Last update: 10 May 2013
Let be a hyperplane system. In this chapter we consider the question under which circumstances one can prove that the space is a If this is the case we call the system a
(1.1) Definition. A hyperplane system is called simplicial, if the cone on a chamber is a simplicial cone, i.e. of the form for some independent set The cone on a set is defined to be 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
(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 of the universal covering space of To ensure that the abstract simplicial complex has simplices with a finite number of vertices we suppose from now on that the configuration satisfies:
(1.4) Assumption. For each facet the number of facets contained in is finite.
Assuming that the system is (3.7), by which we mean that the star subsystems (2.9) are we prove that this cover is simple i.e. that intersections are contractible. Then the geometric realisation of has the same homotopy type as (Weil [Wei1952]). So the is reduced to a contractibility problem for a simplicial complex, if one knows that the system is locally But this 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
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 In the central case the simpliciality takes care for a nice triangulation of a sphere around the origin. The simplicial complex (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 homotopic to Besides this the space 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 conjecture it seems necessary to have knowledge about the minimal representatives of galleries. Hence the connection with word problems.
Let again be a hyperplane system. We need some more notions and properties; first of the configuration
(2.1) Definition. Let then the star of is defined by
(2.2) Proposition. We have
(ii) Since for all is contained in a chamber with respect to
(i) By (ii) The right hand side of (i) equals Now let then by criterion (I.3.5) for each we have so is contained in the right hand side. And conversely.
(2.3) Corollary. is open in
(2.4) Lemma. Let Then the following properties are equivalent:
|(iii)||For each we have|
(i) (ii) Let and So Then so
(ii) (iii) is trivial.
(iii) (i) Let By criterion (I.3.5) we have to show that Suppose not. Then where is the other halfspace. So is an open nonempty set so there exists a chamber which intersects it and since it is also a union of facets, is even contained in it. By (iii) so contradiction with
(2.5) Lemma. Let then:
(i) If then so
(ii) If then for all And with (I.3.5): It follows by definition of facet that
(iii) follows from (i) and (ii).
(2.6) Proposition. Let and Then there is a facet such that
is a union of facets (since is so (I.3.4)). Take with maximal dimension. Suppose To prove: Choose If then finished. If then consider the line through and
Case 1: contains only Since is open (2.3), is open in so there is a Let be the facet containing the point Then but for So by (2.5)(iii) But since is convex, we have so Hence a contradiction.
Case 2: Since is open in is an open interval on containing If then and we are ready. If and for some then there is a such that and So and are at different sides of but is an intersection of hyperplanes in and closed halfspaces. is one of them, so then contradiction.
(2.7) Definition. Let If then we denote the unique facet such that as exists by (2.6) by If and are such that there exists a such that then by locally finiteness there are only finitely may such facets Then by (2.6) thre is a facet such that We denote this facet by
Note that for (respectively and this "meet" (respectively "join") is maximal (respectively minimal) with this property, if it exists.
(2.8) Theorem. If then is nonempty, if and only if is defined and in this case
The equality ($) holds because
(2.9) Definition. (cf. def. (II.4.8)) Let again and define
Note that (or simply if is considered as a map Furthermore note that
(2.10) Proposition. (cf. prop. (II.4.9)). Let Then there exists a map with the properties:
|(i)||The map is a homotopy inverse of the inclusion|
|(ii)||For another such that exists we have|
Choose and let be a closed ball around inside in some metric is open!). For each there is a unique such that Now is a continuous map The rest of the proof is analogous to the proof of proposition (II.4.9) from the point ($). Including the proof of (ii) for which we have to remark that
(2.11) Corollary. The spaces and are homotopy equivalent.
(2.12) Corollary. Let be the inclusion and then the homomorphism is injective.
(2.13) Lemma. Let such that exists and let Then
where these fundamental groups are seen as subgroups of which is done by (2.12).
This follows from (2.10)(ii) in the same way as (II.4.12) follows from (II.4.9)(ii).
