The Goresky-MacPherson formula for toric arrangements
A subspace arrangement is a finite collection of affine subspaces in . One of the main problems associated to arrangements asks up to what extent the topological invariants of the union of these spaces, and of their complement are determined by the combinatorics of their intersection. The most important result in this direction is due to Goresky and MacPherson. As an application of their stratified Morse theory they showed that the additive structure of the cohomology of the complement is determined by the underlying combinatorics.
In this paper we consider toric arrangements; a finite collection of subtori in . The aim of this paper is to prove an analogue of the Goresky-MacPherson’s theorem in this context. When all the subtori in the arrangement are of codimension- we give an alternate proof of a theorem due to De Concini and Procesi.
Key words and phrases:Goresky-MacPherson formula, arrangements of subspaces, Toric arrangements, simplicial resolution, homotopy colimits
2010 Mathematics Subject Classification:32S22, 52C35, 55R80, 55P15, 14N20
A subspace arrangement in is a finite collection of affine subspaces such that if then . Two important spaces associated with an arrangement are the link which is the union of all the subspaces and the complement which is the complement of the link. The formal data associated with an arrangement is the poset of all non-empty intersections of members of and the dimensions of these intersections. The poset is ordered by reverse inclusion and the ambient space is its least element. An arrangement of hyperplanes, now a subject in itself, is an important class of subspace arrangements.
A prominent research direction, in this area, is to try and express topological invariants of the link and those of the complement in terms of the combinatorics of the intersections. To this effect a seminal result is due to Goresky and MacPherson [11, III.1.5, Theorem A]. They used stratified Morse theory to show that there is a finer gradation of the cohomolgoy groups of the complement indexed by intersections. In particular, they gave a closed form formula for these groups in terms of the homology groups of lower intervals in :
Here varies on all positive codimension intersections and on the right hand side are reduced simplicial homology groups of order complexes. We refer to the above expression as the Goresky-MacPherson formula for subspace arrangements. This result refines a theorem of Brieskorn and Orlik-Solomon concerning cohomology of hyperplane complements [15, Lemma 5.91].
The work of Goresky and MacPherson created a lot of activity; along with applications several authors gave alternate proofs and strengthening. For an extensive survey of this subject (mainly for the work done till the 90’s) we refer the reader to the papers of Bjorner  and Vassiliev . In the last decade the main focus has been the multiplicative structure of . In particular, the arrangement of coordinate subspaces has applications in toric topology. See for example  and [2, Chapter 8].
We should mention the work of Vassiliev  and Ziegler-Z̆ivaljević  where they give an alternate proof of the formula. Observe that the one-point compactification of the link and are complementary subspaces of . In both these papers the homotopy type of is expressed in terms of the combinatorial information. The Goresky-MacPherson formula then follows from Alexander duality. The homotopy type of was determined by Vassiliev using simplicial resolutions and by Ziegler-Z̆ivaljević using homotopy colimits.
The study of toric arrangements received prominence because of the work of De Concini and Procesi, [6, 7]. A toric arrangement is a finite collection of codimension- subtori in . Besides being a natural generalization of hyperplane arrangements they have applications in diverse fields like algebraic geometry, combinatorics, partition functions and box splines. The last decade has seen a lot of activity in this field. We refer the reader to [4, Section 1] for an excellent account of the state of the art in this area. However, we do mention some of the relevant results concerning cohomology of the complement of a toric arrangement. The additive structure of cohomology was determined by Looijenga  and also by De Concini and Procesi . It was later shown by D’Antonio and Delucchi in , using discrete Morse theory, that for the class of complexified toric arrangements the cohomology is torsion free. Later the result was extended by Callegaro and Delucchi in  and now it is known that in all cases the cohomology groups are torsion free. Recently, a complete multiplicative structure of the cohomology ring was obtained in . See  for examples of toric arrangements with isomorphic intersection data but non-isomorphic cohomlogy rings. The rational formality (in the sense of Sullivan) of the complement was established by Dupont in .
The main aim of this paper is to establish a Goresky-MacPherson type formula in the context of toric arrangements. We consider finite collections of toric subspaces (not necessarily of codimension ). We show that there is a finer gradation of the cohomology groups of the complement. Moreover, this gradation is indexed by the underlying combinatorial data (Theorem 21). In order to prove this we find a closed form formula for the homotopy type of the one-point compactification of the link (Theorem 15). However, in the toric context we consider the ‘modified’ link; the union of all the toric subspaces and the coordinate axes. The complement and the one-point compactification of the modified link are complementary subspaces in . So the formula is now a direct consequence of Alexander duality. Our proof is based on the Ziegler-Živaljević approach for subspace arrangements. Towards the end we also make use of the spectral sequence developed by Petersen in  to get decomposition of the compactly supported cohomology (Theorem 20). We pay special attention to arrangements of toric hyperplanes (Theorem 17). There are two immediate consequences of our result; first there is no torision in the cohomology (Corollary 16) and second is a theorem due to De Concini and Procesi (Corollary 18). We end the paper with some discussion regarding future research directions.
