aa r X i v : . [ m a t h . A T ] D ec COHOMOLOGY OF MANIFOLD ARRANGEMENTS
JUNDA CHEN, ZHI LÜ AND JIE WUA
BSTRACT . We study the cohomology of the complement M ( A ) of a manifold ar-rangement A in a smooth manifold M without boundary. We first give the conceptof monoidal cosheaf on a locally geometric poset L , and then define the generalizedOrlik–Solomon algebra A ∗ ( L , C ) over a commutative ring with unit, which is built bythe classical Orlik–Solomon algebra and a monoidal cosheaf C as coefficients. Further-more, we construct a monoidal cosheaf ˆ C ( A ) associated with A , so that the generalizedOrlik–Solomon algebra A ∗ ( L , ˆ C ( A )) becomes a double complex with suitable multipli-cation structure and the associated total complex T ot ( A ∗ ( L , ˆ C ( A ))) is a differential al-gebra. Our main result is that H ∗ ( T ot ( A ∗ ( L , ˆ C ( A )))) is isomorphic to H ∗ ( M ( A )) asalgebras. Our argument is of topological with the use of a spectral sequence inducedby a geometric filtration associated with A . In particular, we also discuss the mixedHodge complex structure on our model if M and all elements in A are complex smoothvarieties, and show that it induces the canonical mixed Hodge structure of M ( A ) . Asan application, we calculate the cohomology of chromatic configuration spaces, whichagrees with many known results in some special cases. In addition, some explicit for-mulas with respect to Poincaré polynomial and chromatic polynomial are also given.
1. I
NTRODUCTION
P. Orlik and L. Solomon [23] introduced the OS-algebra A ∗ ( A ) of central hyperplanearrangements A over complex number field C , and proved that A ∗ ( A ) is isomorphicto the cohomology ring of the complement of A . Essentially, A ∗ ( A ) only depends onthe intersection lattice L of A , so the OS-algebra may be defined for every geometriclattice L , also denoted by A ∗ ( L ) . E. Feichtner and G. Ziegler [15] calculated the coho-mology ring of the complement of complex subspace arrangements with a geometricintersection lattice.Dupont [12] studied the cohomology of complement of algebraic hypersurface ar-rangements and built an OS-model for hypersurface arrangements. Bibby [3] studiedthe abelian arrangements, and also gave a kind of OS-model by Leray spectral se-quence. Their works lead many applications. For example, F. Callegaro [7] and Pagaria[24] studied the cohomology of complements of a toric arrangements with integer co-efficients, and B. Berceanu et al. [2] studied the cohomology of partial configurationspaces of Riemann surfaces. Also see [6, 25] for more recent work along this direction.In this paper we are concerned with manifold arrangements, introduced in [14], i.e.,arrangements of smooth submanifolds (may have codimension greater than one andneed not to be algebraic) with "clean" intersection. The notion of manifold arrange-ments is a generalization for some classical arrangements, such as hyperplane (or hy-persurface) arrangements and the configuration spaces of manifolds. We study the Key words and phrases.
Manifold arrangement, OS-algebra, Cosheaf, Spectral Sequence.The second author is partially supported by the grant from NSFC (No. 11971112). ohomology ring of complement of manifold arrangements, and give a model with"monoidal cosheaf" as coefficients of manifold arrangements by an elementary andpure topological approach. In particular, we discuss the mixed Hodge structure on ourmodel if every manifold is a complex smooth variety.Let A = { N i } be a manifold arrangement with locally geometric quasi-intersectionposet L in a smooth manifold M without boundary, where each N i is a smooth subman-ifold and a closed subset in M . Let M ( A ) be the complement of A in M , see Section2 for precise definitions. Then every interval [0 , p ] as a lattice defines an OS-algebra A ∗ ([0 , p ]) in the sense of Orlik and Solomon, which encodes the local combinatorialdata of L and plays an important role in our discussion. Furthermore, we will carryout our work over a commutative ring R with unit as follows:(A) First we generalize the OS-algebra for lattices into a "global" version A ∗ ( L , C ) ,which is built by A ∗ ([0 , p ]) and a "cosheaf" C as coefficients, where the cosheafon a poset is a concept defined in Section 2.5. For any manifold arrangement A , there is an associated cosheaf C ( A ) as a cochain complex, as we will see inSection 3, which encodes the topological data of A . Moreover, two differentialson A ∗ ([0 , p ]) and C ( A ) induce a double complex structure on A ∗ ( L , C ( A )) withdifferentials ∂ and δ . In particular, this double complex also becomes a totalcomplex with differential ∂ + δ of degree 1, denoted by T ot ( A ∗ ( L , C ( A ))) , whichcan be regarded as a cochain complex. We first show that H ∗ ( T ot ( A ∗ ( L , C ( A )))) is isomorphic to H ∗ ( M ( A )) as modules. This result can be understood as a"categorification" of Möbius inversion formula, which is essentially based upona filtration of A ∗ ( L , C ( A )) induced by a geometric filtration associated with A .In particular, if N i and M are all complex smooth varieties, then we can modify C ( A ) into K ( A ) , such that the model T ot ( A ∗ ( L , K ( A )) becomes a mixed Hodgecomplex, inducing the canonical mixed Hodge structure of H ∗ ( M ( A )) .(B) Generally, A ∗ ( L , C ) is not a differential algebra unless the cosheaf C has a suit-able product structure (in this case, we call it a "monoidal coseaf" in Section 2.6).Indeed, the cochain complex C ( A ) is not a monoidal cosheaf under the naturalcup product in general. We construct a monoidal cosheaf ˆ C ( A ) on L as a ’flat’version of C ( A ) such that their cohomology groups H ∗ ( ˆ C ( A )) and H ∗ ( C ( A )) areisomorphic. Then we obtain a generalized OS-algebra A ∗ ( L , ˆ C ( A )) , abbreviateit as A ∗ ( A ) . We will prove that A ∗ ( A ) is a double complex with suitable multi-plication structure such that the associated total complex T ot ( A ∗ ( A )) is a differ-ential algebra, whose cohomology can be realized as H ∗ ( M ( A )) as algebras.Our main result is stated as follows: Theorem 1.1.
As algebras, H ∗ ( T ot ( A ∗ ( A ))) ∼ = H ∗ ( M ( A )) . Furthermore, the spectral sequence associated with double complex A ∗ ( A ) is a spec-tral sequence of algebra. Corollary 1.2.