The following corollary is the key to an important result of this chapter (theorem (3.11)). It makes it possible to prove this result for so called non central (def. (3.2)) systems. This in contrast to Deligne [Del1972]. Compare remark (3.14).
(2.13) Corollary. For let and a curve in Suppose is homotopic to in Then exists and the curves are homotopic in to a curve in
It is easy to see that we may restrict ourselves to the case that and are loops, say from a point From it follows that 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 where is a curve in
(2.16) Definition. Let Define:
(2.17) Lemma. We have
This is a simple inspection of the proof of (I.1.5).
(2.18) Corollary of (2.15). For let If and then exists and for some gallery
Let be a curve in such that as exists by (2.17). From it follows that hence So by (2.15) there is a curve in such that Then there is by (2.17) a such that Then hence
(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 for With the help of (I.3.9) and (I.3.10) we have a map This induces a map which respects gallery equivalence. (This map corresponds in a sense with the map in prop. (2.10)). Now let Then
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.
After these preliminaries we are going to construct an open cover of the universal covering space of where is a hyperplane system as before.
(3.1) Definition. We call a facet minimal if it is so for the natural order relation i.e. if there is no facet contained in the closure
(3.2) Definition. We call a hyperplane system central if and 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 Then since each facet contains at least one minimal facet, we have for all that Hence Hence The only hyperplanes, which meet are the elements of so By (2.11) we have Since we are only interested in the homotopy type of we see that indeed we can suppose that the system is central in the sense of def. (3.2), if there exists a unique minimal facet. Note that we have:
(3.5) Definition. We agree that in the non central case (this fits in with (3.4)!). We define the set of proper facets by:
(hence 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
(3.6) Definition. Let be the universal covering. For define: the collection of components of and:
If the system is central, then is finite, hence is locally finite on For we define: the collection of components of and and we agree that if the system is non central. Furthermore we define:
(3.7) Definition. We call a hyperplane system if is a for each This means that and hence (by (2.11)) are
(3.8) Example. A simple, simplicial system is (central or non central). This follows from the theorem of Deligne (1.2) and the observation that the local system 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 (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
(3.10) Example. (The "triangle"). Let be a plane and let consist of three different lines with but (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 So the system is But it is not This is well known and can be seen by the method of §4 (see (4.17)).
(3.11) Theorem. Let be a hyperplane system which is Then as defined in (3.6) is a simple covering of the universal covering space of In particular this universal covering space is homotopy equivalent to the geometric realisation of the abstract simplicial complex the nerf of the cover
First we prove the following remark: If for and then exists and i.e. is closed under intersections. Now because so it equals (theorem (2.8)). Let Then we have
The right hand side is a union of intersections of elements in and and each intersection is in turn a union of components. So, for instance is a union of elements in and we have to show that this union consists of just one element, that is we have to see that is connected.
For this purpose suppose and let be a curve in from to (each is connected). We have in because is simply connected. So in where Now is in so by corollary (2.15) we know that where is a curve in Let be the unique lift of to with then in in particular Furthermore is inside so we have connected and by a curve in hence is connected. This proves the first remark i.e. that is closed under intersections.
It can be verified that for is a universal covering. For instance the simply connectedness of follows from the fact that induces an injection of fundamental groups (2.12). The assumption ensures now that is a if Hence is contractible then. Together with the first remark it now follows that all intersections of elements of are contractible. In the non central case the elements of cover and so this is then indeed a simple cover.
In the central case we still have to cover the points of with This can be done with the elements of Since these elements are mutually disjoint, we still have only to prove that is contractible (if nonempty) for and where and We have:
Argueing as after (3.12) we can conclude from (3.13) that 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 is connected. The only thing we have to do is to prove the following analog of corollary (2.15): Let (respectively be a curve in (respectively in where we consider and suppose in Then there is a curve in such that in To prove this we define a curve by Let be a homotopy from to then gives a homotopy from to So as we wanted.
