text{.}$ We have to show, that ${\kappa }_1\left(b(x,g)\right),$ which equals $b(d\left(x\right),g)$ by definition, is contained in $Z^n,$ where $d:\bar{A}\to \bar{A}$ is the map constructed in the proof of lemma (6.32). By this lemma $d\left(x\right)\in M$ for some $M\in {ℳ}^A\cap ℳ(A,B),$ so by (DEPP)(ii), we know that $g$ has a minimal representative of the form $G_1{G}_2,$ where $\left|G_2\right|=1$ and $G_2$ passes through $M\text{.}$ So $d\left(x\right)$ is element of the closure $\bar{F}$ of a common face $F$ of $A$ and $\text{end}\left(G_1\right)\text{.}$ The pair $(G_1,F)$ determines an element $h\in {\text{gal}}_{CF}$ and $G_1$ and $G_1{G}_2$ are $F\text{-equivalent,}$ so $(d\left(x\right),h)\sim (d\left(x\right),g)$ and since $\left|h\right|=n,$ it follows that $b(d\left(x\right),h),$ hence also $b(d\left(x\right),g)$ is contained in $Z^n\text{.}$ $\square$

(6.33) Corollary. Let we have a system as in theorem (6.31). Suppose the system is also $\text{locally-}K(\pi ,1)\text{.}$ Then the system is $K(\pi ,1)\text{.}$

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 $\text{locally-}K(\pi ,1),$ then it is $K(\pi ,1)\text{.}$

		Proof.
	In the noncentral case this is corollary (6.33). So suppose the system is central. Choose a hyperplane $N\in ℳ$ and a form $\varphi \in {𝕍}^{*},$ such that $N={\varphi }^{-1}\left(0\right)\text{.}$ Let $H≔{\varphi }^{-1}\left((0,\infty )\right)\text{.}$ Consider $\varphi$ as an element of ${\left({𝕍}_{ℂ}\right)}^{*}\text{.}$ Let $Y\prime ≔{\varphi }^{-1}\left(1\right)\cap Y\text{.}$ Then it is easily seen that $Y\prime$ is a strong deformation retract of $Y^{\prime \prime }≔Y(ℂ,H,ℳ-\left\{N\right\})\text{.}$ It is also not difficult to see, that ${ℂ}^{*}\times Y\prime \ni (t,z)\to tz\in Y$ defines a homeomorphism. So the system $(𝕍,𝕍,ℳ)$ is $K(\pi ,1)$ if and only if $(𝕍,H,ℳ-\left\{N\right\})$ is $K(\pi ,1)\text{.}$ And the latter is $K(\pi ,1)$ by (6.33). $\square$

§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 $\text{locally-}K(\pi ,1)\text{-property}$ one can apply induction on the codimension of $\bigcap _{M\in ℳ}M$ in $𝕍\text{.}$

In this paragraph we want to give a sketch of a possible proof of the conjectured implication: simpliciality $\Rightarrow$ 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 $G_d$ of (i) minimal. Now we need a definition:

(7.2) Definition. We say that a gallery $G$ reduces to a gallery $G\prime ,$ notation $G\prime \angle G,$ if we can obtain $G\prime$ from $G$ by length reducing cancel operations. If we can take these cancel operations as follows: $(G_{dm}·G_1·A^U{B}^{U}A·G_2,G_{dm}·G_1·G_2),$ where $G_{dm}$ is the maximal direct starting piece of the second gallery, we say that $G$ reduces to $G\prime$ respecting the direct starting pieces, notation $G\prime \angle !G\text{.}$

If $g_{\text{fl}}$ is a flip class of galleries, then $G\prime \angle g_{\text{fl}}$ (resp. $G\prime \angle !g_{\text{fl}}\text{)}$ means that there is a $G\in g_{\text{fl}},$ such that $G\prime \angle G$ (resp. $G\prime \angle !G\text{).}$

Now by very general reasoning it can be seen that if $\{G_1,\dots ,G_k\}$ is a finite collection of representatives of a gallery class $g,$ then there exists a flip class $g_{\text{fl}}\subset g$ such that $G_i\angle g_{\text{fl}}$ for $i=1,\dots ,k\text{.}$ In particular there exists such a class for all the minimal representatives of $g\text{.}$ Now (6.28) would be proved if we have:

(7.3) Conjecture. By the minimality of the representatives it follows that we can choose $g_{\text{fl}}$ such that we even have $G_i\angle !g_{\text{fl}}$ for all the minimal representatives $G_i$ of $g\text{.}$

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. $\square$

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 $G_1·A^U{B}^{U}A·G_2\text{.}$

Note that minimal implies irreducible, but not conversely:

(7.6) Example. Consider $G≔1\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}\underset{\_}{2}$ in figure 12, page 40. Then no flip operations nor length reducing cancel operations are possible, so $G$ is irreducible. But $G\sim \underset{\_}{3}\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}\underset{\_}{2}\approx \underset{\_}{3}\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}\underset{\_}{2}\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}3\sim \underset{\_}{3}\hspace{0.17em}2\hspace{0.17em}\underset{\_}{1}\hspace{0.17em}3,$ 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 $G_d,$ such that direct starting pieces of irreducible representatives of $g$ are left contained in $G_d$ and moreover if $g_i\subset g$ for $i=1,2$ is a flip class of irreducible galleries and $G_i$ is the maximal direct starting piece of $g_i,$ then $\left|G_1\right|+\left|G_2\right|>\left|G_d\right|\text{.}$

(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 $G\in \text{Gal}$ and $\text{beg}\left(A\right)≕A\text{.}$ If $G\sim {\varnothing }_A,$ then $G$ is reducible.

		Proof.
	If $G$ is irreducible, then $G=G_1·G_2^{-1}$ for two irreducible and equivalent galleries $G_1$ and $G_2\text{.}$ Using the second condition in def.(7.7), we get a contradiction. The details are left to the reader. $\square$

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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