In this section we sketch the proof, due to Ziegler-Živaljević , of the Goresky-MacPherson formula. We begin by a brief review of topological aspects of posets; the reader’s familiarity with posets and lattices is assumed. In particular, we will make frequent use of the order complex of a poset so we state relevant definitions and facts here, one can find more details in .
The Möbius function of a poset is defined recursively on closed intervals of as follows:
The order complex is an (abstract) simplicial complex whose -simplices correspond to chain of length . In this paper, we do not distinguish between abstract simplicial complexes and their geometric realizations. The reduced homology of , , is defined as the reduced simplicial homology of with integer coefficients. The cohomology of is defined analogously. We will often consider the order complex of an open interval . We denote its homology by .
It is a basic fact, due to P. Hall, that the Möbius function is the reduced Euler characteristic of :
A class of lattices called geometric lattices plays an important role in this article. The intersection lattice of a central hyperplane arrangement (more generally, any proper interval in the intersection poset of a non-central arrangement) is a geometric lattice. A fundamental result due to Folkman states that for a geometric lattice , the reduced homology groups vanish in all but the top dimension (i.e., dimension equal to ). In that dimension the homology is free of rank . In fact, the homotopy type is that of a wedge of spheres of dimension (see [15, Section 4.5] and [21, Section 3.2.3]).
As introduced earlier we denote by
an arrangement of subspaces in . The intersection semilattice is the collection of all non-empty intersections of members of . The elements of will be denoted by small Greek letters and the corresponding intersections by respectively. The intersection semilattice is ordered by reverse inclusion: . This is a meet-semilattice with the least element . In general is not even graded. The dimension function assigns to each the dimension of .
The top element exists if and only if the arrangement is central, i.e, the intersection of all the subspaces in is non-empty. In such a situation the joins are also defined and then becomes a lattice. If the context is clear we write instead of . Finally, use the notation for . We now (re)state the Goresky-MacPherson formula [11, III.1.5].
Let be a subspace arrangement in , be its complement. Then for every :
Suppose that is an arrangement of hyperplanes in . In this case each closed interval is a geometric lattice of rank . The homology groups on the right are nonzero in the dimension , which implies . Consequently, one recovers the celebrated theorem of Zaslavsky [15, Chapter 2].
Suppose is a complex hyperplane arrangement in . Again each closed interval is a geometric lattice of rank . And we get the following reformulation of Brieskorn’s Lemma [15, Lemma 5.91]. For
The first proof of the Goresky-MacPherson formula was given as an application of stratified Morse theory. Some later reproofs involved analyzing, first, the topology of the link, which has more combinatorial structure, and then passing to the cohomology of the complement via Alexander duality.
We present a sketch of these ideas since the proof of our main result is based them. We begin by recalling some relevant notions from homotopy theory. The interested reader can refer to [12, Chapter 15], [23, Apendix] and .
We view a poset as a category in the standard way; elements of being objects and non-identity morphisms pointing down. Given a poset a -diagram of topological spaces is a covariant functor from , i.e., an assignment of spaces for every element and continuous maps for every order relation . The colimit of a diagram is the topological space constructed from the disjoint union of all of ’s by identification of with , for all and all .
Let be another -diagram of topological spaces. A morphism of -diagrams is a collection of continuous maps indexed by such that .
Denote by the poset of chains in and by the subposet of elements below .
The homotopy colimit of a diagram , denoted by , is the colimit of the functor defined by:
on the elements: ;
on the morphisms:
Note that a morphism of -diagrams induces a continuous map between the corresponding homotopy colimits. We are mainly interested in the following three properties of homotopy colimits.
The homotopy lemma [22, Proposition 3.7]; it says that the homotopy type of the homotopy colimit doesn’t change if spaces in a diagram are changed up to homotopy.
Let be a morphism of two -diagrams such that each is a homotopy equivalence. Then the map induced by on the homotopy colimits is also a homotopy equivalence.
The projection lemma [22, Lemma 4.5]; identifies a situation in which the natural map from is a homotopy equivalence.
Let be a diagram indexed by a poset such that all the induced maps are inclusions that are also closed cofibrations. Then the collapsing map from is a homotopy equivalence.
