Arrangements of submanifolds and the tangent bundle complement
Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold we consider a finite collection of locally flat, codimension- submanifolds that intersect like hyperplanes. To such a collection we associate two combinatorial objects: the face category and the intersection poset. We also associate a topological space to the arrangement called the tangent bundle complement. It is the complement of union of tangent bundles of these submanifolds inside the tangent bundle of the ambient manifold. Our aim is to investigate the relationship between the combinatorics of the arrangement and the topology of the complement. In particular we show that the tangent bundle complement has the homotopy type of a finite cell complex. We generalize the classical theorem of Salvetti for hyperplane arrangements and show that this particular cell complex is completely determined by the face category.
Key words and phrases:Hyperplane arrangements, Salvetti complex, Nerve lemma, Acyclic categories
2010 Mathematics Subject Classification:52C35, 57N80, 05E45
An arrangement of hyperplanes is a finite set of hyperplanes in . These hyperplanes and their intersections induce a stratification of the ambient space. These strata form a poset when ordered by topological inclusion known as the face poset. The set of all possible intersections also forms a poset known as the intersection poset (usually ordered by reverse inclusion). These two posets contain combinatorial information about the arrangement. The topological spaces associated with an arrangement are the real complement and the complexified complement . The real complement is the complement of the union of hyperplanes in , whereas the complexified complement is the complement of the union of the complexified hyperplanes in . 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 these complements. For example, one would like to comprehend to what extent the combinatorial data of an arrangement control the topological invariants, such as (co)homology or homotopy groups etc., of these complements.
In this paper we introduce a generalization of real hyperplane arrangements which we call the arrangements of submanifolds of codimension-. We consider situations in which finitely many submanifolds of a given manifold intersect in a way that the local information is same as that of a hyperplane arrangement but the global picture is different. Intuitively, it means that for every point of the manifold there exists a coordinate neighborhood homeomorphic to an arrangement of real hyperplanes. We also introduce an analogue of the complexified complement in this new setting and call it the tangent bundle complement. This paper is an attempt to answer the following question: how does the combinatorics of the intersections of these submanifolds help determine the topology of the tangent bundle complement?
The topological study of the complement of a complexified hyperplane arrangement can be traced back to the work of Arnold on braid groups and configuration spaces. His results led Brieskorn to conjecture that the (complexified) complement of a real reflection arrangement is a space (i.e., its universal cover is contractible). In 1973, Deligne settled this conjecture in . His first step was to parametrize the universal cover of the complement by sequences of adjacent chambers. He showed that these galleries can be expressed in a particular normal form. This normal form was the main ingredient in proving that the constructed universal cover was contractible.
The combinatorial nature of Deligne’s arguments led to the search for combinatorial models for the complement, i.e., cell complexes built using the combinatorial data of an arrangement that are homotopy equivalent to the complement. One such model, the Salvetti complex, was introduced by Salvetti in . He showed that the incidence relations in the underlying face poset are sufficient to build this CW-complex. His construction was generalized to complex arrangements by Björner and Ziegler  where the authors also showed that the face poset determines the homeomorphism type of the complement. Combinatorial models for the covering spaces have also been constructed, see for example Delucchi  and Paris .
The crux of this paper is the generalization of Salvetti’s result. We prove that the incidence relations among the faces of a submanifold arrangement determine a cell complex that has the homotopy type of the tangent bundle complement (Theorem 3.5). We should also note that in the initial version of this paper we only considered the so-called combinatorially special arrangements (Section 4). Using a categorical formulation of the Salvetti complex by d’Antonio and Delucchi in  and also using the language of cellularly stratified spaces developed by Tamaki in [19, 20] we have extended our proof to the current setting.
The paper is organized as follows. We introduce the necessary background material in Section 1. In Section 2 we define the arrangement of submanifolds and provide some examples. We introduce the tangent bundle complement in Section 3 and prove that it has the homotopy type of a finite cell complex. We show that this cell complex is determined by the combinatorics of the incidence relations in the stratification induced by submanifold intersections. In Section 4 we focus on a particularly nice class of submanifold arrangements for which the combinatorial properties are similar to those of a zonotope. In particular, in Section 4.2, we use the theory of metrical-hemisphere complexes to describe the combinatorics. We end the paper in Section 5 by laying out future research and potential applications.