So is indeed connected. is again a universal covering. The base space is convex, hence contractible (and is even a homeomorphism). So is indeed a simple open cover.
As said before, the second statement of the theorem follows from Weil [Wei1952].
(3.14) Remark. Note that we did not suppose simpliciality nor centrality. This in contrast with Deligne. His ingredient which is comparable with corollary (2.15) is [Del1972] 1.31. And there he uses the existence of a gallery which depends on the fact that the situation is central.
(3.15) Remark. We have the same result for the open cover and the abstract simplicial complex as easily can be seen. The space looks like a "thickening" of the space
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 a point and a point above where is the universal covering.
(4.1) Definition. For define:
(Note that and recall (2.16):
(4.2) Definition. Let for We write if
|(ii)||There is a such that (gallery equivalence (I.4.7))|
Note that by (i), so so (ii) make sense. Note also that this relation is transitive.
(4.3) Definition. If then we call and notation: if (then automatically i.e. if there is a such that The classes are denoted by If then Note that for all where is gallery equivalence. Furthermore (note that this is a disjoint union).
(4.4) Definition. Let for Then means that where is in the class It is easily checked that this indeed does not depend on the choice of the representatives.
|(i)||The relation is transitive.|
|(ii)||Suppose and then there is a unique such that|
|(iii)||If and there exists a such that then there exists a unique such that which is the smallest, i.e. if then|
|(iv)||If and there exists a such that then there exists a unique such that which is the biggest, i.e. if then|
|(v)||If then if and only if as sets.|
|(vi)||If then if and only if there is a such that|
(i) is trivial.
(ii) Let Then so also determines a If also and then for some by the definition of But this says at the same time that so
(iii) Suppose and Then so is defined. By (ii) there is a unique such that so by (ii) there is a unique such that By (i) and again by (ii) we have We still have to prove that this construction of does not depend on the choice of so suppose has also the property that Choose representatives and of and respectively and let (respectively be an element of such that (respectively as exists, since (respectively Then so we have Then by corollary (2.18) we have for some Then hence so
(iv) If there are only finitely many such that as follows from (ii) and (1.4). Therefore (iv) follows from (iii).
(v) Suppose and So Let then If then since there is a such that so Hence So Now suppose First we prove For each chamber there is a such that Since we also have By lemma (2.4) it follows that Now let then so Hence there is a such that So
(vi) If there is a such that then by (v), so If then So exists and determines a such that
(4.6) Remark. and is not necessarily an element of
(4.7) Definition. We define a mapping as follows: Let be a curve in such that Then Let be the unique lift with Then is determined by and is contained in a unique In the central case we define also a mapping as follows: Let and suppose is a loop in such that Then the unique lift of where that starts in has his endpoint in a uniquely determined
(4.8) Lemma. Let for Then if and only if
Let then clearly so Let be as in definition (4.7). Let be a curve in from to and Then is a curve in so by lemma (2.17) we have for some Now so by lemma (2.17) we have for some Now so by (I.4.10) So indeed The reverse implication can also easily be seen.
|(i)||The mapping induces a bijection between and the vertices of|
|(ii)||The mapping induces a bijection between and the vertices of which are not vertices of|
(i) Let and a point in above a curve from to and According to (I.4.10) there is a such that Then so is surjective. From lemma (4.8) it follows that if and only if and Hence can be defined on and is then injective.
(ii) Let Let be a curve from to a point in Now is a curve in from to a point We may assume that for some Then is a loop in on so by (I.4.10) there is a such that hence the surjectivity. It is also straightforward to see that if and only if and Hence we have a bijection
So from now on we can identify the vertices of with the set We proceed by describing the simplices of
(4.10) Theorem. A set constitutes a simplex in if and only if there is a such that for
Let for and Then by definition constitutes a simplex if and only if Then for some hence for some Since it follows from lemma (4.8) that
(4.11) Corollary. The simplicial complex is isomorphic to
This follows from (4.10) with the help of proposition (4.5)(vi).