The wedge lemma [22, Lemma 4.9], we include the technical statement instead of its intuitive description.
Suppose is a poset with maximal element . Let be a -diagram such that all the maps between spaces are constant. Then
where “” denotes the topological join of two spaces.
Given a subspace arrangement the subspace diagram is defined as follows:
an object is sent to the corresponding intersection ;
a morphism is sent to the corresponding inclusion .
Now we can proceed with the proof of the Goresky-MacPherson formula. The first step is to determine the homotopy type of the link.
Theorem 5 ([23, Theorem 2.1]).
For a subspace arrangement the homotopy type of the link is given by
In order to arrive at the above result, first note that because of the projection lemma is homotopy equivalent to . On the other hand the order complex of is homotopy equivalent to because of the homotopy lemma.
The homotopy type of the compactification is also combinatorially determined. Here denotes the -sphere. Also, it is to be understood that for all spaces .
Theorem 6 ([23, Theorem 2.2]).
For every subspace arrangement:
Consequently the homology is given by
The proof of the above theorem involves modifying the diagram . Add an extra element to which will be the index of the point at infinity. Now to each element assign the sphere which is homeomorphic to the one-point compactification of . To assign the point at infinity; it gets included in all the spheres. The maps in this diagram are null homotopic; in fact, the union is contained in a closed disc in . Consequently one may replace this diagram with a new diagram consisting of spheres of appropriate dimensions and base point preserving constant maps. The result now follows from the application of the wedge lemma (Lemma 4 above).
Note that the compactified link and the complement are complementary subspaces of . Hence the Goresky-MacPherson formula is a consequence of Alexander duality:
The approach taken by Vassiliev in  to determine the homotopy type of is slightly different. Vassiliev constructed a ‘simplicial blow up’ of the intersections of subspaces: Take sufficiently large number such that each subspace can be embedded in a generic position. Now for every let denote the convex hull of the images of in . The simplicial resolution of is the union of all ’s. He goes on to prove that . Then he uses the stratified Morse theory to arrive at the closed form formula for (see also [20, Section 8]). It was later proved by Kozlov in  that the simplicial resolution itself is a homotopy colimit. Moreover he constructed an explicit deformation retraction from to the homotopy colimit constructed by Ziegler-Živaljević.
3. Toric arrangements
We now turn to the main object of this paper; toric arrangements. The standard -dimensional complex torus is the space of -tuples of nonzero complex numbers. The torus is a group under multiplication of coordinates. In fact, it is an affine algebraic variety with the ring of Laurent polynomials as its coordinate ring.
The character of a torus is a multiplicative homomorphism given by the evaluation of Laurent monomials
The following are some well-known facts regarding tori (see [7, Section 5.2]). The set of characters of is a free abelian group isomorphic to . Conversely, for a finitely generated abelian group of rank the variety is isomorphic to the product of an -torus and a finite abelian group isomorphic to the torsion subgroup of .
The connected, topologically closed subgroups (called the toric subgroups) of are isomorphic to -tori for some . In general a closed subgroup of is isomorphic to where is a finite abelian group. Any coset of a toric subgroup is homeomorphic to the group; thus topologically it is a torus. Let be a closed subgroup of and be the connected component containing identity. Then is a toric subgroup, the quotient is a finite abelian subgroup and is the union of the distinct cosets.
There is a one-to-one correspondence between closed subgroups of and subgroups of . Such subgroups are determined by integer matrices. An integer matrix of size determines a mapping from to . The kernel of this mapping is a closed subgroup of and every subgroup arises in this manner. Consequently depends only on the subgroup (i.e., the sub-lattice) of generated by the rows of . Without loss of generality one can assume that rows of furnish a basis for this sub-lattice so that . By a toric subspace we mean a translate of a closed subgroup.
Because of their importance, we first focus on collections consisting of codimension- subtori. Given a character and a non-zero complex number a toric hypersurface is defined as the level set of , i.e., . A toric hypersurface is a translate of a toric subgroup of codimension-.
A toric arrangement is a finite collection
of toric hypersurfaces in .
For notational simplicity we write instead of . Without loss of generality we assume that each toric hypersurface in is connected, i.e., each character is primitive (recall that a character is primitive if all the exponents are relatively prime).
The intersection poset (or the poset of layers) is the set of all connected components of all nonempty intersections of the toric hypersurfaces in ordered by reverse inclusion. The elements of are called components of the arrangement.
