Last update: 3 May 2013
(1.1) Equivalence relations on wordsets. We need terminology used in connection with free semigroups, relations, presentations, etc.
Let Al and Ob be sets, called the alphabet and the objects respectively. Let two maps be given: A word is a finite sequence with and for
Let be the set of all words. Mostly we add to for each an empty word Define by and If are words and then by juxtaposition we can define This operation is associative and has, if empty words are added left and right neutral elements. This way we get a category with the property that each map from Al to the morphisms together with a map from Ob to the objects such that the image of and are the domain and range respectively of the image of extends uniquely to a functor. Therefore we call this construction the free category generated by the quadruple (Al, Ob; beg, end). If then this is nothing else then the usual free semigroup on (or generated by) Al.
Now let be any subset of such that each pair satisfies and Define an equivalence relation by: iff there exists a sequence such that and for some and with or If then we say that and are equivalent by means of the operations (or relations) in Remark that in that case and This is the smallest equivalence relation with the property and hence we call the equivalence relation generated by
(1.2) A fundamental topological theorem.
Let be a topological space and a cover of consisting of open and simply connected sets. We choose in each a point and now we want to give a description of the groupoid which has as objects the set and for consists of the homotopy classes of paths from to This subgroupoid of the fundamental groupoid of essentially describes the latter, because of the simply connectedness of the sets in
(1.3) Definition. A pregallery consists of a sequence of elements of such that for together with a sequence where is a connected component of for We set and This pregallery is denoted:
If then obviously we can define the pregallery This way we get a category with objects and whose morphisms are the pregalleries.
A pregallery of the form is called direct. Remark that the category is in fact the free category on the direct pregalleries, so we can use the terminology of (1.1).
Now we assign to each direct pregallery an element of in the following way: Choose and a path in from to and a path in from to Now the homotopy class of doesn't depend on the choice of and For let also and be chosen. Choose also a path in from to is connected). Because and stay inside the simply connected set we have Also, So The class defined this way is denoted by Because of freeness extends in a unique way to a functor
(1.4) Definition. Consider the pairs
and let be the equivalence relation on generated by
The following fundamental theorem says that can be described as the quotient category This observation is the basis for all other presentations of fundamental groupoids and groups in the sequel.
(1.5) Theorem. The functor is surjective and iff
For the proof it is convenient to have the following:
(1.6) Definition. Let be a continuous curve. We call a pregallery linked with by a division notation: if:
If there exists a division such that then we say that is linked with notation
(1.7) Remark. If is a sequence of elements of and a curve and a division of such that condition (i) of (1.6) is satisfied, then it follows that so for some component of Of course then where
|Proof of (1.5).|
First the surjectivity of Let be a continuous curve from to The collection forms an open cover of so by compactness of there is a division such that is contained in for some As remarked in (1.7) we now know that there is a pregallery such that The surjectivity of follows now from the
Claim. If then
Now we prove: It is sufficient to show that for each pair from i.e. Choose then a point Let be curves from and respectively to inside and respectively. Then
We also have to remark that but this is true by the simply connectedness of
We prove the converse: in four steps:
Step 1. If and then
Step 2. If and is a refinement of then there exists a such that and
Step 3. If then
Step 4. Let If and then
To finish the proof of theorem (1.5), suppose For the defining curves and (i.e. and certainly and holds. And from it follows that so by step 3 (Suggested correction: 4) we have
Let be a real finite dimensional vector space.
(2.1) If is a hyperplane and the set is contained in one of the two open halfspaces determined by (i.e. components of then we denote this halfspace by For singletons we also write
Let be an open set (so "open in and "open in is the same thing) and a collection of hyperplanes of (not necessarily containing the origin), which is locally finite on Recall that this means, chat each has a neighbourhood such that for only finitely many
(2.2) Proposition. Let and be as above. Then is open.
Let There is a neighbourhood of such that is finite. Then is a neighbourhood of inside
(2.3) Corollary. The components of are open.
(2.4) Definition. These components are called chambers in with respect to We denote the collection of chambers by or by if no confusion is caused by the ambiguity.
If then is defined for all Let also be a chamber. If then there are two possibilities:
After these preliminaries we come to the main definitions of this paragraph.
(2.6) Definition. Let and be as in (2.1). If, moreover, is convex then the triple is called a hyperplane configuration.