There is a spectral sequence of algebra with E ∗ , ∗ ∼ = A ∗ ( L , H ∗ ( C ( A ))) convergesto H ∗ ( M ( A )) as algebras. From now on, we always use H ∗ ( A ) to denote H ∗ ( C ( A )) for convenience. f M and every N i ∈ C are complex smooth varieties, we modified C ( A ) a little,denoted by K ( A ) , such that K ( A ) is a cosheaf of mixed Hodge complex and has thesame cohomology as H ∗ ( A ) . Then we have Theorem 1.3.
The model
T ot ( A ∗ ( L , K ( A ))) is a mixed Hodge complex and H ∗ ( T ot ( A ∗ ( L , K ( A )))) ∼ = H ∗ ( M ( A )) with mixed Hodge structure. Corollary 1.4. If M is projective and using Q coefficients, then Gr Wn H n − i ( M ( A )) ∼ = H − i ( A ∗ ( L , H n ( A )) , ∂ ) and H ∗ ( M ( A )) ∼ = H −∗ ( A ∗ ( L , H ∗ ( A )) , ∂ ) as algebras (probably with different mixed Hodge structures). As an application, we further study the cohomology ring of the chromatic configu-ration space defined as F ( M, G ) = { ( x , ..., x n ) ∈ M n | ( i, j ) ∈ E ( G ) ⇒ x i = x j } where M is a smooth manifold without boundary, G is a simple graph on vertex set [ n ] = { , , ..., n } and E ( G ) is the edge set of G . Clearly F ( M, G ) would be the classicalconfiguration space F ( M, n ) while G is a complete graph. This concept of F ( M, G ) was first introduced by Eastwood and Huggett [13] and many authors used differentnames, see [1, 2, 30, 20], where we prefer to call it the "chromatic configuration space" asused in [20] since it is the space of all vertex-colorings of G such that adjacent verticesreceive different colors if we consider M as a space of colors. This generalizes theconcepts of graphic arrangements and configuration spaces, and it is also an exampleof manifold arrangements.Historically, F. Cohen [9] determined the cohomology ring H ∗ ( F ( R n , q )) for n ≥ .Fulton and MacPherson [16] gave a compactification of the configuration space of aalgebraic variety and use that to compute H ∗ ( F ( M, n )) . Kriz [19] and Totaro [29] im-proved Fulton and MacPherson’s result by different methods, showing that if M is asmooth complex projective variety, the rational cohomology ring of F ( M, n ) is deter-mined by the rational cohomology ring of M . The homology group of chromatic space F ( M, G ) has been studied by Baranovsky and Sazdanovi´c [1]. The compact supportcohomology of F ( M, G ) was studied by Petersen [27]. These works in [1] and [27] ac-tually coincide from the viewpoint of Poincaré duality. On the singular cohomologyring of chromatic configuration spaces, it seems that the only work is what Berceanu [2]does do recently, where Berceanu studied H ∗ ( F ( M, G )) by Dupont’s result [12] when M is a Riemann surface. If M is a manifold without boundary, we show in Section 6that F ( M, G ) is the complement of a manifold arrangement A G . Then using the ap-proach developed in this paper, we obtain an immediate corollary about cohomologyring H ∗ ( F ( M, G )) . Corollary 1.5. H ∗ ( T ot ( A ∗ ( A G ))) is isomorphic to H ∗ ( F ( M, G )) as algebras, where A G is theassociated manifold arrangement such that M ( A G ) = F ( M, G ) . In some special cases, we can express H ∗ ( T ot ( A ∗ ( A G ))) more explicitly . orollary 1.6. Assume that M is a complex projective smooth variety and we use Q as coeffi-cients. Then Gr Wn H k ( F ( M, G )) ∼ = H − ( n − k ) ( A ∗ ( L G , H n ( A G )) , ∂ ) and H ∗ ( F ( M, G )) ∼ = H −∗ ( A ∗ ( L G , H n ( A G )) , ∂ ) as algebras. Corollary 1.7.
Assume that the diagonal cohomology class of M is zero and we use Z ascoefficients ( M may not to be a variety). Then A ∗ ( L G , H ∗ ( A G )) is completely determined by G and H ∗ ( M ) , and there exists a filtration of H ∗ ( F ( M, G )) such that Gr ( H ∗ ( F ( M, G ))) isisomorphic to A ∗ ( L G , H ∗ ( A G )) as algebras. See Theorem 6.4 for an explicit algebra structureof A ∗ ( L G , H ∗ ( A G )) By calculating the Poincaré polynomial of each algebra, we have
Corollary 1.8.
With the same assumption as in Corollary 1.7, let G be a simple graph on vertexset [ n ] and M be a manifold of dimension m . Then P ( F ( M, G )) = ( − n t n ( m − χ G ( − P ( M ) t − m ) where P ( − ) denotes the Z -Poincaré polynomial with variable t , and χ G ( t ) is the chromaticpolynomial of G . In a more special case, we can overcome the gap between Gr ( H ∗ ( F ( M, G ))) and H ∗ ( F ( M, G )) . Theorem 1.9.
