Arrangements of spheres and projective spaces

Arrangements of spheres and projective spaces

Priyavrat Deshpande Chennai Mathematical Institute
India
pdeshpande@cmi.ac.in
Abstract.

We develop the theory of arrangements of spheres. Consider a finite collection of codimension- subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections and the topology of the tangent bundle complement. In the present paper we find a closed form formula for the homotopy type of the complement and express some of its topological invariants in terms of the associated combinatorial information.

Key words and phrases:
Sphere arrangements, Salvetti complex, Artin groups
2010 Mathematics Subject Classification:
20F36, 52C35

1. Introduction

An arrangement of hyperplanes is a finite set consisting of codimension- subspaces of . These hyperplanes and their intersections induce a polyhedral stratification of . The combinatorial information of an arrangement is contained in two posets, namely, the face poset which consists of all the strata and the intersection poset which contains all possible intersections of hyperplanes in . A topological space associated with , denoted , is the complement of the union of the complexified hyperplanes in . It is an open submanifold of with the homotopy type of a finite-dimensional CW complex [16, Section 5.1]. The study of this complement was initiated in the works of Fox and Neuwirth, Arnol’d, Brieskorn and Deligne in the ’s and ’s (see [16, Section 5.1]). One of the aspects of the theory of arrangements is to understand the interaction between the combinatorial data of an arrangement and the topology of . For example, the cohomology ring of the complement, known as the Orlik-Solomon algebra is completely determined by the intersection data [16, Section 5.4]. A pioneering result by Salvetti in [18] states that the homotopy type of the complement is determined by the face poset.

A generalization of hyperplane arrangements was introduced by the author in [9] where a study of arrangements of codimension- submanifolds in a smooth manifold was initiated. In this paper we focus on a particular example: arrangements of spheres. Given a smooth sphere we consider a finite collection of codimension- sub-spheres, denoted by , which satisfy reasonably nice conditions. For example, these sub-spheres are tamely embedded, their intersections are hyperplane-like and they induce a stratification of the ambient sphere such that all the strata are contractible. Consequently, one can define face and intersection posets in this context. The topological space associated with such a collection is the complement of the union of tangent bundles of these sub-spheres in . We call this space the tangent bundle complement and denote it by . We ask the same question, to what extent does the combinatorics of help determine the topology of ?

We explore this interaction of combinatorics and topology by first describing a regular cell complex that has the homotopy type of . The construction of this complex relies on the order relations in the face poset and is a generalization of the classical Salvetti complex. We then concentrate only on those arrangements which exhibit certain antipodal symmetry. For these so-called mirrored arrangements we find a closed form formula for the homotopy type of . We then show that the cohomology groups of are determined by the intersection data. Moreover, the coholomogy ring of can be expressed as a direct sum of an Orlik-Solomon algebra and a free abelian group in the top dimension. The rank of this top-dimensional free abelian group is equal to the number of graded pieces in the Orlik-Solomon algebra. We also identify a class of arrangements for which the word problem for is solvable.

In case of mirrored arrangements, as a consequence of the antipodal symmetry, we can define projective arrangements, i.e., a finite collection of subspaces homeomorphic to in . We exploit this antipodal symmetry to its full extent and derive similar results regarding the tangent bundle complement. For example, the antipodal map helps understand the homotopy type of the tangent bundle complement as well as its fundamental group.

An important motivation to study hyperplane arrangements comes from their natural connection with the Coxeter groups and the associated Artin groups. Let be a finite, irreducible Coxeter group of rank . It acts linearly (in fact as origin-fixing isometries) on a real vector space of dimension . Such a group is generated by reflections and has the following presentation -

Its action on is not free; each reflection in fixes a hyperplane. The union of these reflecting hyerplanes is the reflection arrangement, denoted , associated to . The complement of these fixed hyperplanes is a disjoint union of open simplicial cones called (Weyl) chambers. Under the action these chambers are permuted freely (see [7, Chapter 6]).

Complexifying this situation we get a finite arrangement of complex hyperplanes in . The complement of the union of these hyperplanes, denoted , is connected and admits a fixed point free action of . Brieskorn [3] showed that the fundamental group of the orbit space has the following presentation -

This group is known as the Artin group associated to and is denoted by . There is a natural surjection from onto whose kernel is the so-called pure Artin group . It is the fundamental group of . If is the symmetric group (i.e., type Coxeter group) then is the braid group and is the pure braid group.