The intersection poset is in general not a semilattice. However, it is certainly a graded poset; the rank of every element is the codimension of the corresponding intersection. The ambient torus is the least element . As before, the elements of are denoted by and the corresponding intersections by respectively. We also have the (complex) dimension function . Note that for every the intersection isomorphic to . Every proper interval in is a geometric lattice, see [10, Theorem 3.11]. A few examples before we proceed.
An arrangement in is just a collection of finitely many points.
Let be a toric arrangement in . The toric hypersurfaces and intersect in three points and . Where is a cube root of unity. Figure 1 shows the “real part” of the arrangement, i.e., intersection with the compact torus . It is visualized as a quotient of the unit square. Along with it is the Hasse diagram of .
Now consider the arrangement formed by adding the toric hypersurface given by equation to the previous arrangement. The hypersurfaces and intersect in the same three points as above. Figure 2 shows the “real part” of the arrangement and the associated intersection poset.
The braid arrangement in is the collection of toric hypersurfaces given by the following equations
The intersection poset in this case is the partition lattice .
The complement of a toric arrangement in is
For example, the complement of a toric arrangement in has the homotopy type of wedge of circles. The complement of the braid arrangement is the configuration space of ordered points in .
Let be a component of a toric arrangement . For any let denote the hyperplane arrangement in a coordinate neighborhood of . More intrinsically, in the tangent space . Note that if is a point such that it does not belong to any other component contained in then the intersection data of is independent of the choice of . One can safely disregard the reference to as we are mainly interested in the combinatorics. We refer to as the local arrangement at . The complement is the local complement at . Finally, for define numbers as follows:
i.e., the sum of -th Betti numbers of all local complements at codimension components of . The reader familiar with hyperplane arrangements will realize that the summands are precisely cardinality no-broken-circuits of local arrangement . Further details can be found in [15, Section 3.1] and in [6, Section 3.4] (in the context of toric arrangements).
The cohomology groups of were computed and by De Concini and Procesi in [6, Theorem 4.2] using an algebraic De Rham complex. We state an equivalent version of their theorem below.
Given a toric arrangement the Poincaré polynomial of its complement is given by
In particular, the -th Betti number is .
Let us apply the above theorem to the arrangement in Example 12. Observe that the local arrangement at every point is the arrangement of concurrent lines.Whereas the local arrangement at every is isomorphic to the arrangement of a point in . Hence, and . Plugging these numbers in the above formula we get .
We now proceed to prove the analogue of the Goresky-MacPherson formula in this context. First, a simple observation. Given a toric arrangement in we have
The observation motivates the following definition.
Given a toric arrangement in the (modified) link is defined as
Forgetting the complex structure on and passing on to the compactification we get
Hence our strategy is to express the homotopy type of in terms of the underlying combinatorics and then use the Alexander duality to arrive at the Goresky-MacPherson formula. We begin by proving a couple of lemmas.
Let denote the union of all the coordinate axes in . Then
Notice that is the link of the arrangement of coordinate hyperplanes in . The associated intersection poset is the Boolean lattice, which we denote by . Each lower interval in is homotopic to the sphere of appropriate dimension. The result follows from Theorem 6 as join of two spheres is again a sphere. ∎
The homotopy type of the one-point compactification has the following closed form
Recall that for two locally compact, Hausdorff spaces we have the following homeomorphism
that maps the compactification point to the smash point and is identity on .
Observe that is homotopic to . To see this, first consider as a subset of the Riemann sphere; i.e., the sphere with both poles missing. Now add the missing two points and and then identify them. As the smash product distributes over the wedge, the following homotopy equivalence establishes the result
For notational simplicity we denote by the wedge of spheres on the right hand side of the above formula. Also, note that .
Given a toric arrangement in the modified intersection poset is defined as
here denotes the Boolean lattice.
The toric subspace diagram is defined as follows:
If then is , the corresponding intersection. If then is .
The maps are the inclusions of subspaces.
The homotopy colimit of is homotopy equivalent to the modified link .
Follows from the projection lemma. ∎
The point has a compact cone neighborhood in .
Each is a neighborhood of in and there are inclusions: . For every there is a deformation retraction of on to that fixes . Paste these retractions to contract on to in . ∎
In fact, the point has arbitrarily small contractible neighborhoods which are wedge of cones with as the common cone point. For example, let
The closures ’s form an exhaustion by compact sets and the complement of each them has connected components (i.e., the -torus has ends). By similar arguments as above one can construct a cone over components of .
Suppose is a toric subspace. Then its one point compactification is contained in a compact cone neighborhood of in .
Recall that a toric subspace can be specified by a set of characters. Assume that the given is the intersection of toric hyperplanes whose defining characters are for .