Using Z as coefficients, assume that M is a manifold of dimension m such thatthe diagonal cohomology class of M is zero and H i ( M ) = 0 for all i ≥ ( m − / . Then H ∗ ( F ( M, G )) is isomorphic to A ∗ ( L G , H ∗ ( A G )) as algebras. Comparation to known results. • Theorem 1.9 agrees with the classical result of H ∗ ( F ( R m , n )) . • Using the proof process of Theorem 1.9, we may reprove Orlik-Solomon’s resultfor the complement of complex hyperplane arrangements. • We may reprove the Zaslavsky’s result [33] for the f -polynomial of hyperplanearrangements in R d (see Corollary 4.5). • Theorem 5.4 largely generalizes Dupont’s result [12] to general submanifoldswithout the hypothesis of codimension one and being algebraic. Furthermore,Theorem 5.6 coincides with Dupont’s result if we restrict to the case of algebraichypersurface arrangements. • Let G be a complete graph on [ n ] . Then Corollary 6.3 also agrees with the Kriz[19] and Totaro [29]’s model (denoted it by E ( n ) ) of F ( M, n ) for smooth complexprojective variety M . • We may reprove Eastwood and Huggett’s result [13] for Euler characteristic of F ( M, G ) by the spectral sequence in Theorem 6.2. Corollary 1.5 extends thisresult to the Poincaré polynomial in a special case. See Section 6.2 Baranovsky and Sazdanovi´c [1] defined a complex, denoted it by C BS ( G ) , as E page of a spectral sequence converge to H ∗ ( F ( M, G )) . Recently, M. Bök-stedt and E. Minuz [4] studied the relation between C BS ( G ) and Kriz-Totaro’smodel E ( n ) , defined a dual C BS ( G ) ∗ and generalized E ( n ) to a model R n ( A, G ) .They proved that C BS ( G ) ∗ and R n ( A, G ) are quasi-isomorphic, and C BS ( G ) ∗ is E page of a spectral sequence converge to H ∗ ( F ( M, G )) as modules, giving aquestion whether the spectral sequence has a product structure . Our work affirma-tively answers this question. As discussed in Section 5, our spectral sequenceis equipped with a product structure and the E page of spectral sequence inTheorem 6.4 coincides with R n ( A, G ) as algebras. The product structure of ourspectral sequence seems to be new.This paper is organized as follows. In Section 2, we review the concepts of manifoldarrangements and geometric lattice, and introduce our concept of "momoidal cosheaf".In Section 3, we introduce some filtration induced by manifold arrangements, and useit to prove our main theorem of module structure and discuss the mixed Hodge com-plex structure of our model in Section 4. In Section 5, we discuss the product structureon our construction, and then prove an algebra version of our main theorem. We intro-duce the chromatic configuration space in Section 6. Then, as an application of mainresult, we give some explicit formulas in some special cases. In appendix, we reviewsome results for the spectral sequence of filtrated differential algebra, and then givethe proof of Theorem 6.4.Throughout Sections 2–5 of this paper, we always choose a commutative ring R withunit as default coefficients. 2. P RELIMINARIES
Geometric lattice.
Here we are only concerned with finite posets. A lattice is aposet L in which any two elements have both a supremum (join ∨ ), and an infimum(meet ∧ ). Denote the unique minimal element (resp. maximal element) by (resp. ).A lattice is ranked if all maximal chains between two elements have the same length.By r ( p ) we denote the length between and p , which is called the rank of p . Of course, r ( ) = 0 . An element of rank is called an atom . If p > q and r ( p ) = r ( q ) + 1 , then wesay that p covers q , written as p : > q or q < : p . A geometric lattice (or matroid lattice ) is aranked lattice subject to r ( p ∧ q ) + r ( p ∨ q ) ≤ r ( p ) + r ( q ) and every element in L − { } is the join of some set of atoms. For more details aboutlattices and geometric lattices, see [17]. Example . Let G denote a undirected simple graph with vertex set [ n ] = { , ..., n } . Aspanning subgraph of G is the subgraph such that its vertex set is [ n ] and its edge setis a subset of all edges of G . We use L G to denote the lattice consisting of all partitionsof vertices induced by connected components of spanning subgraphs of G , so ∈ L G is the partition containing every single vertex and ∈ L G is the partition induced byconnected components of G . It is well-known that this L G is a geometric lattice, alsocalled the bond-lattice by Rota [28]. For any partition p ∈ L G , by | p | we denote thelength of p as a partition, so r ( p ) = n − | p | . There is a well-known relation between the öbius function µ of L G and the chromatic polynomial χ G ( t ) of graph G as follows: χ G ( t ) = X p ∈ L G µ (0 , p ) t n − r ( p ) where the Möbius function µ is an integer valued function on L G × L G defined recur-sively by µ ( p, p ) = 1 , P p ≤ l ≤ q µ ( p, l ) = 0 if p < q and µ ( p, q ) = 0 otherwise. Given apartition p ∈ L G , we can remove those edges in G of crossing different components of p , and then get a subgraph of G , denoted by G | p . It is easy to check that the associatedlattice L G | p of this subgraph is the interval [0 , p ] ⊂ L G . This fact will be used in Section6. The following property is important for geometric lattice, see [17, Chapter IV]. Proposition 2.1.
Let L be a geometric lattice. Then every interval [ a, b ] of L is also a geometriclattice. For some elements p i of L , r ( ∨ i p i ) ≤ P i r ( p i ) by definition of geometric lattice. Wesay that the p i ’s are independent if r ( ∨ i p i ) = P i r ( p i ) and dependent otherwise. Here isan easy lemma we will use later. Lemma 2.1. (i) If a is an atom and p ∈ L with a (cid:2) p and p = , then a, p are independent,i.e., r ( p ∨ a ) = r ( p ) + 1 . (ii) Every element p = is the join of some independent atoms a i , ≤ i ≤ r ( p ) . (iii) For dependent p, q and write q = ∨ a i as join of independent atoms, thereexists an integer s such that a s ≤ p ∨ a ∨ a ∨ · · · ∨ a s − .Proof. (i) First r ( a ∧ p ) + r ( a ∨ p ) ≤ r ( a ) + r ( p ) by definition. Since a is an atom and a (cid:2) p ,we have that a ∧ p = so r ( a ∨ p ) ≤ r ( p ) + 1 , and a ∨ p > p so r ( a ∨ p ) > r ( p ) . Thus, r ( p ∨ a ) = r ( p ) + 1 . (ii) Clearly p is always the join of some atoms { a i } by definition.Remove the elements a s with a s ≤ a ∨ a ∨ · · · ∨ a s − , we get the required independentset of atoms by (i). (iii) If not, r ( p ∨ a ∨ a ∨· · ·∨ a s ) = r ( p )+ s by induction. Furthermore, r ( p ∨ q ) = r ( p ) + r ( q ) , which contradicts to the condition that p, q are dependent. (cid:3) Locally Geometric lattice.Definition 2.1.
A poset L is locally geometric if it has a minimal element and the inter-val [ , p ] is a geometric lattice for any p ∈ L . Remark . It is known that the intersection lattice of hyperplane arrangements is a geo-metric lattice. For more general case, the poset of layers (connected components ofintersections) of hypersurface arrangements are always locally geometric since thesearrangements locally "looks like" hyperplane arrangements. The above definition is acombinatorial abstraction of poset of layers.From now on, we always denote geometric lattice (resp. locally geometric poset) as L (resp. L ) and write [ p, ∞ ) = { q ∈ L | q ≥ p } . The poset [ p, ∞ ) is also locally geometricsince interval [ p, q ] of geometric lattice [ , p ] is also geometric lattice.There are some easy properties of locally geometric poset. Proposition 2.2.