(4.12) Remark. If we define then it can be seen also, that the simplicial complex is isomorphic to
(4.13) Definition. An element (or a representative as in theorem (4.10) is said to point to that simplex Let and a set of facets in Then by proposition (4.5)(ii) there is a uniquely determined with and they constitute a simplex. In particular, if is the set of all the facets in then this simplex is denoted by The set of all simplices to which points is exactly the set of all subsimplices of The maximal simplices of are the simplices of the form for for some
(4.14) Definition. Let the system be central. For a vertex of which is not a vertex of (i.e. an element of we denote by the subcomplex of consisting of all simplices that form with a simplex of The corresponding subcomplex of is denoted by For we write (respectively instead of (resp. We define and analogously.
In remark (6.21) we will see that (resp. has the homotopy type of a sphere (respectively cell). Now we still have to finish our description of in the central case:
(4.15) Theorem. Suppose that the hyperplane system is central and let and Then the galleries that point to simplices of are equivalent to galleries of the form where is a direct replacing gallery of with and
Let and let the curve and the point be as in the proof of (4.9)(ii). Let point to a simplex of (definition (4.13)); suppose that this simplex forms a simplex in with This means that if then For we may suppose that, if then hence Now where is a curve in from to with Let be a curve in from to a point in above and put Now define the "straight" curves:
Then ?? and Hence Now and by the definition of the functor (I.1.3) the latter equals where So we have as has to be proven.
If moreover the hyperplane system is simple, then we have still another description of the extension
(4.16) Theorem. Suppose we have a simple central hyperplane system. Then there is a bijection between and the set of equivalence classes of galleries starting in So the subcomplexes can also be labelled by the latter. If and then the galleries that point to simplices of are equivalent with galleries of the form where is a direct positive gallery.
Let us have the situation of theorem (4.15) and suppose moreover that the system is simple. Then we have so Let and be direct Tits galleries and and Then so
(where denotes pregallery equivalence, see (I.1.4)). By proposition (I.4.8) the first and the last gallery are also gallery equivalent, so where is the positive gallery and Now each gallery is equivalent to a gallery of this form and this induces the desired bijection between and
(4.17) Example. As an illustration let us determine the complex of the hyperplane system "the triangle", example (3.10) (equals in this non central case). It is easily seen that the group (which is in general the fundamental group of !) is isomorphic to by the isomorphism where is the equivalence class of the gallery (in the notation used in this kind of examples, see §(I.5)). For instance the commutation relation is illustrated by flip relations around the plinth
The minimal facets are the points: For each there is a unique representative So the vertices of can be labelled by a disjoint union of three copies of The two-simplices are the maximal simplices of The system is so by theorem (3.11) or in fact remark (3.15) we have Now we can see why the system is not Take a subcomplex consisting of the simplices with where and are two different integers. Then is a 2-sphere (the surface of the platonic solid in figure 28). Since the simplicial complex is two dimensional generates a nontrivial element in
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 be a group and a finite collection of subgroups. We define an abstract simplicial complex, called the cosetnerf of the group and the collection as follows:
(5.2) Remark. Since the collection itself is a (maximal) simplex of This simplex serves as a fundamental domain for the natural action of on
(5.3) Lemma. the map induces a bijection between and the set of maximal simplices of
(5.4) Lemma. If then the space is homeomorphic to where we give the discrete topology. Hence is a disjoint union of spaces homeomorphic to
Let Then each left coset of in is a disjoint union of left cosets of So for each element of we have a subcomplex of The space 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 as can be seen from the action of on
(5.5) Lemma. The space is connected if and only if is generated by
If is connected, then by (5.3) the index of the subgroup generated by the elements of is one. Conversely, let be generated by Then an arbitrary vertex of is of the form and Write for and let Then, since the sequence is a sequence of 1-simplices connecting the vertex to a vertex of the fundamental simplex.