This paper is a part of the author’s doctoral thesis . The author would like to thank his supervisor Graham Denham for his support. This particular generalization of hyperplane arrangements and the complexified complement is due to Tduesz Januskeweisz and Richard Scott. It is a pleasure to acknowledge the discussions with Tduesz Januskeweisz, without which this work would not have been possible. Sincere thanks to Mario Salvetti for being the external examiner of the thesis and also for the warm hospitality in Pisa.
We start by briefly reviewing some basic facts about hyperplane arrangements.
1.1. Hyperplane Arrangements
Hyperplane arrangements arise naturally in geometric, algebraic and combinatorial instances. They occur in various settings such as finite dimensional projective or affine spaces defined over a field of any characteristic. Here we formally define hyperplane arrangements and the associated combinatorial and topological data in a setting that is most relevant to our work.
A real arrangement of hyperplanes is a collection of finitely many hyperplanes in , .
The rank of an arrangement is the largest dimension of the subspace spanned by the normals to the hyperplanes in . We assume that all our hyperplane arrangements are essential, i.e., their rank is equal to the dimension of the ambient vector space. An arrangement is called central if the intersection of all the hyperplanes in is non-empty. For a subset of , the restriction of to is the sub-arrangement .
There are two posets associated with , namely, the intersection poset and the face poset which contain important combinatorial information about the arrangement.
The intersection poset of is defined as the set of all intersections of hyperplanes ordered by reverse inclusion.
is a ranked poset with the rank of an element being the codimension of the corresponding intersection. In general it is a (meet) semilattice; it is a lattice if and only if the arrangement is central.
The hyperplanes of induce a stratification of such that each stratum is an open polyhedron; these strata are called the faces of .
The face poset of is the set of all faces ordered by topological inclusion: if and only if .
The Codimension- faces are called chambers. The set of all chambers is denoted by . A chamber is bounded if it is a bounded subset of . Two chambers and are adjacent if their closures have a non-empty intersection containing a codimension- face. By a complexified hyperplane arrangement we mean that an arrangement of hyperplanes in for which the defining equation of each hyperplane is real. Hence to every hyperplane arrangement in there corresponds an arrangement of hyperplanes in . An important topological space associated with a real hyperplane arrangement is the following
Let denote a hyperplane arrangement in . The complexified complement of such an arrangement is denoted by and defined as
where is the hyperplane in with the same defining equation as .
Note that since is of real codimension in , it is connected. In fact it is an open submanifold of with the homotopy type of a finite dimensional CW complex [14, Section 5.1].
1.2. The Salvetti Complex
For a hyperplane arrangement in there is a construction of a regular CW-complex, introduced by Salvetti . This complex has the same homotopy type as that of the complexified complement. The construction relies on the incidence relations among the faces.
We explain the construction of the 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 .
Theorem 1.5 (Salvetti ).
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.
1.3. Topological Combinatorics
We briefly review order complexes, the Nerve lemma and cell complexes. We assume the reader’s familiarity with the basic notions related to abstract simplicial complexes and posets. The main reference is Kozlov’s book .
A standard -simplex is defined as the convex hull of standard unit vectors in . The following construction will be used as a path to move from combinatorics to topology.
Given a finite abstract simplicial complex , its geometric realization is the topological space obtained by taking the union of standard -simplex in , for all simplices of dimension . Any topological space that is homeomorphic to the realization of is called the geometric realization of , and is denoted by .
A topological statement about an abstract simplicial complex is in fact a statement about its geometric realization. For the sake of simplicity, in this paper, we will not differentiate between an abstract simplicial complex and its geometric realization. We will let the context decide. A simplicial map between two simplicial complexes induces a continuous map between the corresponding geometric realizations.
The order complex of a poset is the abstract simplicial complex on vertex set whose -faces are the -chains in .
We will not differentiate between the order complex and its geometric realization. A poset map between two posets induces a simplicial map between the corresponding order complexes. Given a simplicial complex , the set of its faces ordered by inclusion forms a poset which is called the face poset. The order complex of this face poset is the (first) barycentric subdivision denoted by . The spaces and are homeomorphic. Thus from a topological viewpoint simplicial complexes and posets can be considered equivalent notions.