(2.7) Definition. Let be a hyperplane configuration and let be given, where is a locally finite collection of hyperplanes in Then we call the quadruple a hyperplane system. Let us summarize the defining properties of a hyperplane system
|(i)||is a finite dimensional real vector space.|
|(ii)||is an open convex subset of|
|(iii)||is a collection of hyperplanes of locally finite on|
|(iv)||is an assignment where is a locally finite collection of hyperplanes of|
We associate a space (or briefly to a hyperplane system as follows:
In the sequel the complex structure of is not relevant. We merely use the notations for and Re and Im for the projections as it is convenient. Note that we can assume that if then and for otherwise doesn't occur in the definition of the space Remark also that is locally finite on hence is open in
(2.10) Definition. Together with each hyperplane configuration there is a canonical simple hyperplane system defined, also denoted by the triple by setting For instance, if the origin of is contained in all then can be considered as a complement of complex hyperplanes:
The theory we are going to develop has two main applications
In the latter case, the collection consists of hyperplanes all parallel to hence the use of the letter
Now we construct a cover of such that the result of §1 is applicable.
(2.11) Definition. For we define:
(2.12) Proposition. is open in
Let Choose and put Due to the compactness of and locally finiteness of on is finite. Then is locally finite on since the finite union of a locally finite collection is again locally finite. If then Thus so which means Therefore lies in a chamber Choose also a chamber such that This is possible, because is connected and for any we have Now we have for if and then so which implies that So is a neighbourhood of inside
Let Choose an open neighbourhood of such that is finite. By deleting from (which is closed because of the finiteness of we see that we can suppose that Moreover we may assume that is connected. Then is defined for all and Because the interior of is empty there has to be a such that For this we have for all Now let Then so Then So we have proved that
(2.14) Corollary (from 2.12) and (2.13)). The space is open in and is an open cover of
In order to prove that is simply connected (even contractible) and to study the intersections we consider now the space for certain First observe:
In particular, if and then The latter is equivalent to and this means that for So if we give the following:
(2.15) Definition. Let and
($) Then we have Now note first that is locally finite on for we have the following:
(2.16) Lemma. If then
By compactness of where and and locally finiteness of on we see that is finite. But it is easy to see that
From ($) we see that the projection maps the set into the union of chambers with respect to In fact we have the following:
(2.17) Proposition. The projection maps each connected component of onto a chamber w.r.t. This sets up a bijective correspondence between the connected components of and Furthermore, these components are contractible.
The projection Re maps each component of into a component of the union of the chambers w.r.t. i.e. into a chamber. Choose then for each Then for each holds. So ($$) and the latter is nonempty. We want to show that it is contractible. If then For if then for and so also for So we have i.e. From this we can see that is a deformation retract of itself is contractible (for so is so is indeed contractible. Hence for each there is one and only one connected, even contractible, component of namely that is mapped onto (this follows from ($$)) by the projection Re.
(2.18) Corollary If then is contractible, in particular simply connected.
(2.19) Corollary If then the components of are in a bijective correspondence with the elements of This correspondence is given by
Denote such a component for the moment by From (2.18) it follows that we have the situation of (1.2). A pregallery is now given by a sequence together with (by (2.19)) a sequence where This pregallery is denoted by and from now on all the pregalleries we are considering, are of this type. We need one more corollary of (2.17):
(2.20) Corollary Let and and Then
is easy. For choose Then for all hyperplanes we have so there is a with According to proposition (2.17) the component is mapped onto by Re, so it contains a with Now we have
Combining the results of §1 and §2 we now get the main result of this section:
(2.21) Theorem. Let be a hyperplane system. Let be the free category on the direct pregalleries where and Then there is a surjective functor where is the subgroupoid of the fundamental groupoid of consisting of all homotopy classes of paths from and to points of the form where in each there is chosen a point
Two pregalleries have the same if and only if they are equivalent by means of the relations
This is a restatement of theorem (1.5) for the case of hyperplane systems. By (2.20) the equivalence relations are indeed the same.