(5.6) Remark. It is convenient to restrict ourselves to collections which have the following two properties:
|(ii)||is generated by|
By lemma (5.4) we do not lose much generality by supposing (ii). If (ii) is supposed we will suppress in the notation:
Property (i) ensures with the help of lemma (5.3) that we have a bijection
In all the cases we will consider, the properties (i) and (ii) will be satisfied.
(5.8) Example. Let be a finite, affine or compact hyperbolic system i.e. the subgroups for are finite. Then the collection satisfies (5.6) (i) and (ii). Now is the well known Coxeter complex. Note that if is finite, then is homeomorphic to the
(5.9) Theorem. Suppose a simple hyperplane system of a linear Coxeter group is given. Let be the associated Artin system. Then:
|(i)||The simplicial complex of the hyperplane system as defined in (3.6) is isomorphic to the cosetnerf of the collection of subgroups of of the form where and is finite. This collection satisfies (5.6) (i) and (ii).|
|(ii)||The system is central if and only if is finite and in this case there is a bijection ("midpoint") from to the vertices of which are not vertices of If then the maximal simplices of the subcomplex of are the simplices of the form: where has to be interpreted as a subset of by remark (II.3.19).|
The fact that the collection of subgroups of as in (i) satisfies (5.61)(i) follows from theorem (II.4.13)(ii). We have a surjective map as the composite 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 induces a bijection between and the cosets of the subgroups of mentioned in (i). For, if then the elements of are equivalent to galleries of the form with After dividing out the action of we can suppose then where is of the type i.e. is finite because and is a proper subset, because Now is the coset Conversely, if is finite, but and it is easy to see that for some proper facet determined by and (i) now follows from corollary (4.11). (ii) follows from (4.16).
(5.10) Definition. Let be an extended Artin group of rank as in definition (III.2.13). For we define the following subgroups:
(5.11) Remarks. (i) We have but the other inclusion need not hold (take for instance What we do have is:
(ii) Analogous to theorem (II.4.13) it can be proved that the flip relations form a complete set of relations for and also:
(iii) From (ii) it can be seen that (as the notation already suggests) forms an extended Artin system and we have
(5.13) Theorem. Suppose a hyperplane system of an extended Coxeter group of rank is given. Let be the associated extended Artin group of Then:
|(i)||The simplicial complex of the hyperplane system is isomorphic to the cosetnerf of the collection subgroups of of the form where and is finite. This collection satisfies (5.6)(i) and (ii).|
|(ii)||The system is central if and only if is finite and in this case there is a bijection ("midpoints") from to the vertices of which are not vertices of If is the unique way to write an element of as product of and then the maximal simplices of the subcomplex are the simplices of the form where has to be interpreted as a subset of which can be done by remark (II.3.19) and lemma (III.4.1) and is the pseudo conjugation the action of on the set as defined in (III.4.14).|
(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 and and prove (ii) with the help of (4.15). But let us give a more direct and instructive proof: Let be a vertex of which is not a vertex of Here for unique and as in the proof of (III.3.5)(iii).
A maximal simplex of is determined by a for a unique This simplex forms with a simplex in (i.e. is a simplex of iff We may assume, that there is a point in this inter¬section, which projects (by on where and (recall that Let be a curve in from to this point in Then is a curve in from to Define the "straight" curves as follows
Note that and Now the curve projects on a loop in representing an element in of the form as above. Conversely each element in determines a curve in from to a point of the form Composing this curve with and lifting to the endpoint lies in a This way we get the desired bijection
The curve projects on a loop representing an element of such that This describes the bijection between and the maximal simplices of given by (i).
Since we have
for some represented by the projection of To analyse this element we note that projects on the same loop in as where
the transforms of under Inspection of these formulas gives that projects on a loop representing the element
The curve projects on the same loop as its transform under so it projects on a loop representing we conclude that
The reasoning has shown that if is a simplex of then where Conversely, it can be checked that is a simplex of for all and
(5.14) Remark. In the situation of a linear Coxeter group as well the situation of an extended Coxeter group the simplicial complex can be described by adding to the conditions on in (5.9)(i) (respectively (5.13)(i), namely that and finite, that is maximal with respect to these properties.