Let L be a locally geometric poset.(1) Set r ( p ) = r ([ , p ]) for p ∈ L . Then r ( p ) is a grading of poset L , where r ([ , p ]) in theright side is the rank of geometric lattice [ , p ] .
2) In general, two elements p, q ∈ L may have more than one minimal upper bounds, wedenote this set by p ˚ ∨ q as "global" join of p, q . Notice that for any s ∈ p ˚ ∨ q , s = p ∨ q inlattice [ , s ] .(3) Given p , q , p , q ∈ L such that p ≤ p , q ≤ q , then for every s ∈ p ˚ ∨ q , there isone and only one element t ∈ p ˚ ∨ q such that t ≤ s . This property gives us a canonicalmap λ p q p q : p ˚ ∨ q → p ˚ ∨ q .Proof. (1) and (2) are obvious. It suffices to prove (3). For any s ∈ p ˚ ∨ q , choose anelement t = p ∨ q in lattice [ , s ] , we claim that it is a minimal upper bound of p , q in L . Otherwise, we will have another upper bound t ≤ t , a contradiction to t = p ∨ q in lattice [ , s ] . For the "only one" part, assume that t , t ∈ p ˚ ∨ q with t , t ≤ s and t = t . It is easy to see that t , t are also minimal upper bound of p , q in [ , s ] , butthis is impossible since [ , s ] is a lattice. (cid:3) Orlik-Solomon algebra.
P. Orlik and L. Solomon introduced the OS-algebra A ∗ ( L ) in [23] for any finite geometric lattice L .Let Atom( L ) be the set of all atoms of L . Recall a subset S ⊂ Atom( L ) is said to bedependent if r ( ∨ S ) < | S | . Let E ∗ ( L ) be the exterior algebra over a commutative ring R with unit 1, generated by the elements e a , a ∈ Atom( L ) with the basis e S = e a · · · e a k , S = { a , .., a k } ⊂ Atom( L ) , where e S = 1 if S is empty. E ∗ ( L ) admits the naturalderivation ∂ : E ∗ ( L ) −→ E ∗ ( L ) given by ∂e S = if S is empty if S = { a } P kj =1 ( − j − e a · · · c e a j · · · e a k if S = { a , .., a k } . Let I ( L ) be the ideal in E ∗ ( L ) , generated by ∂e S for all dependent sets S ⊂ Atom( L ) .Then the Orlik-Solomon algebra of L is defined as the graded commutative quotient R -algebra A ∗ ( L ) = E ∗ ( L ) / I ( L ) .We list some well known properties of the OS-algebra here. For more details, see[32, 11]. Proposition 2.3. (1) The OS algebra A ∗ ( L ) is a L -graded algebra, i.e., A ∗ ( L ) = L p ∈ L A ∗ ( L ) p , where A ∗ ( L ) p denotes the homogeneous submodule of order p , which is also a free R -module.(2) A ∗ ( L ) p · A ∗ ( L ) q ⊆ A ∗ ( L ) p ∨ q . If r ( p ∨ q ) < r ( p ) + r ( q ) , then A ∗ ( L ) p · A ∗ ( L ) q = 0 .(3) A ∗ ([ , p ]) is naturally isomorphic to the sub-algebra L q ≤ p A ∗ ([ , s ]) q ⊂ A ∗ ([ , s ]) forevery p ≤ s . Denote the imbedding map A ∗ ([ , p ]) ֒ → A ∗ ([ , s ]) by i s .(4) The derivation ∂ on E ∗ ( L ) induces a derivation (still denoted by ∂ ) on the OS-algebra A ∗ ( L ) , which maps A ∗ ( L ) p to L p : >q A ∗ ( L ) q . In particular, ( A ∗ ([ , p ]) , ∂ ) is an exactcomplex for every p = , p ∈ L .(5) dim A ∗ ( L ) p = ( − r ( p ) µ (0 , p ) for any field as coefficients, where µ ( − , − ) is the Möbiusfunction of L . Since we mainly consider the case of locally geometric posets in this paper, naturallywe wish to know whether OS-algebra can still be defined on locally geometric posets or not.
We will answer this in Sections 2.5–2.6. .4. Manifold arrangements.
Given a connected manifold M and a finite collection ofsubmanifolds A = { N i } , where M and each N i are smooth without boundaries. Asdefined in [14], A is said to be a manifold arrangement if it satisfies the Bott’s clean inter-section property that for every x ∈ M , there exist a neighborhood U of x , a neighbor-hood W of the origin in R n , a subspace arrangement { V i } in R n and a diffeomorphism φ : U → W such that φ maps x to the origin and maps { N i ∩ U } to { V i ∩ W } .Roughly speaking, a manifold arrangement is ’locally diffeomorphic’ to a subspacearrangement in an Euclidean space. There are some different combinatorical structuresassociated with manifold arrangements A (e.g. the poset of all possible intersection ofeach A ). In the case of hypersurface arrangements, many authors prefer the poset oflayers. R. Ehrenborg and M. Readdy [14] introduced the concept of intersection poset as a flexible tactic that there could be several suitable intersection posets dependingon particular circumstances. Roughly speaking, they combined some layers togetherby a suitable way. However, some condition in their definition of intersection posetis unnecessary in this paper, we remove these condition and define quasi-intersectionposet .Recall that an intersection of A is the intersection of a subset of A , and we always letthe intersection of empty set be M . A connect component of a nonempty intersectionis called a layer. Definition 2.2.
For any nonempty intersection with connected components c , ..., c k ,any disjoint union c i ⊔ c i ⊔ · · · ⊔ c i s is called a quasi-layer . All quasi-layer form a poset P ordered by reverse inclusion. For any p ∈ P , let M p be the associated quasi-layer (itis the same thing of p , we use the symbol M p to emphasize that it is a submanifold of M rather than an element of poset). A quasi-intersection poset P of manifold arrangement A is a sub-poset of P such that:(1) M is the minimum element;(2) M p ∩ M q equals disjoint union ⊔ s ∈ p ˚ ∨ q M s for all p, q ∈ P .In this paper, we always assume the following two additional conditions for manifoldarrangement A that are enough for our application: • every submanifold N i is a closed subset of M ; • there exists a quasi-intersection poset L of A such that L is locally geometricposet.Given p ∈ L , let M p be the associated quasi-layer. Set S p = M p − S q>p M q and write M ( A ) = S . By A p we denote the collection { M q | q : > p } . Then A p is a manifoldarrangement in M p with an intersection poset [ p, ∞ ) and S p = M ( A p ) .2.5. Cosheaf and sheaf on poset.