Deligne [8] showed that the universal cover of is contractible. Hence is a space. Subsequent study of these groups is much influenced by Deligne’s work. Some of the important properties of Artin groups were proved by expanding on his ideas, notably the biautomatic nature of these groups [4]. Simply put, it says that the Artin groups have solvable word and conjugacy problem. We refer the reader to [5, Section 1.2] for details and references.

We investigate sphere arrangements with a similar motivation. It is well known that the finite subgroups of isometries of a sphere generated by reflections are in fact Coxeter groups (see [7, Chapter 10] and [12]). Each reflection in this Coxeter transformation group fixes a codimension- subsphere giving rise to a sphere arrangement. The complement of this arrangement is a disjoint union of ‘spherical’ simplices and they are freely permuted by the action. Since the group acts via isometries the action extends to the tangent bundle of the sphere. The complement of the union of the tangent bundles of the fixed sub-spheres serves as the analogue of the space introduced above. The Coxeter transformation group acts fixed point freely on this complement. The fundamental group of the orbit space is the desired generalization of Artin groups. The main aim of this paper is to lay topological foundations for the study of these “Artin-like” groups. We illustrate with an example.

Example 1.1.

Consider the -sphere . In this case a Coxeter transformation group is a dihedral group of order with the presentation . The reflections in fix -spheres i.e., points. Declaring one of the chambers as the fundamental chamber all others can be labeled by elements of . The points in this arrangement can be labeled by conjugates of the two standard parabolic subgroups of . See [7, Chapter 5] for details regarding such labeling.

The tangent bundle complement is an infinite cylinder with punctures. The Salvetti complex (see Section 3.6 for its construction) has -cells with labels for every . There are -cells with labels of the form where is one of the standard parabolic subgroups and . The reader can verify that the boundary of this -cell is such that . The ‘labeled’ Salvetti complex inherits the free -action on the tangent bundle complement. The orbit complex consists of exactly one -cell and two -cells with both their end points joined at the -cell. It has the homotopy type of wedge of two circles.

The generalized pure Artin group in this case is the free group on generators. The generalized Artin group is and we get the following exact sequence

If we were to denote the generators of as and then has the following presentation -

where is or depending on the parity of .

In a joint work with Ronno Das [6] we extend this correspondence to higher-dimensional spheres (in fact, to smooth manifolds). In particular, we show that the homotopy equivalence between the tangent bundle complement and the Salvetti complex is -equivariant. Further we explain how the group theoretic data can be used to define this complex. We also describe a presentation for the associated groups. Current work in progress, among other things, focuses on computation of the (twisted) cohomology of with group ring coefficients.

The paper is organized as follows. Section 2 is about the preliminaries of hyperplane arrangements. In Section 3 we introduce the new objects of study, arrangements of spheres and the tangent bundle complement. In Section 4 we look at how the combinatorics of intersections determines the topology of the complement. We investigate the fundamental group in Section 5. In Section 6 we look at arrangements of projective spaces.

2. Arrangements of Hyperplanes

Hyperplane arrangements arise naturally in geometric, algebraic and combinatorial instances. In this section we formally define hyperplane arrangements and the combinatorial data associated with it in the setting that is most relevant to our work.

Definition 2.1.

A (real) arrangement of hyperplanes is a collection of finitely many hyperplanes in , .

An arrangement is called central if the intersection of all the hyperplanes in is non-empty. However, we allow our arrangements to be non-central. For a subset of , the restriction of to is the subarrangement .

Analogously one can define hyperplane arrangements in which are called complex arrangements. To every real arrangement there is an associated complex arrangement ; for every there is a hyperplane with the same defining equations as . In this paper we focus on (complexified) real arrangements of hyperplanes.

Associated with there are two posets containing important information about the arrangement, namely, the face poset and the intersection poset.

Definition 2.2.

The intersection poset of is the set of all intersections of hyperplanes, including itself as the empty intersection, ordered by reverse inclusion.

The intersection poset is a ranked poset with the rank of an element being the codimension of the corresponding intersection. The rank of an arrangement is defined to be the rank of its intersection poset. An arrangement is said to be essential if its rank equals the dimension of the ambient space; without loss of generality we will from now assume this to be the case. In general a (meet) semilattice; it is a lattice if and only if the arrangement is central.

Definition 2.3.

The face poset of is the set of all faces ordered by topological inclusion: if and only if .

Codimension- faces are called chambers. The set of all chambers will be denoted by . A chamber is called bounded if it is a bounded subset of . Two chambers and are adjacent if they have a common face in their closure.

The topological space associated with a real hyperplane arrangement is its complexified complement which is defined as follows:

Definition 2.4.

where is the hyperplane in with the same defining equation as .