Given a path in the space (defined in §2), then by a small homotopy we can make the imaginary part avoid intersections of distinct hyperplanes of For the set of such points has codimension bigger then one. This leads to the idea, that it is always possible to choose the sequence of chambers of a pregallerv in such a way, that and are separated by only one hyperplane. Such a sequence we will call a Tits gallery. Tits himself and Deligne call this a gallery, but we prefer to use this word for the pregalleries that arise this way. Much of this paragraph can be found in Bourbaki V §1 [Bou1968] and in Deiigne [Del1972]. But our situation is slighty more general, because we consider only an open part of Let be a hyperplane configuration as defined in (2.6).
(3.1) Definition. Two points of are said to be on the same facet, if for each there are only two possibilities: (i) Both points are contained in (ii) The points are not contained in and they are moreover on the same side of i.e. if and are these two points. This is an equivalence relation and the classes are called facets of with respect to The collection of facets is denoted by or for short
For each (in particular for we define: and the support of Furthermore
From the definition it is clear that for we have
(In general, if is on one side of for instance if is convex and for all then
Remark. is just the closure of as subset of We denote this in the sequel by
|Proof of (3.3).|
From (3.2) it follows at once that the left hand side is contained in the right hand side. Conversely, let be contained in the right hand side and choose then and for all So But that means that
(3.4) Corollary. The closure (in of a facet is a (disjoint) union of facets.
For and each are so.
We also have a useful criterion to see if a facet is part of this union:
(3.5) Criterion. Let Then if and only if for all holds.
Let and then for if then and so a contradiction. So G is contained in one of both open half spaces of Now we have so which also means Conversely, suppose that the condition holds. Then is contained in and so by (3.3) also in which is to be proved). For if then by contraposition of the condition, so If then two possibilities arise: (1) and then certainly (2) and then by the condition so
(3.7) Definition. Let and
If then is called a face of
If then is called a plinth of
If is support of a face of is called a wall of
Let also a configuration be given and let and If then there is one and only one such that For if then for some and if is on the same facet w.r.t. and (definition(3.1)), then and are also on the same facet w.r.t. and so then Uniqueness follows from the fact that is a partition of
(3.8) Definition. The map obtained above is denoted by or if then For we define:
(cf. [Del1972] Deligne 1.5(ii)).
The map restricted to is bijective.
Injectivity. Suppose and For we have iff So iff And if then so then and on the same side of
If then and by criterion (3.5) and the fact that for we see that So also in this case and on the same side of We have seen that for all for or and on the same side of holds, so
Surjectivity. Let Then It follows that so each neighbourhood of a point has points in common with Choose a neighbourhood of such that for all and choose a point Let be the facet such that Then and (what was to be proved). The latter is clear from We prove the fact that holds, by criterion (3.5): Suppose because we have so
(3.10) Remark. and in particular (by (3.9))
If and then it doesn't harm to suppose that the origin is contained in this intersection. This way we can define in this case for each
(3.11) Definition. Let and Then we define the reflection of the facet in the facet by: Remark again if
(3.12) Lemma. If then iff
See Bourbaki [Bou1968] V §1.4.
(3.13) Theorem. If then
See Bourbaki [Bou1968] V §1.4 prop.9.
(These proofs also work in our (more general) situation)
(3.14) Corollary. If and then
If then for all so so conflicting
We need some propositions about separating hvperplanes.
(3.15) Proposition. If then
|(i)||symmetric difference of sets)|
(i) For any consider
(3.16) Proposition. If then the following are equivalent.
(i) (ii) follows from (3.15)(i). (i) and (ii) (iii) is obvious. (iii) (i) follows from (3.15)(ii).
(3.17) Proposition. If is a sequence of elements of then
Remark. For the implication (ii)a. need not hold (it does hold for by proposition (3.16) (ii) (i)), consider for instance the configuration of figure 7.
|Proof of (3.17).|
Trivial for Suppose proved for
Then by (3.15) (i) we have
and by induction we know that the latter is contained in
This proves (i) for also. We always have:
(3.18) Definition. We call a sequence of chambers a Tits gallery from to if for The number is called the length of the Tits gallery. The only element of is often denoted by
(3.19) Remark. This Tits gallery is completely determined by its first chamber and the sequence
Let and Then and so and are both unequal to the latter. By lemma (3.12) this means that Then is the support of a common face of and So is determined by and by the rule The remark follows now.