We see from Proposition 2.3 that ( A ∗ ([ , p ]) , ∂ ) is anexact complex, i.e., its all homologies vanish. If we combine this complex with suitable"coefficients", then the corresponding homologies will be more interesting.For p, q in a poset P , p ≤ q , we may understand that there is a unique morphism p → q , so P may be regarded as a category. Definition 2.3. A cosheaf on a poset P is a contravariant functor C from P to the cat-egory of R -modules (or algebras), by mapping p p and mapping every q → p to f p,q : C p → C q , satisfying f p,p = id ;(2) f q,s ◦ f p,q = f p,s .All cosheaves on P with natural transformations as morphisms also form a category.Similarly, we may also define a covariant functor from P to the category of R -modules(or algebras) by mapping every p → q to f p,q : C p → C q , which is called a sheaf on P . Remark . The definition of a sheaf on P by regarding P as a topological space withorder topology is essentially equivalent to the definition of a covariant functor on P as defined by Yanagawa in [31]. We follow this equivalence statement and use theterminology "sheaf (cosheaf)" as well. Example . Let A be a manifold arrangement in M with a quasi-intersection poset L .The cochains C ( A ) p = C ∗ ( M, M − M p ) combined with inclusion maps f p,q : C ∗ ( M, M − M p ) → C ∗ ( M, M − M q ) for p ≥ q give a cosheaf on L . Similarly, let H ∗ ( A ) p = H ∗ ( M, M − M p ) , H ∗ ( A ) is also acosheaf on L . We will see more details in next section. Example . With the same assumption as Example 2.2, the cohomology rings H ∗ ( M p ) combined with φ ∗ q,p give a sheaf (of rings) on L , where φ ∗ q,p is the cohomology homo-morphism induced by the inclusion map φ q,p : M q → M p for q ≥ p .Now, we define a complex ( A ∗ ( L , C ) , ∂ ) for a cosheaf C on a locally geometric poset L , which is a generalization of the complex structure on OS-algebra. Definition 2.4.
Let C is a cosheaf on L . Define A ∗ ( L , C ) p = A ∗ ([ , p ]) p ⊗ C p A ∗ ( L , C ) = M p ∈ L A ∗ ( L , C ) p with the differential ∂ defined by ∂ ( x ⊗ c ) = X i x i ⊗ f p,p i ( c ) for x ∈ A ∗ ([ , p ]) p , c ∈ C p and ∂x = P i x i in A ∗ ([ , p ]) such that x i ∈ A ∗ ([ , p ]) p i and p covers p i .It needs to check that ( A ∗ ( L , C ) , ∂ ) is a well-defined chain complex. Actually, let ∂x i = P j x i,j such that x i,j belongs to some A ∗ ([ , p ]) p j where r ( p j ) = r ( p i ) − r ( p ) − . Then we see that P i x i,j = 0 for every j since ∂∂ ( x ) = 0 . Thus ∂∂ ( x ⊗ c ) = P i,j x i,j ⊗ f p,p j ( c ) = P j ( P i x i,j ) ⊗ f p,p j ( c ) = 0 , as desired.Put a negative grading on A ∗ ( L , C ) such that A ∗ ( L , C ) − i = L r ( p )= i A ∗ ([ , p ]) p ⊗ C p . Thenegative grading will be convenient for the construction of double complex and theuse of spectral sequence later. For any subset [ p, ∞ ) ⊂ L , we will use A ∗ ([ p, ∞ ) , C ) todenote A ∗ ([ p, ∞ ) , C| [ p, ∞ ) ) for a convenience, where C| [ p, ∞ ) means the restriction of C onposet [ p, ∞ ) . Then we can check easily that all C 7−→ A ∗ ([ p, ∞ ) , C ) define a functor fromcosheaves on L to the category of R -module complexes. emark . Actually we may also define a functor Γ p ( C ) = Coker ( M α>p C α ⊕ f α,p −−−→ C p ) It can easily be checked that Γ p is right exact and the left derived functor L i Γ p is nat-urally isomorphic to H − i ( A ∗ ([ p, ∞ ) , C ) , ∂ ) . However, we do not use this fact in thisarticle.The following definition and lemma will be used later. Definition 2.5.
Let α ∈ L and A be any R -module. Define a cosheaf j α ∗ A on L as ( j α ∗ A ) p = (cid:26) A, p ≤ α , otherwisewhere the map f p,q is the identity if q ≤ p ≤ α and zero otherwise, which is called the sky-scraper cosheaf .The following lemma is a direct result of the exactness of the OS-algebra. Lemma 2.2.
For any sky-scraper cosheaf j α ∗ A , ( A ∗ ([ p, ∞ ) , j α ∗ A ) , ∂ ) is exact if and only if α = p . If α = p , then H − i ( A ∗ ([ p, ∞ ) , j α ∗ A )) = ( A if i = 00 if i = 0 . Monoidal cosheaf on lattice.
With the understanding on a negative grading on A ∗ ( L , C ) , the cohomology H − i ( A ∗ ([ p, ∞ ) , C )) will be an R -algebra if C has a suitableproduct structure on it, as we will describe below. It is exactly our main approach forthe calculation of the cohomology ring of the complement of manifold arrangements. Notation.
Let C be a cosheaf on P and I is a subset of P , denote C I = L p ∈ I C p . Let J ⊆ P and there exists map λ : I → J that ∀ p ∈ I, λ ( p ) ≤ p . Denote f I,J,λ = L p ∈ I f p,λp .In this paper, λ is always clear in context, so we omit it and always write f I,J . Definition 2.6.