2.1. The Salvetti Complex

In [18] Salvetti constructed a regular CW-complex which has the homotopy type of the complexified complement. The construction uses the ordering in .

Let be a hyperplane arrangement in . We construct a regular -complex, called the Salvetti complex and denoted by , by first describing its cells. The -cells, for , of are in one-to-one correspondence with the pairs , where is a codimension- face of and is a chamber whose closure contains .

Since is regular all the attaching maps are homeomorphisms. Hence it is enough to specify the boundary of each cell. A cell labelled is contained in the boundary of another cell labelled if and only if in and are contained in the same chamber of . Now we state the seminal result of Salvetti.

Theorem 2.5 (Salvetti [18]).

Let be an arrangement of real hyperplanes and be the complement of its complexification inside . Then there is an embedding of into . Moreover there is a natural map in the other direction which is a deformation retraction.

The above construction is generalized by Björner and Ziegler in [2] where authors give a CW-complex with the homotopy type of the complement of a complex subspace arrangement.

2.2. Cohomology of the Complement

We begin by associating a combinatorially defined algebra, called the Orlik-Solomon algebra, to a (complex) hyperplane arrangement.

Let be a hyperplane arrangement. For every call a -tuple of hyperplanes to be independent if and call it dependent if the intersection is nonempty and its codimension is strictly less than . Geometrically, the independence implies that the hyperplanes of are in general position.

Let be the free -module generated by the elements for every . Define to be the exterior algebra on and let denote the differential in . For a -tuple of hyperplanes we denote by the intersection of elements in and by we mean . Let denote the ideal of generated by

Definition 2.6.

The Orlik-Solomon algebra of a (complex) arrangement is the quotient algebra and denoted by .

The following important theorem shows how cohomology of depends on the intersection poset. It combines the work of Arnold, Brieskorn, Orlik and Solomon. For details and exact statements of their individual results see [16, Chapter 3, Section 5.4].

Theorem 2.7.

Let be a hyperplane arrangement in . For choose a linear form , such that . Then the integral cohomology algebra of the complement is generated by the classes

The map defined by

induces an isomorphism of graded -algebras.

This theorem asserts that a presentation of the cohomology algebra of can be constructed from the data that are encoded by the intersection poset. Let us mention one more theorem that explicitly states the role of the intersection poset in determining the cohomology of the complement. In particular the result states that there is a finer grading of cohomology groups indexed by the intersections and the rank of each cohomology group is determined by the Möbius function of the intersection poset (see [16, Proposition 3.75, Lemma 5.91]).

Theorem 2.8.

Let be a nonempty complex arrangement. For let denote the complexified complement of the restricted arrangement . There are following isomorphisms for each

induced by the inclusions . Moreover the rank of each cohomology group is determined by the following formula

where is the Möbius function of .

3. Arrangements of Spheres

We now introduce arrangements of codimension- subspheres in a sphere. First we isolate essential properties of a hyperplane arrangement:

  1. there are finitely many codimension subspaces each of which separates into two components;

  2. there is a polyhedral stratification of the ambient space and the face poset of this stratification has the homotopy type of the ambient space.

Remark 3.1.

Recall that associated to every poset there is an abstract simplicial complex known as the order complex. A -simplex of the order complex corresponds to a -chain of the poset. By homotopy type of the poset we mean the homotopy type of the geometric realization of the associated order complex.

In this section we first generalize the above properties in the context of spheres. Then we compare our definition with the topological representation of oriented matroids. Finally, we look at the combinatorics of the sphere arrangements.

3.1. Codimension- tame subspaces of Spheres

We start with a generalization of the property above. By an -sphere we mean a smooth, closed -manifold homeomorphic to the unit sphere in . The -sphere consists of two points and we assume that the empty set is the sphere of dimension .

If is an -sphere embedded in () as a closed subset then has two connected components. Hence codimension- subspheres generalize hyperplanes in this respect. In general codimension- subspheres in a sphere could be very difficult to deal with. For example, consider the Alexander horned sphere. It is an embedding of inside such that one of the connected component of the complement is not even simply connected. In order to avoid such pathological instances we restrict ourselves to a nice class. A codimension- subsphere of is said to be tame (or locally flat) if for every there is a neighbourhood of in such that . For such a subsphere the following statements are equivalent(see [17, Theorem 1.8.2]):

  1. there exists a homeomorphism of onto the standard unit sphere in such that is the equator cut out by the coordinate hyperplane ;

  2. is homeomorphic to a piecewise-linearly embedded -subsphere;

  3. the closure of each connected component of is homeomorphic to the -ball.