(5.15) Remark. In the case that is finite the sets with the properties mentioned in (5.14) are precisely the sets with hence of the form for a Hence for a given (respectively we have a bijection between the maximal simplices of the Coxeter complex of (example (5.8)) and the maximal simplices of (respectively This bijection induces a homeomorphism between the and (respectively
If the space is contractible and the system is then the system is as follows from theorem (3.11). To investigate the question of contractibility of we construct a space in terms of facets and galleries, which turns out to be homotopy equivalent to as follows:
(6.1) Definition. Let Note that in the non central case and in the central case which has the homotopy type of a sphere, hence the use of the symbol
(6.2) Definition. For we define (hence, by (3.5) in the non central case).
(6.3) Definition. We first define the sets:
We give the topology of and we give the topology of the disjoint union of the spaces (Note that the set is a disjoint union of the sets Furthermore for Note that this fits in with the notation, because is indeed the closure of in Now we come to the definition of the space
(6.4) Definition. We define an equivalence relation on as follows: If for then if and if is the facet containing and the unique element of (see (4.5)(ii)) such that then
Let denotes the quotient space and the projection.
(6.5) Remark. From the definition it follows that each equivalence class contains a unique element with the property that So where This is a disjoint union. The sets are the facets of the space
(6.6) Remark. To clarify the picture of the space (see also fig. 30) we note the following: Since each point in is contained in for some chamber we could have restricted ourselves to elements such that (and is then nothing but an equivalence class of galleries). I.e. can also be considered as a quotient space
(6.7) Definition. Let We define the star of the facet (and denote it by Str to avoid confusion with the notations St and Star) by:
(6.8) Proposition. is open.
Let It suffices to show that
if the l.h.s. is nonempty, because the r.h.s. of (6.9) is open in Let and By the definition of (6.7) we have for some with Then and which entails So first of all we see that, if the l.h.s. of (6.9) is nonempty, then Furthermore from it follows that so 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 and prove that the r.h.s. is contained in the l.h.s.: Suppose and Then for some facet with Furthermore so there is a unique such that so there is a unique element such that So but we also know that so and hence From and we conclude that Hence is element of the r.h.s. of (6.9).
(6.10) Proposition. Let Then:
|(ii)||is closed in hence|
(i) is clear from def. (6.4) and remark (6.5).
(ii) Suppose is cuh that then by (i): We show that which proves (ii): Suppose then and hence contradiction.
(6.11) Remark. We have and a map defined on is continuous, if and only if the restrictions to are continuous. If we wish we even can restrict ourselves to with
(6.12) Proposition. Let Then is nonempty if and only if exists and then it equals
If nonempty, then there is a contained in the intersection and then So then exists. And conversely.
(6.13) Proposition. Let Then is contractible.
Choose a point Then and we are going to contract on Let and suppose, that the left hand side of (6.9) is nonempty (i.e. By the equality (6.9) we see that we can define a continuous map by where is the set in (6.9). It is not difficult to check, that then So we have a continuous map which is the desired contraction on
(6.14) Theorem. The space is homotopy equivalent to the space
It follows from (6.8) and (6.12) that is an open cover of whose nerf is isomorphic to This cover is simple by (6.13) and (6.12), so the theorem follows.
(6.15) Discussion. We have a map sending the class of to Suppose we have a central system. For and we define a section of as follows:
If then for some Now let be the class of where and is a direct replacing gallery of the direct pregallery with Then We have to check that this does not depend on the choices involved. We sketch this check for the choice of another such that Let be the facet such that Then, as we saw in (I.3.9) and (I.3.10) the map restricted to is a bijection on So we can take Then a direct Tits gallery from to followed by a direct Tits gallery from to is a direct Tits gallery from to So can be chosen of the form and Then and are
The map is continuous, because it is so when one restricts it to a set and is a finite union of such in closed sets. Furthermore it is clear that and where
So we conclude from this discussion, that we have a collection of spheres in in the central case, labelled by Now the hope is that we can prove in some cases that 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 we are going to attach for each a cell Now, if it could happen that with then the contractibility of would be doubtful. But we have:
(6.16) Lemma. If then
Let then so which implies From it follows that where So from which it can be proved that by a simple application of an invariant technique, which we do not want to develop here.