Let C be a cosheaf on a locally geometric poset L . Then C is said tobe monoidal if L p C p is an associative algebra satisfying that C s · C t ⊆ C s ˚ ∨ t for every s, t ∈ L and b · f p,q ( a ) = f p ˚ ∨ s,q ˚ ∨ s ( b · a ) f p,q ( a ) · b = f p ˚ ∨ s,q ˚ ∨ s ( a · b ) for q ≤ p, a ∈ C p , b ∈ C s . Notice that in above notation of f p ˚ ∨ s,q ˚ ∨ s , we need a map λ : p ˚ ∨ s → q ˚ ∨ s , we always let it be the canonical map λ psqs given in Proposition 2.2(3)with no confusion. Remark . By Definition 2.6, a monoidal cosheaf C on a geometric lattice L is actuallya monoidal functor from L to the category of R -modules if there is a unit ∈ C , butwe will not use this general terminology for simplicity. In this case, L is regarded as amonoidal category and ∨ is the monoidal product. Example . Let A be a manifold arrangement in M with a quasi-intersection poset L . Then the cohomology rings H ∗ ( A ) p := H ∗ ( M, M − M p ) form a monoidal cosheafwith cup product H ∗ ( M, M − M p ) ⊗ H ∗ ( M, M − M q ) ∪ −→ L s ∈ p ˚ ∨ q H ∗ ( M, M − M s ) . Thismonoidal cosheaf H ∗ ( A ) will appears many times later. or a monoidal cosheaf C on L , making use of the OS-algebra A ∗ ([ , p ]) and themonoidal product of C we can define a product structure on A ∗ ( L , C ) as follows. Definition 2.7.
Assume that C is a monoidal cosheaf on a locally geometric poset L .Let j s : L p C p → C s be the projection on C s . Define the product on A ∗ ( L , C ) as ( x ⊗ c ) · ( y ⊗ c ) = ( − deg( c ) r ( q ) X s ∈ p ˚ ∨ q ( i s x · i s y ) ⊗ j s ( c · c ) for x ∈ A ∗ ([ , p ]) p , y ∈ A ∗ ([ , q ]) q , c ∈ C p , c ∈ C q , where the first " · " in the right side isthe product of OS-algebra A ∗ ([ , s ]) and i s is the imbedding map in Proposition 2.3(3). Remark . The algebra A ∗ ( L , C ) can be viewed as a "global" OS-algebra with C as coef-ficients, and it is actually a differential graded algebra. We will prove it later.3. A COSHEAF AND ITS FILTRATION FOR MANIFOLD ARRANGEMENTS
Let A be a manifold arrangement in M with quasi-intersection poset L . Recall that M p is the quasi-layer associated with p ∈ L , and S p = M p − S q>p M q , as defined in lastsection.Associated with the manifold arrangement A , there is a natural cosheaf C ( A ) thatencodes topological data of A , such that C ( A ) p = C ∗ ( M, M − M p ) , and C ( A ) : p ( A ) p is a graded cosheaf on L with the inclusion map f p,q : C ( A ) p → C ( A ) q for p ≥ q .3.1. A classical filtration.
We consider a classical filtration F ∗ p M of M for every p ∈ L and then study the E -term of associated spectral sequence. This will be very useful inthe proof of our main theorem. Definition 3.1.
Let p ∈ L . Define an increasing filtration F p M ⊂ F p M ⊂ · · · ⊂ M by F ip M = M − M p , if i < r ( p ) M − [ q ≥ p,r ( q )= i M q , if i ≥ r ( p ) . These filtrations F ∗ p M, p ∈ L induce the decreasing filtrations of the cosheaf C ( A ) ,which are defined as follows. Definition 3.2.
Let p ∈ L . Define a filtration of C ( A ) F C ( A ) p ⊃ F C ( A ) p ⊃ · · · ⊃ F i C ( A ) p ⊃ · · · by F i C ( A ) p = C ∗ ( M, F ip M ) . Let E ∗ , ∗ r,p ( A ) be the E r -term of the spectral sequence associated with the filtration F ∗ C ( A ) p . Clearly, the inclusion map f p,q : C ( A ) p → C ( A ) q gives F i C ( A ) p ⊂ F i C ( A ) q by definition, so f p,q induces the map between two spectral sequences, denoted by f p,q,r : E ∗ , ∗ r,p ( A ) → E ∗ , ∗ r,q ( A ) . Thus E ∗ , ∗ r, ∗ ( A ) is also a cosheaf on L for every E r -page.The following lemma will be useful in the study of the cosheaf on L for the E -page. Lemma 3.1. F ip M is an open subset of F i +1 p M , formed by removing some sub-manifolds ⊔ α ≥ p,r ( α )= i S α as closed subsets. roof. If i ≥ r ( p ) , then F ip M = M − ∪ α ≥ p,r ( α )= i M α = M − ⊔ α ≥ p,r ( α ) ≥ i S α = M − ⊔ α ≥ p,r ( α ) ≥ i +1 S α − ⊔ α ≥ p,r ( α )= i S α = M − ∪ α ≥ p,r ( α )= i +1 M α − ⊔ α ≥ p,r ( α )= i S α = F i +1 p M − ⊔ α ≥ p,r ( α )= i S α If i < r ( p ) , then ⊔ α>p,r ( α )= i S α = ∅ so the above equation follows from the definition of F ip M . A similar calculation shows that S α = M α ∩ F i +1 p M for α ≥ p and r ( α ) = i . Then S α is a closed subset of F i +1 p M since M α is closed in M , so is ⊔ α ≥ p,r ( α )= i S α because L isfinite. (cid:3) The cosheaf of the E -page. We will show that the cosheaf of the E -page has asimple structure. Actually it is just a direct sum of some sky-scraper cosheaves.In the following discussion, for a submanifold S of some manifold X , by N ( S ) wedenote the tubular neighborhood of S in X and N ( S ) = N ( S ) − S . If S is zero-codimensional, we convention that N ( S ) = S so N ( S ) = ∅ . Remark . Lemma 3.1 tells us that we may use the excision theorem on the couple ( F i +1 p M, F ip M ) where F ip M = F i +1 p M − ⊔ α ≥ p,r ( α )= i S α . Consider the tubular neighbor-hood N p,i of ⊔ α ≥ p,r ( α )= i S α in F i +1 p M , we may write N p,i = ⊔ α ≥ p,r ( α )= i N ( S α ) where N ( S α ) is the tubular neighborhood of S α in F i +1 α M , and then by N p,i, we means ⊔ α ≥ p,r ( α )= i N ( S α ) . Therefore, we have that H ∗ ( F i +1 p M, F ip M ) = H ∗ ( N p,i , N p,i, ) by theexcision theorem. Theorem 3.1.