Tame subspheres need not intersect like hyperplanes. As an example, consider Figure 1, it shows the non-Pappus arrangement on the unit sphere in .

Figure 1. Non-Pappus arrangement

The cone over each of these circles is homeomorphic to a plane passing through the origin. Clearly there is no self-homeomorphism of such that the cone over the image of each of these circles is a -dimensional subspace. This picture can arise as the boundary of a neighborhood of -spheres intersecting in a -sphere. We would like to avoid such situations as we are interested in dealing with the tangent bundle.

We introduce a notion that will guarantee hyperplane-like intersections of subspheres. But first some notation. Let be a collection of tame, codimension- subspheres in . For every and an open neighbourhood of homeomorphic to let,

Denoting by we mean the union of elements of .

Definition 3.2.

Let be a collection of codimension-, tame sub-spheres of . We say that these sub-spheres have locally flat intersections if for every there exists an open neighbourhood and a homeomorphism such that where is a central hyperplane arrangement in with as the common point.

3.2. Cellular stratification

Now we generalize property (2). Let be a collection of codimension-, tame sub-spheres of with locally flat intersections. Let be the set of all possible nonempty intersections of members of and be the subset containing codimension- intersections. We have and . For each Consider the following subset of

Note that each may be disconnected and that the sphere can be expressed as the disjoint union of these connected components. We want these sets to define a ‘nice’ stratification of hence we introduce the language of cellular stratified spaces developed by Tamaki in [21]. Recall that a subset of a topological space is said to be locally closed if every has a neighborhood in with closed in .

Definition 3.3.

Let be a topological space and be a poset. A stratification of indexed by is a surjective map satisfying the following properties:

  1. For , is connected and locally closed;

  2. for ;

  3. .

The subspace is called the stratum with index .

One can verify that the boundary of each stratum, is itself a union of strata. Such a stratification gives a decomposition of . The indexing poset is called the face poset of the stratification.

It is now easy to check that the connected components of define a stratification of . However it is not desirable to consider arbitrary stratifications. For example, in the case of two non-intersecting circles in there are three codimension- strata and two codimension- strata. But the resulting face poset does not have the homotopy type of the -sphere. We need to focus on stratifications such that the strata are cells and the incidence relations between the strata recover the homotopy type of . In order to achieve property (3) we assume that each stratum is a cell and the resulting stratification is a regular CW-complex.

3.3. Definitions and examples

The desired generalization of hyperplane arrangements is the following:

Definition 3.4.

Let be a smooth sphere of dimension . An arrangement of spheres is a finite collection of codimension- smooth subspheres in such that:

  1. the ’s have locally flat intersections (see Definition 3.2);

  2. for all the intersection is a sphere of some dimension;

  3. if , for some and some , then is a codimension- subsphere in ;

  4. the stratification induced by the intersections of ’s define the structure of a regular CW-complex.

If there exists a fixed-point free, involutive diffeomorphism of the sphere such that for each we have and then we call a centrally symmetric arrangement of spheres.

As in the case of hyperplane arrangements the combinatorial information associated with sphere arrangements is contained in the two posets which we now define.

Definition 3.5.

The intersection poset denoted by is the set of connected components of all possible nonempty intersections of ’s ordered by reverse inclusion. By convention as the least element.

The intersection poset is a ranked poset. The rank of each element in is defined to be the codimension of the corresponding intersection.

Definition 3.6.

The intersections of these ’s in define a stratification of . The connected components in each stratum are called faces. The collection of all the faces ordered by topological inclusion i.e., is called the face poset. The top-dimensional faces are called chambers and the set of all chambers is denoted by .

For a face define its support as the least-dimensional intersection containing . The dimension of a face is the dimension of its support. It is straightforward to see that the dimension function makes the face poset a ranked poset.

We now look at two examples of sphere arrangements.

Example 3.7.

Let be the circle , a smooth one-dimensional manifold. The codimension- subspheres are the pairs of (diametrically opposite) points in . Consider the arrangement of such points. For both these points there is an open neighbourhood which is homeomorphic to an arrangement of a point in . Figure 2 shows this arrangement and the Hasse diagrams of the face poset and the intersection poset.

Figure 2. Arrangement of points in a circle.
Example 3.8.

As a -dimensional example consider an arrangement of great circles in . Figure 3 shows this arrangement and the related posets. The face poset has two -cells, four -cells and four -cells. Also note that the geometric realization of the face poset has the homotopy type of .