(6.17) Definition. We define a space as follows: If the system is non central then If the system is central, then we attach a cell to each sphere of Precisely: is a quotient of where points of are identified with points of by
Note that we can extend in the central case, so we have a projection in both cases.
(6.18) Theorem. The space is homotopy equivalent to the space
For the non central case this is theorem (6.14). So suppose central. Then: The in open sets for induce open sets in This follows from the fact that is open in These open sets still do not cover But for each induces an open set in and together with the first sets this gives an open cover of It can be verified, that this cover is simple and that its nerf is isomorphic to
(6.19) Corollary. If the system is then the universal covering space is homotopy equivalent to
(6.20) Remark. It can be proved that the homotopy equivalence of (6.19) can be realised by the following map which makes the diagram shown commutative. Let and set If then and is the point in determined by and If then for an Then and is the point in determined by (Recall the maps and from (4.9)).
(6.21) Remark. The spaces are homotopy equivalent to a sphere, since they can be seen to correspond to the spheres by the homotopy equivalence of theorem (6.14).
(6.22) Remark. If the system is simplicial and if then the hyperplanes cut out a triangulation of a sphere around the origin. Deligne uses this triangulation to define his complex (which coincides with 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 (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 is contractible in the non central case (for a simple system it follows then for a central system too).
(6.23) Definition. Let Recall (I.4.1) that is the length of The gallery is called if for all in the class of A gallery is called minimal, if it is i.e. if its length is minimal for the gallery equivalence class to which it belongs. Note that implies minimality, but in general not conversely.
If we define the length of notation also to be the length of an representative of
(6.24) Definition. Let is said to be left (resp. right) contained in if (resp. for some Recall that denotes flip equivalence (I.4.5).
(6.25) Definition. Let be an equivalence class of galleries. Then a gallery is called a direct starting (resp. ending) piece of if is direct and left (resp. right) contained in some minimal representative of
(6.26) Example. The galleries 1 2 and 2 are direct starting pieces of the gallery class that is depicted in figure 12 (example (I.5.5)). The galleries 2 and 2 3 are the direct ending pieces.
(6.27) Example. Let us consider the configuration of figure 32. The galleries starting in where
(Example is due to the computer) are all the minimal representatives of a gallery class
We see that the (maximal) direct starting nieces of are: 2 5, 5 and 4 5. The (maximal) direct ending pieces: 5 1, 5 3 1 and 5 3, ending in chamber This example is interesting because this equivalence class splits up in three flip equivalence classes: and
(6.28) Definition. A gallery class (with is said to satisfy the direct starting pieces property (DSPP), if there exists a direct gallery such that:
|(i)||Each direct starting piece of is left contained in|
|(ii)||For each there is a minimal representative of which starts through i.e. which Deligne gallery is of the form (definition of (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: for (6.26) and for example (6.27): for the direct starting pieces (note that is direct, because the underlying Tits gallery is so and chamber prescribes the signs). And for the direct ending pieces (now chamber prescribes the signs of
(6.30) Definition. A hyperplane system is said to satisfy the facet minimality property, if for each and is and is minimal, then is minimal (if defined).