(3.20) Corollary of (3.17). For Tits gallery are equivalent:
and in that case we have
(3.21) Definition If the conditions a. and b. of (3.20) are fulfilled then we call such a Tits gallery a direct Tits gallery. (Deligne uses the word minimal but we want to use that word for other things)
(3.22) Theorem (Deligne [Del1972 1.3]). If then there exists a direct Tits gallery from to
Induction on If then then then trivial. Now let and let the theorem be proved in cases where the number of separating hyperplanes is less. By corollary (3.14) we know that Let and let be the face of supported by Set Then we have by (3.15)(i) since By induction there is a direct Tits gallery from to Then is a direct Tits gallery from to
(3.23) Theorem. Let If for some plinth of then there are exactly two direct Tits galleries from to
First we remark that, if and are chambers with a common plinth then For let so Also so by criterion (3.5) both and are contained in hence It follows that where is a (direct) Tits gallery iff is a (direct) Tits gallery. The theorem follows from
(3.24) Lemma. If and then there are exactly two direct Tits galleries from a chamber to the chamber
(3.25) Definition. The equivalence relation in the set (or better: the free category) of all Tits galleries, generated by the pairs and are the two direct Tits galleries from to where is a plinth of some chamber is called plinth equivalence and denoted
(3.26) Theorem. Let and suppose that and are the two direct Tits galleries from to Then
Remark. Both theorem and proof are essentially due to Deligne [Del1972] 1.12. We don't want to use his condition that the chambers are simplicial cones. Therefore we need the simple but technically somewhat complicated lemma (3.28). For the proof we first give a definition:
(3.27) Definition. If and are two faces of a chamber then we call and plinthed if there exists a plinth of such that The supporting walls are also said to be plinthed w.r.t.
|Proof of (3.26).|
If then there is nothing to prove. Else, let be the direct Tits gallery and Suppose and
Case 1. If then (see remark (3.19)) and then and are two direct Tits galleries from to By induction they are plinth equivalent. But then also
Case 2. Let and and are plinthed walls of by a plinth Let be the unique direct Tits gallery from to that starts with and the one that starts with Let be a direct Tits gallery from to By (3.15)(i) and the latter equals since So by (3.20) and are direct Tits galleries from to Because and (respectively and start with (respectively by step 1 (respectively holds. It is clear that so
Case 3. General case. By lemma (3.28) (proved in the sequel) there is a sequence such that and are plinthed walls of for For there exists a direct Tits gallery from to starting with Let and By step 2 we have for So There remains:
(3.28) Lemma. If where then there is a sequence such that and are plinthed walls of for and
Theorem (3.26) is proved.
(3.29) Corollary of (3.22) and (3.26). Let be fixed. Then induces a bijection between the plinth equivalence classes of direct Tits galleries with and
In this section we combine the results of §2 and §3 to get the main result of this chapter. Let be again a hyperplane system as in §2.
(4.1) Definition. We call a pregallery a gallery if is a Tits gallery. The number is called the length. The subcategory of we obtain is denoted For we denote the length by
(4.2) Proposition. Let be a direct Tits gallery. Given denote by the unique element of which contains Then Conversely, if for and Then and
(In fact the condition that is a direct Tits gallery, is too strong. It is enough to require that condition (ii)a. of proposition (3.17) holds: if Let define and let be as in the proposition i.e. Then and so we have So
Second part: is convex, so for some if and then for Then also by proposition (3.17). On the other hand we have for because is convex and if and then So Moreover and so as in the first part we conclude the equivalence of the pregalleries.
(4.3) Definition. Such a gallery with and a direct Tits gallery we call a direct gallery.
(4.4) Corollary. Every direct gallery is pregallery-equivalent to a unique direct pregallery. Each direct pregallery is pregallery-equivalent to a direct gallery (which is in general not unique).
Clear from proposition (4.2). For the second statement we also need theorem (3.22).
In this case we call the gallery a direct repacing gallery of the given direct pregallery.
(4.5) Definition. Let and for some plinth of Suppose According to theorem (3.23) there are exactly two direct Tits galleries from to Let and be the two direct replacing galleries belonging to by (4.2). Then we call the pair a flip relation. The equivalence relation in generated by these flip relations is called flip equivalence and denoted by
(4.6) Corollary of proposition (4.2) and theorem (3.26): Let and be two direct replacing galleries of a direct pregallery. Then
(4.7) Definition. Pairs of type we call cancel relations. The equivalence relation in generated by the flip relations and the cancel relations is called gallery equivalence or simply equivalence and is denoted by (note that see (5.5)).