Figure 3. Arrangement of circles in a sphere.

3.4. Topological representation of oriented matroids

We now explore a connection between sphere arrangements and hyperplane arrangements using oriented matroids. The theory of oriented matroids is intimately connected with hyperplane arrangements. This combinatorial structure combines the information contained in face and intersection posets of a hyperplane arrangement. There are several (axiomatic) ways of defining oriented matroids. We refer the reader to the book of Björner et. al. [1] for various aspects related to oriented matroids. We do not intend to define and explain the properties of oriented matroids. Our aim is to compare their topological representation with the sphere arrangements.

The oriented matroids which correspond to hyperplane arrangements are known as the realizable oriented matroids. There are oriented matroids that do not correspond to hyperplane arrangements (e.g., the non-Pappus configuration). Hence for a long time an important question in this field was to come up with the right topological model for oriented matroids. This was settled by Folkman and Lawrence in [10]. The Folkman-Lawrence Topological Representation Theorem states that in general oriented matroids correspond to certain collections of finitely many topological spheres and balls. These so-called pseudo-arrangements not only describe oriented matroids in the same way that and collections of half spaces describe an obvious combinatorial structure but there is a one-to-one correspondence between such arrangements and the oriented matroids.

In their original formulation Folkman and Lawrence introduced arrangements of pseudo-hemispheres. Much simplification of their ideas was achieved by A. Mandel in his thesis [14]. He defined the notion of sphere systems which we now state.

Definition 3.9.

A finite multi-set of codimension-, tame subspheres in is called a sphere system if the following conditions hold:

  1. is a sphere, for all .

  2. If for , and and are the two sides of , then is a subsphere in with sides and .

A sphere system is said to be essential if the intersection of all the sub-spheres is empty. It can be shown that the stratification of induced by an essential sphere system defines regular CW decomposition of the sphere [1, Proposition 5.1.5]. The topological representation theorem states that (loop-free) oriented matroids of rank (upto reorientation and isomorphism) are in one-to-one correspondence with centrally symmetric, essential sphere systems in . However, the sphere arrangements that we want to deal with are not general enough to represent oriented matroids. Note the differences between the definition of a sphere system and Definition 3.4.

  1. We assume that all intersections are locally flat,

  2. the arrangement is repetition-free, i.e., every subsphere appears exactly once.

Given a central and essential arrangement of hyperplanes consider its intersection with the unit sphere; these intersections define a centrally symmetric sphere arrangement in the sense of Definition 3.4. However, the converse need not be true. Figure 1 shows an arrangement of pseudo-circles which is a sphere arrangement in the sense of Definition 3.4 but it does not arise as an intersection with a central hyperplane arrangement.

The rank oriented matroids can be realized as centrally symmetric, repetition-free, essential sphere systems in . The reader can verify that such sphere systems are arrangements of spheres (since intersections of pseudo-circles are locally-flat). Consequently, there is a one-to-one correspondence between rank oriented matroids and the sphere arrangements in . However, not all higher rank oriented matroids can be realized using sphere arrangements. For example, one can construct a sphere system in such that there exists at least one intersection whose spherical neighborhood looks like Figure 1.

Remark 3.10.

We would like to clarify the distinction between Definition 3.4 and Definition 3.9. The concept of sphere system is more general than that of an arrangement of spheres. It is clear that every sphere arrangement is a sphere system. However the converse is not true in general. There are examples of oriented matroids (say, of rank ) which do not correspond to any sphere arrangement in . In light of this observation it would be an interesting problem to obtain a combinatorial characterization of (non-realizable) oriented matroids which correspond to sphere arrangements.

A pseudo-hyperplane is a tame embedding of a codimension- subspace in . Equivalently, it is the cone over a tame subsphere. An arrangement of pseudo-hyperplanes, intuitively, can be constructed by taking the cone over a sphere system.

Definition 3.11.

A finite collection of pseudo-hyperplanes in is called an arrangement of pseudo-hyperplanes if:

  1. For every the set is either empty or homeomorphic to some for .

  2. For every either , or is a locally flat embedding of a codimension- subspace of .

We say that a pseudo-hyperplane arrangement is locally flat if all the (nonempty) intersections are locally flat.

The construction of the Salvetti complex and the Orlik-Solomon algebra hold true in case of pseudo-hyperplane arrangements. In fact, the Orlik-Solomon algebra associated to a pseudo-hyperplane arrangement is isomorphic to the cohomology algebra of the corresponding Salvetti complex (see [1, Section 2.5] and [2, Section 7] for details).