The cosheaf E i,j , ∗ ( A ) on L is the direct sum of some sky-scraper cosheaves asfollows: E i,j , ∗ ( A ) ∼ = M r ( α )= i j α ∗ H i + j ( N ( S α ) , N ( S α ) ) Proof. If p > or i > , using Lemma 3.1 and Remark 6, we have E i,j ,p = H i + j ( F i C ∗ ( M, M − M p ) /F i +1 C ∗ ( M, M − M p ))= H i + j ( F i +1 p M, F ip M )= H i + j ( N p,i , N p,i, ) . Now consider the map f p,q, of E -page as mentioned in Definition 3.2. There is thefollowing commutative diagram of spaces ( N q,i , N q,i, ) φ q −−−→ ( F i +1 q M, F iq M ) ϕ x χ y ( N p,i , N p,i, ) φ p −−−→ ( F i +1 p M, F ip M ) here φ q , φ p are the inclusion maps induced by excision, and ϕ, χ are also inclusionmaps. Then we have the following commutative diagram L α ≥ q,r ( α )= i H i + j ( N ( S α ) , N ( S α ) ) φ ∗ q ←−−− ∼ = E i,j ,qϕ ∗ y x χ ∗ = f p,q, L α ≥ p,r ( α )= i H i + j ( N ( S α ) , N ( S α ) ) φ ∗ p ←−−− ∼ = E i,j ,p . Now we are going to show that φ ∗ q ◦ f p,q, ◦ ( φ ∗ p ) − is an inclusion map. For this, wenotice that the set { α | α ≥ q, r ( α ) = i } can be divided into two disjoint parts: { α | α ≥ p, r ( α ) = i } and { α | α ≥ q, α (cid:3) p, r ( α ) = i } . Let ϕ : ( N ( S α ) , N ( S α ) ) → ( N q,i , N q,i, ) be the inclusion map for any α in the second part. We first show that ϕ ∗ ◦ φ ∗ q ◦ f p,q, =0 . This follows from fact that N ( S α ) is contained in F ip M because F ip M = F i +1 p M −⊔ α ≥ p,r ( α )= i S α and N ( S α ) ∩ N ( S α ) = ∅ for any α lying in the first part; namely χ ◦ φ q ◦ ϕ maps N ( S α ) into F ip M . Combining with the above commutative diagram, we see that f p,q, must be the inclusion map under the isomorphisms φ ∗ q and φ ∗ p , i.e., E i,j ,p ∼ = M α ≥ p,r ( α )= i H i + j ( N ( S α ) , N ( S α ) ) , i ≥ r ( p )0 elseand f p,q, is the inclusion map M α ≥ p,r ( α )= i H i + j ( N ( S α ) , N ( S α ) ) ֒ → M α ≥ q,r ( α )= i H i + j ( N ( S α ) , N ( S α ) ) under the sense of the above isomorphisms. Thus, E ∗ , ∗ , ∗ ( A ) is isomorphic to the directsum of sky-scraper cosheaves as follows E i,j , ∗ ( A ) ∼ = M r ( α )= i j α ∗ H i + j ( N ( S α ) , N ( S α ) ) . (cid:3)
4. C
ONSTRUCTION OF MAIN MODEL
In Section 3, for a manifold arrangement A with a quasi-intersection lattice L , theassociated cosheaf C ( A ) p is chosen as a cochain complex for every p ∈ L , and thestructure map f p,q is commutative with the differential δ of every complex C ( A ) p .4.1. Cosheaf of cochain complex.
Let C be a cosheaf on a locally geometric poset L ina general sense. Definition 4.1.
We call C is a cosheaf of cochain complex if for p ∈ L , C p = L i C ip is acochain complex with the differential δ : C ip → C i +1 p and f p,q is cochain map between cochain complexes. e have seen from Definition 2.4 that A ∗ ( L , C ) is a complex with differential ∂ ,where ∂ is induced by the differential on A ∗ ([0 , p ]) . Now assume that C is a cosheafof cochain complex. Then we can use the differential on C to define another differentialon A ∗ ( L , C ) by δ ( x ⊗ c ) = ( − r ( p ) x ⊗ δ ( c ) where x ∈ A ∗ ([0 , p ]) p , c ∈ C ip and δ in the right side is the differential of complex C p .This means that A ∗ ( L , C ) becomes a double complex.Furthermore, consider the operation ∂ + δ on A ∗ ( L , C ) . We claim that ∂ + δ is adifferential on A ∗ ( L , C ) . It suffices to check that ( δ∂ + ∂δ )( x ⊗ c ) = 0 for x ∈ A ∗ ([ , p ]) p and c ∈ C ip . In fact, let ∂x = P j x j as in Definition 2.4 where x j ∈ A ∗ ([ , p ]) p j and p covers p j . Then δ∂ ( x ⊗ c ) = ( − r ( p ) − X j x j ⊗ f p,p j δ ( c ) and ∂δ ( x ⊗ c ) = ( − r ( p ) X j x j ⊗ f p,p j δ ( c ) by definition, from which follows that ( δ∂ + ∂δ )( x ⊗ c ) = 0 as desired. As well-known, A ∗ ( L, C ) with ∂ + δ is called the associated total complex , denoted by T ot ( A ∗ ( L, C )) , and ∂ + δ is called the total differential , which is of degree +1 . In addition, for x ∈ A ∗ ([ , p ]) p and c ∈ C ip , the total degree of x ⊗ c is defined as deg( x ⊗ c ) = i − r ( p ) .Combining the above arguments, we have that Proposition 4.1.
Let C be a cosheaf of cochain complex on a locally geometric poset L . Then A ∗ ( L , C ) naturally admits a double complex structure with two differential ∂ and δ , and theassociated total complex T ot ( A ∗ ( L , C )) is also a cochain complex with the total differential ∂ + δ of degree +1 . It is well known that for every double complex, there are two filtrations and twospectral sequences associated with it. For the double complex A ∗ ( L , C ) , we choose the"column-wise" filtration(4.1) τ − k T ot ( A ∗ ( L , C )) = M q with r ( q ) ≤ k A ∗ ( L , C ) q It is a decreasing filtration of modules, satisfying the condition of [21, Theorem 2.6].Thus we have
Corollary 4.1.
The spectral sequence associated with τ −∗ satisfying E − i,j = H − i ( A ∗ ( L , H j ( C )) , ∂ ) , which converges to H ∗ ( T ot ( A ∗ ( L , C ))) as modules. Double complex model of manifold arrangements.