It is often the case that given some combinatorial data one would like to synthetically construct a topological space consistent with the data. The geometric simplicial complex is one such example. Here we consider the class of regular cell complexes. A subset of a topological space is called a closed (open) -cell if it is homeomorphic to (the interior of) the standard -ball in .
A regular cell complex is a pair consisting of a Hausdorff space and a finite collection of open cells in such that
the boundary of each cell is a union of some members of .
Given a (regular) CW-complex , the set consisting of its cells ordered by inclusion forms a poset. This poset is the face poset and is denoted by . We state a result concerning regular cell complexes which is important from the combinatorial viewpoint.
Let be a regular cell complex and be its face poset. Then
Furthermore, this homeomorphism can be chosen so that it restricts to a homeomorphism between and , for all .
However, we will need a more general setting which we now define.
A small category is called acyclic if only identity morphisms have inverses, and any morphism from an object to itself is an identity.
A poset can be regarded as an acyclic category as follows. The objects of this category are elements of . There is a unique morphism from to if . In fact, one can think of acyclic categories as generalization of posets in the following sense. Given an acyclic category there is a partial order on its objects given by . is called the underlying poset of . If is an acyclic category with at most morphism between any two objects then and its underlying posets are equivalent as acyclic categories. This is precisely what it will mean when we say that an acyclic category is (equivalent to) a poset.
As stated earlier the face poset of a simiplicial complex captures the incidence relations between simplices. Similarly an acyclic category encodes incidence relations among cells of more general type of cell complexes. Before we introduce those spaces let us look at a generalization of the order complex construction. We refer the reader to Kozlov’s book [12, Definition 10.4] for the precise technical definition.
The nerve (or the geometric realization) of an acyclic category , denoted by , is space constructed as follows:
The ordered vertex set of is the object set of .
For , there is a set of -simplices denoted by . This set contains all the composable morphism chains consisting of nonidentity morphisms.
A -simplex is glued to a face of a -simplex by the canonical linear homeomorphism between them that preserves the ordering of the vertices.
The geometric realization of an acyclic category is a regular trisp (delta complex). Recall that a delta complex is a quotient space of a collection of disjoint simplices obtained by identifying certain of their faces via canonical linear homeomorphisms that preserve the ordering of vertices. In the case of regular trisps there are no self identifications in the boundary of a simplex. See [12, Section 2.4] for details.
For an arbitrary acyclic category , let denote the poset whose minimal elements are objects of , and whose other elements are all composable morphism chains consisting of nonidentity morphisms of . The elementary order relations are given by composing morphisms in the chain, and by removing the first or the last morphism. The poset is called the barycentric subdivision of .
The barycentric subdivision of an acyclic category can also be defined as the face poset of its nerve. Hence there is a regular CW complex homeomorphic to the geometric realization of .
The Nerve Lemma often helps simplify a given topological space for combinatorial applications. Recall that an open covering of a topological space is a collection of open subsets of the space such that their union is the space itself. The nerve of an open covering is a simplicial complex whose vertices correspond to the open sets and a set of vertices spans a -simplex whenever the corresponding open sets have a nonempty intersection. In general the nerve need not reflect the topology of the ambient space, but the following result gives a useful condition when it does.
Theorem 1.14 (Theorem 15.21).
If is a finite open cover of a topological space such that every non empty intersection of open sets in is contractible, then .
1.4. Cellularly stratified spaces
We now introduce the language of cellular stratified spaces developed by Tamaki in [19, 20]. A hyperplane arrangement induces a stratification of the ambient space such that each stratum is a relatively open, convex polyhedron. These strata might be unbounded regions. Consequently such a stratification is not regarded as a CW-complex structure in the usual sense. One needs to appropriately modify the definition so that the induced stratification behaves like a regular CW-complex.
Let be a topological space and be a poset. A stratification of indexed by is a surjective map satisfying the following properties:
For , is connected and locally closed.
The subspace is called as the stratum with index .
A subset of a topological space is locally closed if is open in .