(4.8) Proposition. If then
For the proof we need:
(4.9) Lemma. Let Then for (This means that if two pregalleries are pregallery-equivalent, then this equivalence can be realised by a sharpened sort of pregallery-equivalence relations, i.e. one can suppose that
The part is trivial. For the part it is sufficient to show that if We proceed by induction on If then we are done. Let Choose a (exists by (3.14)) and define Then Put:
Let and suppose Then (for and (for so It follows that So for some So now we have for and Since the induction hypothesis says that for Finally for and
|Proof of proposition (4.8).|
Sufficient for the part is: 1.
and are pregallery-equivalent. Well,
2. If and are as in the definition of flip relation (4.5), then
and To prove:
be a sequence of pregalleries such that we obtain from
by a pregallery-equivalence operation. By lemma (4.9) we may assume that these operations are in the there defined sharpened form. Choose now each
a gallery in the following way:
substitute for each direct piece a replacing direct gallery (4.4). Then automatically
because these were already galleries.
(£) This suffices, because all other replacing direct galleries are (even flip) equivalent by (4.6). For the proof of (£) let and consider two cases: (1) In this case is direct because and so we have even (2) Then is a replacing direct gallery of So and we have
Summarising we have the main result of this chapter:
(4.10) Theorem. Let be a hyperplane system. Let be the category of all galleries i.e. the free category generated by the so-called one step galleries where and are chambers separated by one common wall and Let in each a point be chosen and let be the subgroupoid of the fundamental groupoid of this space (2.9), where for consists of the homotopy classes of paths from to
Then there exists a surjective functor Moreover the equivalence relation on "having the same is generated by the
where and are the two direct replacing galleries of a direct pregallery plinth of and
The required functor is the restriction to of the functor of theorem (2.21). This restriction remains surjective, because each pregallery is pregallery-equivalent to some gallery by (4.4). The assertion about the relations follows from the corresponding one in (2.21) and proposition (4.8).
This theorem describes the fundamental groupoid of the space as a combinatorial object, namely the quotient category which is even more related to the structure of the hyperplane system than (cf. (1.5) and (2.21)).
In the case of a simple hyperplane system we have for all So if is a one-step gallery with then we have two possibilities: or In the first case the one-step gallery is called positive and we denote it by otherwise it is called negative and denoted Therefore an arbitrary gallery is now of the form
where is a Tits gallery and For this reason we will call them sometimes signed galleries. Such a signed gallery is called positive (respectively negative), if (respectively for There is a bijective correspondence between the Tits galleries and the positive galleries.
(5.1) Proposition. A positive gallery is direct if and only if the underlying Tits gallery is so.
To see the if-part of this statement, note that if is a direct Tits gallery and then for all the are different (def.(3.21)). So and thus Hence is direct as a gallery (def. (4.3)).
Corresponding to the two possibilities of the cancel relations are now the pairs of the type:
The flip relations are the pairs:
where and are the two direct Tits galleries from to for some plinth of and in the sequence of signs there is at most one change of sign.
|Proof of this statement.|
If (5.3) is a plinth relation, then there is a where and that "prescribes signs" i.e. such that both galleries are replacing direct galleries of the pregallery Suppose for instance that Then and hence at most one change of sign (no change if or Conversely, if there is at most one change in sign, for instance and then is as above.
Example. To give examples we recall (3.19) that a Tits gallery is given by its first chamber and the sequence of separating walls. A signed gallery can be denoted by this sequence, underlining the elements of corresponding to the negative steps. For instance, if we number the elements of in Figure 10 from 1 to 4 then the galleries of the plinth relation can be denoted as 2 3 4 respectively 4 3 2 (assuming that one knows what the first chamber is).
We depict a signed gallery by drawing an arrow for positive one step galleries and an arrow for negative one step galleries. The following sequence illustrates the next remark
(5.4) Remark. Two signed galleries are equivalent iff they are equivalent by means of the cancel relations and the positive (i.e. with in (5.3)) flip relations.
Consider an arbitrary flip relatlon (5.3) with for instance and Then
is by means of cancel relations equivalent to
and this is by means of a positive flip relation equivalent to
and this is by means of cancel relations equivalent to
(5.5) We conclude this paragraph with an example of two equivalent galleries which are not flip equivalent:
They are equivalent for
This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.