Since is closed we can choose two -tuples and such that they are in the same path component of , and both are rationals for every .
Consider the following compact subset of :
(even though the definition depends on the choice of ’s and ’s we choose to ignore them in the notation). The construction implies that . Hence is contained in a neighborhood of an end of . Equivalently, the compactification is contained in a compact cone on with as its cone point. ∎
We now determine a closed form formula for the homotopy type of the one point compactification of the modified link.
The homotopy type of the compactified link of is given by
where denotes the -fold reduced suspension.
Construct a new diagram such that the spaces indexed by are replaced by their one-point compactifications. In particular, we have given by setting for and . The maps between the spaces are inclusions.
We claim that these inclusions are null homotopic. This is clear for spaces indexed by . For spaces indexed by it follows from Proposition 14. By projection lemma .
Now construct one more diagram, say , indexed by : if then otherwise and is the wedge of spheres homotopic to the one-point compactification of as obtained in Lemma 8. Finally, send to . The maps in this new diagram are the constant maps to the base point.
We now construct a morphism as follows. First, a claim: for a given the union is contained in a compact cone neighborhood of in . In order to prove this claim we employ the same strategy as above. Construct a compact set by choosing two points in (here the union is over all ). The claim now follows since the union of compactifications is contained in the compact cone on . Choose a homotopy equivalence that contracts this compact cone and sends it to the base point of .
For choose a homotopy equivalence that contracts the closed disk containing the union of smaller-dimensional spheres to the base point. The homotopy lemma now implies that is homotopy equivalent to .
The application of the wedge lemma to gives us the following decomposition:
Recall that for well-pointed spaces the join distributes over the wedge. Hence,
The result then follows from the fact that join with an -sphere is homotopic to the -fold reduced suspension. ∎
Let be a toric arrangement and be the associated complement. Then the cohomology groups are torsion free.
Note that since each interval is a geometric lattice the corresponding order complex is homotopic to a wedge of spheres [15, Theorem 4.109]. Their dimension being and their number being . Consequently we have
It means that the homotopy type of is again a wedge of spheres. ∎
Finally, the toric analog of the Goresky-MacPherson formula.
Let be a toric arrangement in . Then we have the following decomposition of the cohomology groups of the complement .
By Alexander duality we have . The homology groups of are direct sum of homology of factors in the wedge decomposition in Equation 4. In particular we have,
Hence we determine each summand separately and plug it back in Equation 6. First, observe that and are complementary subspaces, hence
We now turn to the join of spaces, for notational simplicity let .
Due to Folkman’s theorem homology of is concentrated in degree . Hence the summand corresponding to survives and all others vanish. Hence,
which is the required decomposition. ∎
The Poincaré polynomial of the complement of a toric arrangement is
Observe that the interval is isomorphic to the intersection lattice of the local arrangement . It follows from Example 3 that ; hence its rank is a summand in . Finally, the other factor is the rank of . ∎
Consider the arrangement in Example 11. Here, . For each rank element there is a copy of . For each rank element there is a copy of . Hence,
The reduced cohomology of the complement is in degree , in degree and zero everywhere else.
We now give an alternate description of the Goresky-MacPherson formula in terms of compactly supported cohomology. In a recent paper  Petersen constructed a spectral sequence for stratified spaces which computes the compactly supported cohomology of an open stratum. The ingredients needed for this spectral sequence are the compactly supported cohomology groups of closed strata and the reduced cohomology groups of the poset of strata.
Let be a toric arrangement and for define
Consequently, we have a stratification . Note that and for . Since each stratum is an affine algebraic variety defined over we have the following result by Petersen [17, Theorem 3.3]. For notational simplicity denote by the set of codimension- components of .
In the situation described above there is a spectral sequence of mixed Hodge structures
Since in our case all the intervals in are geometric lattices [17, Example 3.10] the first page of the spectral sequence simplifies as follows:
These groups are nonzero only when and . Moreover, all these columns have different weights so there can be no differentials in the spectral sequence. Recall that for all the group is pure of weight whereas the group is pure of weight . Consequently we have the following decomposition of the compactly supported cohomology of the complement.
Let be a toric arrangement in and be its complement. Then we have following analog of the Goresky-MacPherson formula using compactly supported cohomology:
Note that the above decomposition is Poincaré dual to the earlier one in Equation 5.
We consider the toric arrangement in Example 11. Here, an order complex of intervals ending in rank elements consists of discrete points hence its reduce cohomology is in dimension . An easy calculation shows that for respectively and for all other values of .
Now consider the toric arrangement in Example 12. For a space denote by the configuration space of ordered points in . Using above result and Poincaré duality we get