The reader can verify that the boundary of each stratum is itself a union of strata. Such a stratification defines a decomposition of . The poset is called the face poset of . A morphism of stratified spaces is a pair consisting of a continuous map and a poset map making the obvious diagram commute. is strict if for every . A subspace is a stratified subspace if the restriction is a stratification. When the inclusion of into is a strict morphism, is called a strict stratified subspace.
In general, each stratum of a stratified space could have non-trivial topology, however, we focus on those stratifications in which strata are cells.
A globular -cell is a subset of (the unit -disk) containing . We call the boundary of and denote it by . The number is called the (globular) dimension of .
Let be a Hausdorff space. A cellular stratification of is a pair of a stratification and a finite collection of continuous maps , called cell structures, satisfying following conditions:
and is a homeomorphism.
is a quotient map for every .
For each -cell the boundary contains cells of dimension for .
A cellularly stratified space is a triple , where is a cellular stratification.
We assume that all our cellularly stratified spaces (CS-spaces for short) have a finite number of cells. Consequently the cell structure is CW (i.e., each cell meets only finitely many other cells and X has the weak topology determined by the union of cells). Morphisms of CS-spaces and strict cellular stratified subspaces can be defined analogously. A cell is said to be regular if the structure map is a homeomorphism. A cell complex is regular if all its cells are regular.
Let be a CS-space. is called totally normal if, for each -cell ,
there exists a structure of regular cell complex on containing as a strict cellular stratified subspace of and
for any cell in , there exists such that and .
In particular, the closure of each -cell contains at least one cell of dimension , for every . Consider ; its boundary does not contain a -cell. This is an example of a CS-space which is not totally normal.
The face category of a cellularly stratified space is denoted by . The objects of this category are the cells of . For each pair , define (i.e., the set of all morphisms) to be the set of all maps such that .
Note that, in general, the set of morphisms between two objects in the face category has a non-trivial topology (see [19, Definition 4.1]). However, we assume, without loss of generality, that the set morphisms between any two objects is always finite.
Lemma 1.21 (Lemma 4.2 ).
If is a totally normal CS-space then its face category is acyclic. Moreover if is regular then is equivalent to the dual of the face poset of .
For CS-spaces the poset underlying its face category coincides with the (classical) face poset of a cell complex. The reason we are focusing on these spaces is the following lemma.
Let be an arrangement of hyperplanes in . Then the stratification induced by the hyperplanes in defines the structure of a totally normal, cellularly stratified space.
As an example consider the arrangement of coordinate axes in . For the closed cell the subset can serve as the globular cell.
We state one more property of these spaces which we wish to use in order to define arrangements of submanifolds. In particular, the following result generalizes Theorem 1.9.
Lemma 1.23 (Corollary 4.17).
For a totally normal cellularly stratified space , the nerve of its face category embeds in as a strong deformation retract.
If is a totally normal CS-space then one can also define its dual . The construction is such that is also a totally normal CS-space. There is an order-reversing poset isomorphism between the face poset of and that of . The attaching maps in are defined such that the face category of is the opposite category of .
1.5. Homotopy theoretic techniques
Now we state relevant results from homotopy theory applied to combinatorial settings. The approach we want to take here is more concrete; we follow the paper of Welker, Ziegler and Z̆ivaljević , see also [12, Chapter 15]. All of their results are for diagrams indexed by posets. We modify the results to our setting. The crucial observation is that in most of their results one can replace posets by acyclic categories.
A diagram of spaces indexed over an acyclic category is a covariant functor from an acyclic category to the category topological spaces and continuous maps (in this case will be called a -diagram).
Consider a diagram , then the homotopy colimit of is defined as
where the disjoint union is taken over all simplices in . The equivalence relation is defined by following condition: for , let be the inclusion map; then
for , is identified with the subset of by the map induced by ;
for , we have for any and .
Recall that in a small category and for some the comma category, denoted , is the category whose objects are the morphisms of ending at and morphisms are commuting triangles. In its simplest form the next theorem provides sufficient conditions for a map between two posets to be a homotopy equivalence at the level of order complexes.
Theorem 1.26 (Quillen’s theorem A).