(6.31) Theorem. Let be a non central hyperplane system, which satisfies the direct pieces property and the facet minimality property. Then 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 Set
Then is a strong deformation retract of
Choose for is of the form for some unique We prove that First remark that,
Let If there is a with then for So then Conversely, if there is no such then for all we have So then This shows that The continuity of is now also clear. By the convexity of we see that gives a deformation retraction of on
|Proof of theorem (6.31).|
We construct a filtration of For we define union of all where and Then and so it suffices to show that is a (strong) deformation retract of Therefore we define a continuous map which is a (strong) deformation retraction of on Let us define this first on for a and If then we define to be the identity on
Now let Let be the direct gallery, such that all the direct ending pieces of are right contained in (so in particular Let Then for Since the system is non central, we have so So we can define on with the help of the deformation retraction of lemma (6.32). Let us check that is now well defined on by proving that the definitions coincide on
There is only something to prove in the case that at least one, say of the gallery classes has length So and we use the same notations for and its attributes as above. Suppose implies that so this intersection equals where (2.7). Now there is a unique such that and Choose an representative of and let be a minimal gallery such that is a representative of Let also be minimal, such that is a representative of It follows from the facet minimality property, that and are minimal, in particular Now we may suppose that for otherwise hence so then and and there is nothing to prove, go we can write with and let pass through the wall Since is a direct ending piece of we have by (DEPP). Now so also and hence So where is as in lemma (6.32). It follows that restricted to the intersection is the identity for each in particular coincidence of the definition on and on So is veil defined and it is continuous by remark (6.11).
We still have to prove Let and its attributes be again as above and suppose We have to show, that which equals by definition, is contained in where is the map constructed in the proof of lemma (6.32). By this lemma for some so by (DEPP)(ii), we know that has a minimal representative of the form where and passes through So is element of the closure of a common face of and The pair determines an element and and are so and since it follows that hence also is contained in
(6.33) Corollary. Let we have a system as in theorem (6.31). Suppose the system is also Then the system is
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 then it is
In the noncentral case this is corollary (6.33). So suppose the system is central. Choose a hyperplane and a form such that Let Consider as an element of Let Then it is easily seen that is a strong deformation retract of It is also not difficult to see, that defines a homeomorphism. So the system is if and only if is And the latter is by (6.33).
(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 one can apply induction on the codimension of in
In this paragraph we want to give a sketch of a possible proof of the conjectured implication: simpliciality direct pieces property. First we remark that it is only necessary to prove (6.28)(i), because in the simplicial case it can be proved that (ii) is satisfied if one takes the length of of (i) minimal. Now we need a definition:
(7.2) Definition. We say that a gallery reduces to a gallery notation if we can obtain from by length reducing cancel operations. If we can take these cancel operations as follows: where is the maximal direct starting piece of the second gallery, we say that reduces to respecting the direct starting pieces, notation
If is a flip class of galleries, then (resp. means that there is a such that (resp.
Now by very general reasoning it can be seen that if is a finite collection of representatives of a gallery class then there exists a flip class such that for In particular there exists such a class for all the minimal representatives of Now (6.28) would be proved if we have:
(7.3) Conjecture. By the minimality of the representatives it follows that we can choose such that we even have for all the minimal representatives of
(7.4) Proposition. (6.28) is true for flip equivalence.
This can be proved analogously as the key lemma of Deligne [Del1972] 1.19.
Here is the essential use of the simpliciality.
It is interesting to ask if a somewhat stronger form of the direct pieces property is true, namely if we replace the word "minimal" by "irreducible", where:
(7.5) Definition. We call a gallery irreducible, if it is not flip equivalent to a gallery, that contains a cancellation, i.e. that is of the form
Note that minimal implies irreducible, but not conversely:
(7.6) Example. Consider in figure 12, page 40. Then no flip operations nor length reducing cancel operations are possible, so is irreducible. But so 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 there exists a direct gallery such that direct starting pieces of irreducible representatives of are left contained in and moreover if for is a flip class of irreducible galleries and is the maximal direct starting piece of then
(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 and If then is reducible.
If is irreducible, then for two irreducible and equivalent galleries and Using the second condition in def.(7.7), we get a contradiction. The details are left to the reader.
If simplicial systems would satisfy the strong direct pieces property, then theorem (7.8) would give a very simple solution to the word problem for Artin groups. Note that for braid groups it would mean the following: "If a braid is trivial, then this can be shown without increasing the number of crossings".
This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.