3.5. Combinatorics of sphere arrangements

We now take a closer look at the combinatorics of the incidence relations among the faces. Let denote a sphere arrangement in . A hypersphere in is said to separate two chambers and if they are contained in the distinct connected components of . For two chambers the set of all the hyperspheres that separate these two chambers is denoted by . The following lemma is now evident.

Lemma 3.12.

Let be an arrangement of spheres in , an -sphere. Let be three chambers of this arrangement. Then,

The distance between two chambers is defined as the cardinality of and denoted by . Given a face and a chamber of a sphere arrangement define the action of on as follows:

Definition 3.13.

A face acts on a chamber to produce another chamber satisfying:

  1. ,

  2. .

Lemma 3.14.

With the same notation as above, the chamber always exists and is unique.

Proof.

Clear. ∎

It is easy to check that if is a chamber and are two faces such that . Moreover if then .

3.6. The tangent bundle complement

Recall that for a real hyperplane arrangement the complexified complement is the complement of the union of complexified hyperplanes inside the complexified ambient vector space (Definition 2.4). If one were to forget the complex structure on then, topologically, it is just the tangent bundle of . Same is true for a hyperplane and its complexification . Hence the complexified complement of a hyperplane arrangement can also be considered as a complement inside the tangent bundle. We use this topological viewpoint to define a generalization of for sphere arrangements.

Definition 3.15.

As before let be a sphere arrangement in . Let denote the tangent bundle of and let . The tangent bundle complement of is defined as

The above space was introduced in [9, Chapter 3] in the context of submanifold arrangements. We now construct a regular CW-complex, in the spirit of Salvetti’s construction, that has the homotopy type of the tangent bundle complement.

We denote by the regular cell structure of induced by . We are interested in the dual cell structure which is obtained as follows. For every face fix a point call it the barycenter of . Note that is homeomorphic to an appropriate-dimensional disc . Then there exists a regular cell structure of whose face poset is isomorphic to that of . For every the barycenter determines a point of . Moreover, if is a chain of faces of then form a simplex of which is the convex hull of the vertices . Denote by the image of under the given homeomorphism. Note that need not be the convex hull of . Finally, denote by the union of all those ’s which arise from chains ending in and call it the dual cell of . The collection of all the dual cells defines a regular cell structure since link of each vertex is a sphere. We denote by this dual cell structure. Here is the face poset of this cell structure with the partial order . Note that is dual poset of , i.e., . Every -face in corresponds to an -cell in for .

For the sake of notational simplicity we will denote the dual cell complex by (and by if the context is clear). Note that a -cell is a vertex of a -cell in if and only if the closure of the corresponding chamber contains the -face . The action of the faces on chambers that was introduced in Definition 3.13 is also valid for the dual cells. The symbol will denote the vertex of which is dual to the unique chamber closest to .

Given a sphere arrangement in construct a regular -complex as follows:

The -cells of correspond to -cells of , which we denote by the pairs .
For each -cell with vertices , take two homeomorphic copies of denoted by and . Attach these two -cells in (the -skeleton) such that

for . We put an orientation on the -skeleton by directing each -cell such that the initial vertex is .

By induction assume that we have constructed the -skeleton of , . To each -cell and to each of its vertex assign a -cell whose face poset is isomorphic to that of . Let be the same characteristic map that identifies a -cell with the -cell . Extend the map to the whole of and use it as the attaching map, hence obtaining the -skeleton. The boundary of every -cell is given by

(3.1)

Now we state the theorem that justifies the construction of this cell complex.

Theorem 3.16.

The regular CW-complex constructed above has the homotopy type of the tangent bundle complement .

Proof.

This is a special case of [9, Theorem 3.3.7]. We only sketch the proof here. The first step is to identify an open cover of the sphere indexed by the faces. There are three key properties that these open sets satisfy. First, for every face the corresponding open set is a regular neighborhood of . Second, for another face the intersection if and only if . Finally, all these open sets and their non-empty intersections are contractible. Now for any point on the manifold the tangent space at that point contains a arrangement of hyperplanes combinatorially equivalent to the local arrangement. Using the local trivialization one can construct an open covering of which is indexed by the pairs such that each of the open set is contractible and so are their intersections. This type of open covering satisfies the hypothesis of the nerve lemma. The final step is to establish the condition when two such open sets have a non-empty intersection. Thus providing an isomorphism between the nerve of this open cover and the face poset of the Salvetti complex constructed above. ∎

Example 3.17.