Let be a functor of acyclic categories such that is contractible . If is any -diagram and the corresponding (pull-back) -diagram then,
We now look at an explicit connection between the nerve construction and the homotopy colimit construction.
Let be an acyclic category. A diagram of acyclic categories is a diagram indexed over that assigns an acyclic category to each .
Given a diagram of acyclic categories we construct a new acyclic category, called as the acyclic limit of . It is denoted by and defined as follows:
The morphisms in this category can be described as follows:
For a diagram of acyclic category its homotopy colimit is indeed a nerve as the following lemma suggests.
Lemma 1.28 (Simplicial Model Lemma).
Let be a diagram of acyclic categories. Then
The above lemma is a special case of the Grothendieck construction, which is used to define the homotopy colimit of a diagram of categories.
2. Arrangements of Submanifolds
In this section we propose a generalization of hyperplane arrangements. In order to do this we isolate the following characteristics of a hyperplane arrangement:
there are finitely many (translated) codimension- subspaces each of which separates into two components,
there is a stratification of the ambient space into open polyhedra,
the face poset of this stratification has the homotopy type of .
Any reasonable generalization of hyperplane arrangements should posses these properties. Since smooth manifolds are locally Euclidean they are obvious candidates for the ambient space. In this setting we can study arrangements of codimension- submanifolds that satisfy certain nice conditions. For example, locally, we would like our submanifolds to behave like hyperplanes.
2.1. Locally flat submanifolds
We now generalize property (1) mentioned above. Throughout this paper a manifold always has empty boundary. Our focus is on the codimension- submanifolds that are embedded as a closed subset of finite-dimensional smooth manifold. This type of submanifold behaves much like hyperplanes. We begin by exploring some of its well-known separation properties.
If is a connected -manifold and is a connected -manifold embedded in as a closed subset, then has either or components. If in addition then has exactly two components (i.e., separates ).
The lemma follows from the following exact sequence of pairs in mod homology:
Let be a connected -manifold and let be a connected, -submanifold embedded in as a closed subset. Then separates if and only if the inclusion induced homomorphism (on the cohomology with compact supports) is trivial.
The proof is a simple diagram chase:
In general a codimension- submanifold need not separate ; there are weaker separation conditions.
A connected codimension- submanifold in is said to be two-sided if has a neighbourhood such that has two connected components; otherwise is said to be one-sided. is locally two-sided in if each has arbitrarily small connected neighbourhoods such that has two components.
Note that being two-sided is a local condition. For example, a point in does not separate , however, it is two-sided. The following corollary follows from the above definition and the lemmas.
If is an -manifold and is an -manifold embedded in as a closed subset such that is trivial then is two-sided.
Every codimension- submanifold is locally two-sided in .
An -manifold contained in an -manifold is locally flat at ; if there exists a neighbourhood of in such that . An embedding such that is said to be locally flat at a point if is locally flat at . Embeddings and submanifolds are locally flat if they are locally flat at every point.
It is necessary to consider the locally flat class of submanifolds otherwise one could run into pathological situations. For example, the Alexander horned sphere is a non-flat embedding of inside such that the connected components of its complement are not even simply connected see Rushing [16, Page 65] for details.
Locally flat submanifolds need not intersect like hyperplanes. A simple example comes from the non-Pappus arrangement of pseudolines in . Corresponding to this arrangement there is a rank oriented matroid which realizes an arrangement of pseudocircles in , see Figure 1 below.
Now consider cones over these pesudocircles. The cones are pseudoplanes in . Each pseudoplane, individually, is homeomorphic to a -dimensional subspace of . However there does not exist a homeomorphism of which will map all of these to planes passing through a common point. For a detailed discussion about pseudo-arrangements see Björner et. al. [1, Chapter 5].
We would like to avoid such situations. Hence we introduce a condition that will guarantee hyperplane-like intersections. Let be a collection of locally flat, codimension- submanifolds of . For every and an open neighbourhood (homeomorphic to ) of let . By we mean the union of elements of .
Let be a manifold of dimension . Let be a collection of codimension-, locally flat submanifolds of . We say that these submanifolds have a locally flat intersection if for every there exists an open neighbourhood and a homeomorphism such that where is a central hyperplane arrangement in with as a common point.
2.2. Cellular Stratifications
Now we generalize properties (2) and (3). Let be a collection of codimension- submanifolds of having locally flat intersection. Let denote the set of all non-empty intersections and be the subset of codimension- intersections. We have and . For each consider the following subsets of
Note that each may be disconnected and that can be expressed as the disjoint union of these connected components. We want these components to define a ‘nice’ stratification of . For example, in case of hyperplane arrangements each stratum is an open polyhedron of appropriate dimension. We need to focus on stratifications such that the strata are cells otherwise there is no hope of generalizing property (3). For example, consider two non-intersecting longitudinal circles in the -torus ; there are two codimension- strata and two codimension- strata. The resulting face poset does not have the homotopy type of the torus. In view of Lemma 1.23 it is desirable to assume that the induced stratification is cellular and totally normal (Definition 1.19).
Imposing this condition on the stratification implies that the resulting face category has the homotopy type of the ambient manifold (Lemma 1.23). Moreover, each stratum is locally polyhedral (see [19, Section 3.3] and [20, Lemma 2.42]). Locally polyhedral means that the cell structure of each stratum is defined using a convex polyhedron and PL maps (see Lemma 1.22). At first glance this might seem restrictive. For example, the stratification of induced by the braid arrangement is not even cellular. However, there are ways to deal with such situations. We discuss this in detail in Remark 3.2.
2.3. Definition and examples
The desired generalization of hyperplane arrangements is the following
Let be a connected, smooth, real manifold of dimension . An arrangement of submanifolds is a finite collection of codimension- smooth submanifolds in such that:
the ’s have locally flat intersection,
the stratification induced by the intersections of ’s defines the structure of a totally normal cellularly stratified space on .
Let us look at how we can associate combinatorial data to such an arrangement and at a few examples.
The intersection poset, denoted by , is the set of connected components of all possible intersections of ’s ordered by reverse inclusion. The rank of each element in is defined to be the codimension of the corresponding intersection.
By convention is the intersection of no submanifolds, hence it is the smallest member.
The cells of the stratification will be called the faces of the arrangement. The face category of the arrangement is the face category of the induced stratification and is denoted by . Codimension- faces are called chambers and the set of all chambers is denoted by .
Note that the face category of an arrangement is acyclic. It is equivalent to the face poset if and only if the cell structure induced by is regular.
If there is no confusion about the arrangement, we will omit and denote the above two objects by and .
The obvious examples of these submanifold arrangements are the hyperplane arrangements. Let us look at different examples. We start with the unit circle ; here the codimension- submanifolds are just points.
Consider the arrangement in . The strata define the minimal cell decomposition of ; . The cell structure map for the -cell can be given by . Moreover, there are two lifts of the cell structure map for the unique -cell. Hence the face category contains two objects, the -cell and the -cell. Apart from the identity morphisms there are two morphisms from the -cell to the -cell which we denote by and . Figure 2 shows the arrangement and the nerve of the corresponding face category.
Consider the arrangement (i.e., a -sphere) in . The stratification defines a regular cell structure. Hence the face category is equivalent to the face poset. Figure 3 shows this arrangement and the Hasse diagrams of the face poset and the intersection poset.
As a -dimensional example consider an arrangement of great circles in . Figure 4 shows this arrangement and the related posets. The face poset has two -cells, four -cells and four -cells. Also note that the order complex of the face poset has the homotopy type of .
Now consider the -torus . A codimension- submanifold of is the image of a straight line (with rational slope) passing through the origin in under the quotient map . Let be the arrangement obtained by projecting the lines . These toric hyperplanes intersect in points, namely . There are chambers. Figure 5 shows the arrangement ( is considered as the quotient of the unit square) with its intersection poset.
Example 2.15 is an example of so-called toric arrangements. This is an area which recently has seen an increase in interest. See, for example, De Concini and Processi  for applications to partition functions, Ehrenborg et al.  for combinatorial perspective and also Moci .