As an example consider the arrangement of points in a circle (Example 3.7). The left side of the Figure 4 below illustrates the arrangement with the induced dual cell structure drawn using dotted lines. The right hand side shows the associated Salvetti complex with the cell labeling.

Figure 4. Arrangement in and the associated Salvetti complex

We now look at some obvious properties of the above defined CW structure and also infer some more information about the tangent bundle complement.

Theorem 3.18.

Let be a sphere arrangement in and let denote the associated Salvetti complex. Then

  1. there is a natural cellular map given by . The restriction of to the -skeleton is a bijection and in general

  2. For every chamber there is a cellular map taking to which is an embedding of into , and

  3. The absolute value of the Euler characteristic of is the number of chambers.

  4. Let denote the union of the tangent bundles of the submanifolds in then,

Proof.

Proofs of (1) and (2) are fairly straightforward. It follows that is homeomorphic to a retract of .

We prove (3) by explicitly counting cells in the Salvetti complex. The Euler characteristic of a CW-complex is equal to the alternating sum of the number of cells of each dimension. Given a -dimensional dual cell there are as many as -dimensional cells in . Hence for a -cell the number of -cells of with this particular vertex is equal to the number of -cells of that contain . The alternating sum of number of cells that contain a particular vertex of is equal to , where is the link of in . Applying this we get,

Since is compact all the chambers are bounded we have . Thus,

Hence,

Let denote the union of hyperspheres in . Since induces a regular cell decomposition has the homotopy type of wedge of -spheres. The claim (4) follows from the homeomorphism of pairs . ∎

Corollary 3.19.

Let be a sphere arrangement in . Then can not be an aspherical space.

Remark 3.20.

Let be a positive-dimensional intersection of . The restriction of to , i.e., the collection defines a sphere arrangement in . The reader can check using the inclusion (its restriction to ) from Theorem 3.18 that is homeomorphic to a retract of .

4. Topology of the Complement

The aim of this section is to investigate how the combinatorics of the associated posets affects the topology of the tangent bundle complement. Our investigation is based on a simple observation; if for a given centrally symmetric sphere arrangement there is an equator generically intersecting the sub-spheres then the restriction of the arrangement to the two hemispheres (i.e., components of the complement of the equator) gives combinatorially identical pseudo-hyperplane arrangements. We claim that these restricted pseudo-arrangements play a central role in understanding the topology of the complement. We identify a class of sphere arrangements for which it is easy to derive a closed form formula for the homotopy type of the complement. Then we establish a connection between the intersection poset and the cohomology groups.

4.1. The homotopy type of the complement

First we look at arrangements in . An arrangement in consists of copies of , i.e. points. The tangent bundle complement of such an arrangement is homeomorphic to the infinite cylinder with punctures. Thus we have the following theorem.

Theorem 4.1.

Let be an arrangement of -spheres in . If then

From now on we assume that all our spheres are simply connected. We say that two arrangements are combinatorially isomorphic if their corresponding face posets and intersection posets are isomorphic.

Let be an arrangement of pseudo-circles in . Then as a consequence of the Levi’s enlargement lemma [1, Proposition 6.4.3] there exists a pseudo-circle such that it is the equator with respect to the given antipodal map and it meets every member of in exactly one point. Let denote the connected components of . As the equator intersects every generically, is a pseudo-line in , respectively for . Denote by , the pseudo-line arrangements in the respective hemispheres. Then and are combinatorially isomorphic arrangements of pseudo-lines.

For the rest of the section we assume that is centrally symmetric and there exists a hypersphere in general position such that the restriction of to both the hemispheres (obtained by deleting ) result in combinatorially isomorphic, locally-flat pseudo-hyperplane arrangements (all the intersections are locally flat). This assumption motivates the following definition.

Definition 4.2.

A sphere arrangement in is said to be mirrored if it is centrally symmetric and there exists a pseudo-sphere such that restriction of to one of the connected components of is results in a pseudo-hyperplane arrangement combinatorially isomorphic to the restriction of to the other connected component.

Note that all sphere arrangements in dimensions are mirrored. Not all sphere arrangements in higher dimension are mirrored since the enlargement lemma fails in general. We refer the reader to [1, Proposition 10.4.5] for an example of pseudo-sphere arrangement in such that there is no equator in general position. However our assumption is not too restrictive, for example, arrangements corresponding to realizable oriented matroids are mirrored. If is a mirrored sphere arrangement then we denote by the restriction of to one of the hemispheres and by the restriction to the other. We note here that a lot of properties of hyperplane arrangements that we are interested in are also true for pseudo-hyperplane arrangements. For example, if all the intersections are locally flat then the construction of the associated Salvetti complex is same as described in Section 2.1 and it has the homotopy type of the tangent bundle complement [9, Theorem 3.3.7]

Here are two well known facts that we need.