There is also a ‘multiplicative’ way of defining toric arrangements. One can view an -torus as ; i.e., a multiplicative group. Recall that the group homomorphisms from onto can be described as vectors in . For example, if are coordinates of and then the corresponding character is given by . The kernel of such a character is a codimension- submanifold. In general this kernel need not be connected; the component containing the identity is a subgroup whereas the other components are its translates. A toric arrangement is a finite collection of hypersurfaces where is a character and .
Let be an arrangement of submanifolds. For a point the local arrangement at is
For a face the local arrangement at is
The restriction of a local arrangement to an open set is
Let be a face of an arrangement . Then there exists an open set containing and a map such that
for every ;
is a homeomorphism;
maps to a central arrangement of hyperplanes in .
Each submanifold has an open neighborhood in which is homeomorphic to (this follows from Brown’s bicollared theorem [16, Theorem 1.7.5]). We call such an open neighborhood the bicollar of that submanifold. A face is homeomorphic to an open cell of some dimension and is an intersection of finitely many codimension- submanifolds. Take the open set to be the intersection of all the bicollars containing . Moreover this can be chosen such that it intersects only those faces which intersect .
Since is a smooth manifold, there exists a homeomorphism , say a coordinate chart. If necessary can be composed with a homeomorphism of so that for every containing , the set is mapped to a codimension- subspace of . As is a homeomorphism it preserves the incidence relations between the faces, thus preserving the combinatorial structure of the local arrangement . ∎
Intuitively every point has an open neighborhood in which is homeomorphic to a central arrangement of hyperplanes. The hyperplanes in that arrangement correspond to for every .
In general need not be an arrangement of submanifolds in the sense of Definition 2.8. However it defines a stratification of the manifold. Let be the map of the stratified spaces. It induces a map between the corresponding face categories which we again denote by .
3. 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 1.4). If one were to forget the complex structure on then, topologically, it is just the tangent bundle of . The 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 its generalization to submanifold arrangements.
Let be an -dimensional manifold and be an arrangement of submanifolds. Let denote the tangent bundle of and let . The tangent bundle complement of the arrangement is defined as
At this point we would like to revisit the totally normal CS-space assumption in Definition 2.8. We start with a familiar class of examples. Suppose that an arrangement of hyperplanes in is not essential which clearly means that the induced stratification is not totally normal (however it is cellular). If the rank of is then there exists a subspace of dimension (the subspace spanned by the normals to the hyperplanes in ). Projecting onto we get a new hyperplane arrangement in which we denote by . The arrangement is the essentialization of . It is easy to see that the intersection and face posets of and respectively are isomorphic. Moreover the projection map, extended to , gives a trivial vector bundle and hence
Thus it is always assumed that the hyperplane arrangements are essential.
Such a process can be imitated in some important examples of arrangements. We explain one such class: the toric arrangements. First note that the tangent bundle of an -torus is homeomorphic to . Recall that a toric arrangement is a collection of codimension- subtori. These subtori are the hypersurfaces defined by the vanishing of characters. The same is true for tangent bundles of the subtori: they are kernels of characters from onto . We explain the essentialization of toric arrangements with the help of one particular example the braid arrangement.
Let be the arrangement in . The stratification induced by is not totally normal and cellular. Consider the codimension- subtorus defined by
Let denote the restriction of to . The stratification of induced by is the one with the required properties. Let us see what happens to the tangent bundle complement. Define a map as . Check that this map is a locally trivial fiber bundle and . This idea extends to any toric arrangement. The fiber bundle is never globally trivial; this observation corrects a statement in [4, Remark 3.6]. Example 2.15 above is the essentialization of the braid arrangement in .
The totally normal CS-space assumption is not very restrictive; it covers most of the important examples. Given an arbitrary collection of codimnesion- submanifolds (with locally flat intersection) it might be possible, at least in some cases, to change it to a submanifold arrangement while keeping track of the combinatorial and topological data.
For a point let . Define the tangent space complement at as
Then can be rewritten as
We start the study of by first understanding the tangent space complement .
Let and let be a coordinate chart for an open neighborhood of . Then for every ,
Observe that the linear map is an isomorphism and for the same reasons for every . The map defined by sending is a homeomorphism independent of the choice of coordinates.∎
Since is an arrangement of hyperplanes in (Lemma 2.18), is an arrangement of hyperplanes in . The two arrangements have isomorphic face posets as well as intersection lattices.
As a consequence of the above lemma chambers of the tangent space complement can be indexed by the chambers of . Let us identify them. First note that if is a face such that then the connected components of are in one-to-one correspondence with the morphisms from to . We denote these components by for every morphism . Now for a chamber , a morphism and for every define the following subset of -
We construct an open cover of the tangent bundle complement. The collection of open sets constructed in Lemma 2.18 is an open cover of . These open sets can be chosen to trivialize the tangent bundle. Let denote the local trivialization, where is the projection map. Choose a chamber such that is nonempty. For every point the map is a linear isomorphism. Let be an arbitrarily chosen point in and be the connected component of corresponding to . Define open subsets of as
Note that the above subset is open and does not depend on the choice of . For the sake of notational simplicity we will identify with its image and will treat as the product .
Let be an arrangement of submanifolds in an -manifold with as its face category and as the associated tangent bundle complement. Then there exists a finite open cover of such that these open sets are indexed by morphisms between a face and a chamber whose closure contains . Moreover, each of these open sets is contractible and so are their intersections.
We show that the subsets constructed above satisfy the requirements. Note that by construction these are open and contractible subsets of .
Let be an arbitrary point. Suppose that for some face . As we have , where is some chamber whose closure contains . Therefore for some . Hence the collection
is an open covering of .
The intersection of any two such open sets is given by
Exactly one of the following two situations can occur.
The trivial situation being that .
is contained in the boundary of . In that case is a disjoint union of finitely many open connected subsets of ; their number being the cardinality of .
In the second situation we have for every the intersection is a convex subset of . Consequently the intersection is at most a disjoint union of finitely many contractible subsets. A refinement of the open cover establishes the theorem. ∎
Consider an arrangement in , it divides the real line in two chambers and . The tangent bundle complement of this arrangement is the punctured plane, since is the only point whose tangent space is disconnected. Observe that the face category of this arrangement is equivalent to the face poset. For we take and for the chambers we take and as the coordinate neighborhoods. Since there is at most one morphism between any two objects we simplify the notation for the open sets and describe them as follows:
Figure 7 is a sketch of the corresponding open cover.
The open covering constructed in Theorem 3.5 satisfies the hypothesis of the Nerve Lemma. The nerve of this open cover has the homotopy type of . In order to identify the simplices of this nerve we need to understand how these open sets intersect.
if and only if and .
From the proof of Theorem 3.5 we have if and only if . We also need the other intersection in the product to be nonempty.
Note that the connected components of can be indexed by the morphisms from to .
Let be a smooth -manifold and be an arrangement of codimension- submanifolds. The Salvetti category is defined as follows
The objects of are the morphisms in from faces to chambers, i.e., such that is a face and is a chamber.
A morphism of defines a morphism from to in if and only if .
The composition is defined as:
whenever and are composable.
The Salvetti complex is defined as the geometric realization of .
Such a categorical description of the Salvetti complex first appeared in  where the authors deal with complexified toric arrangements.
The reader can verify that the Salvetti category is acyclic. Hence the Salvetti complex is a regular trisp. In the case when the face category is equivalent to the face poset the Salvetti category is also equivalent to a poset and the Salvetti complex is a simplicial complex.
Let the tangent bundle complement of an arrangement and let denote the associated Salvetti complex. Then .
The theorem follows from the observation that the nerve of the open covering constructed in Theorem 3.5 is the barycentric subdivision of the Salvetti complex.∎
Intuitively speaking, a submanifold arrangement is constructed by gluing central hyperplane arrangements in a compatible way. It is natural to ask whether the associated tangent bundle complement is also made up by gluing local (complexified) complements. We now address this point by showing that the complement is homotopy equivalent to the homotopy colimit of a diagram of spaces over the face category. For the local arrangement let denote the associated Salvetti category and let denote its geometric realization.
With the notation as before, if are two faces then for every there is an inclusion as trisps.
It suffices to prove that there is an injective functor from into . This follows if the face category of injects into the face category of . But this is clear since and for any face the face category is equivalent to the comma category