Lemma 4.3.

If is a CW pair such that the inclusion is null homotopic then , where is the suspension of .

Proof.

See [13, Chapter 0].∎

Lemma 4.4.

Let be an essential and non-central arrangement of pseudo-hyperplanes in with locally flat intersections. Then the cell complex which is dual to the induced stratification is regular and homeomorphic to a closed ball of dimension .

Proof.

See [18, Lemma 9] and [19, Proposition 9]. ∎

Let be a chamber of and for any other chamber let denote the set of all those faces such that if and only if . Clearly all the chambers of are in and it is a disconnected set. In fact, if then is just the set of all chambers. Moreover, note that if and some face such that then . Let denote the regular cell complex dual to the stratification induced by and let denote the dual of . It is straightforward to verify that each connected component of is contractible and deformation retracts onto a subset of .

For a chamber of let denote the inclusion that takes a dual cell to the cell of the associated Salvetti complex. This inclusion is same as the one introduced in Theorem 3.18. In fact, most of Theorem 3.18 is true for pseudo-hyperplane arrangements (with appropriate modification in claim 3). For two distinct chambers of we define the following subset of the associated Salvetti complex

Lemma 4.5.

The subset is non-empty and disconnected.

Proof.

First observe that a cell if and only if . Which means that there is no hyperplane which contains and separates and . None of the cells obtained from a hyperplane can belong to and hence, since all the vertices of the Salvetti complex are in , the set is disconnected. In fact, .∎

Theorem 4.6.

Let be a mirrored sphere arrangement in . Let and be the pseudo-hyperplane arrangements in the two hemispheres and be the set of chambers of . Then the tangent bundle complement

Proof.

The assumption that the arrangement is mirrored implies that and are combinatorially isomorphic, non-central, essential pseudo-hyperplane arrangements with locally flat intersections. Let and let denote the dual cell complex . Define the map as:

Claim 1: The image of the map in is homeomorphic to .

Observe that is the restriction of the map defined in Theorem 3.18, which is an embedding. Hence maps homeomorphically onto its image.

Thus is the characteristic map which attaches the boundary to the -skeleton of . For notational simplicity let denote the restriction of to .

Claim 2: The image of is the boundary of an -cell in .

Consider the subcomplex of given by the cells . By Lemma 4.3 above this subcomplex is homeomorphic to the closed -ball. The boundary of this closed ball is precisely the image of .

The characteristic map is the extension of to the cone over (which is ). Hence is null homotopic. Let be any other chamber. In view of Lemma 4.5 the intersection set retracts onto the boundary . Then it follows from Lemma 4.3 that has the homotopy type of . Repeating these arguments for every chamber of establishes the theorem. ∎

We state the following obvious corollary for the sake of completeness.

Corollary 4.7.

Let be a mirrored sphere arrangement in . With notation as as before we have:

Example 4.8.

Consider the arrangement of circles in introduced in Example 3.8. It is clear that the arrangement in this case is the arrangement of two lines in that intersect in a single point. Hence

The Salvetti complex consists of four -cells, eight -cells and eight -cells. The -torus in the above formula corresponds to .

Example 4.9.

Finally, consider the arrangement of three s in that intersect like co-ordinate hyperplanes in . The in this case is the arrangement of co-ordinate hyperplanes hence , the -torus. This arrangement has chambers. So we have the following

Example 4.10.

Consider the arrangement of three circles in that intersect in general position. This arrangement arises as the intersection of with the coordinate hyperplanes in . In this case is the arrangement of three lines in general position. Figure 5 shows and the dual cell complex .

Figure 5. Restricted arrangement and the associated dual complex .

The intersection (see Lemma 4.5 above) contains the -cell . The boundaries of and collapse to a point hence has the homotopy type of . In general the tangent bundle complement has the homotopy type of .

4.2. Cohomology of the Complement

We now establish a relationship between the cohomology of the tangent bundle complement and the intersection poset. Let be a mirrored sphere arrangement in , let be the pseudo-hyperplane arrangement in the positive hemisphere. Let and denote the corresponding intersection posets. Observe that the map from to that sends to is one-to-one up to rank . If and denote the sub-posets consisting of elements of rank less than or equal to then the previous map is a poset isomorphism. For notational simplicity we use for .

Theorem 4.11.

With notation as above, we h