For a manifold arrangement A in M , clearly C ( A ) is a cosheaf of cochain complex.Then A ∗ ( L , C ( A )) has a double complex structure. Furthermore, A ∗ ([ p, ∞ ) , C ( A )) is alsoa double complex for any p ∈ L since [ p, ∞ ) is also a locally geometric poset. Theorem 4.2. H ∗ ( T ot ( A ∗ ([ p, ∞ ) , C ( A )))) and H ∗ ( N ( S p ) , N ( S p ) ) are isomorphic as mod-ule. roof. Consider the filtration of C ( A ) in Definitin 3.2. Denote F i C ( A ) : p F i C ( A ) p .Then F i C ( A ) is also a cosheaf of cochain complex on L since the inclusion map f p,q iscompatible with the filtration F i C ( A ) p . Moreover, we may use F i C ( A ) to give a filtra-tion T ot ( A ∗ ([ p, ∞ ) , F i C ( A ))) of T ot ( A ∗ ([ p, ∞ ) , C ( A ))) , also denoted by F i T ot ( A ∗ ([ p, ∞ ) , C ( A ))) .Now we calculate H ∗ ( T ot ( A ∗ ([ p, ∞ ) , C ( A )))) by this filtration. By Definitin 3.2, wesee that F i C ( A ) p /F i +1 C ( A ) p = C ∗ ( F i +1 p M, F ip M ) which is also a cosheaf of cochain complex on L when p runs over L . Then E -term is F i T ot ( A ∗ ([ p, ∞ ) , C ( A ))) /F i +1 T ot ( A ∗ ([ p, ∞ ) , C ( A ))) = T ot ( A ∗ ([ p, ∞ ) , F i C ( A ) /F i +1 C ( A ))) We may calculate the homology of this total complex by "calculating homology twice"as seen in [21, Theorem 2.15]. Firstly let us calculate the homology under differential δ : H i + j ( A ∗ ([ p, ∞ ) , F i C ( A ) /F i +1 C ( A )) , δ ) = A ∗ ([ p, ∞ ) , E i,j , ∗ ( A )) where E i,j , ∗ ( A ) ∼ = L r ( α )= i j α ∗ H i + j ( N ( S α ) , N ( S α ) ) is the direct sum of some sky-scrapercosheaves as we calculated in Theorem 3.1. Secondly we calculate the homology underdifferential ∂ as follows: H −∗ ( A ∗ ([ p, ∞ ) , E i,j , ∗ ( A )) , ∂ ) = 0 for all i = r ( p ) by Lemma 2.2,which means that H ∗ ( T ot ( A ∗ ([ p, ∞ ) , F i C ( A ) /F i +1 C ( A )))) vanishes for all i = r ( p ) ; inother words, the E i,j -term of filtration F i T ot ( A ∗ ([ p, ∞ ) , C ( A ))) vanishes for all i = r ( p ) .So the quotient map T ot ( A ∗ ([ p, ∞ ) , F r ( p ) C ( A ))) → T ot ( A ∗ ([ p, ∞ ) , F r ( p ) C ( A ) /F r ( p )+1 C ( A ))) induces an isomorphism of cohomologies. When restricted on [ p, ∞ ) , since F r ( p ) C ( A ) α always equals C ∗ ( M, M − M p ) for all α ≥ p by the definition of F ∗ C ( A ) , we see that F r ( p ) C ( A ) /F r ( p )+1 C ( A ) can only have nonzero elements of C ∗ ( F r ( p )+1 p M, F r ( p ) p M ) on p ,and zero otherwise. So T ot ( A ∗ ([ p, ∞ ) , F r ( p ) C ( A ) /F r ( p )+1 C ( A ))) = ( C ∗ ( F r ( p )+1 p M, F r ( p ) p M ) , δ ) . Combining this equality and above quotient map, we conclude that the map
T ot ( A ∗ ([ p, ∞ ) , C ( A ))) → C ∗ ( F r ( p )+1 p M, F r ( p ) p M ) induces an isomorphism of cohomologies. On the other hand, we know by Remark 6that H ∗ ( F r ( p )+1 p M, F r ( p ) p M ) ∼ = H ∗ ( N ( S p ) , N ( S p ) ) . This completes the proof. (cid:3) Remark . Theorem 4.2 gives the equivalence expression of H ∗ ( N ( S p ) , N ( S p ) ) for every p rather than only H ∗ ( M ( A )) .Let p = be minimal element in L . Then we have following corollary Corollary 4.3. H ∗ ( T ot ( A ∗ ( L , C ( A )))) and H ∗ ( M ( A )) are isomorphic as modules. In partic-ular, this isomorphism is actually induced by the quotient map A ∗ ( L , C ( A )) → C ∗ ( M ( A )) that maps A ∗ ( L , C ( A )) q to zero for q > and A ∗ ( L , C ( A )) = C ∗ ( M ) to C ∗ ( M ( A )) (natu-rally induced by the inclusion M ( A ) ֒ → M ). roof. Let p = in the proof of Theorem 4.2. We note that N ( S ) = S = M ( A ) , N ( S ) = ∅ , F M = M ( A ) , and F M = ∅ . Then the quotient map in the proof ofTheorem 4.2 T ot ( A ∗ ([ p, , F r ( p ) C ( A ))) → T ot ( A ∗ ([ p, , F r ( p ) C ( A ) /F r ( p )+1 C ( A ))) becomes T ot ( A ∗ ( L, C ( A ))) → T ot ( A ∗ ( L, C ( A ) /F C ( A ))) . Now, since the quotient cosheaf C ( A ) p /F C ( A ) p is zero if p > and be C ∗ ( M ( A )) if p = , we see that the above quotient map would be a zero-morphism if p > and thequotient map C ∗ ( M ) → C ∗ ( M ( A )) induced by the inclusion M ( A ) ֒ → M if p = . (cid:3) Consider the spectral sequence associated with the double complex A ∗ ([ p, ∞ ) , C ( A )) ,we have an immediate corollary. Corollary 4.4.
Associated with double complex A ∗ ([ p, ∞ ) , C ( A )) , there is a spectral sequencewith E − i,j = A ∗ ([ p, ∞ ) , H j ( A )) − i E − i,j = H − i ( A ∗ ([ p, ∞ ) , H j ( A )) , ∂ ) converges to H ∗ ( N ( S p ) , N ( S p ) ) as modules. In particular, for p = , there exists a spectralsequence with E − i,j = H − i ( A ∗ ( L , H j ( A )) , ∂ ) converges to H ∗ ( M ( A )) as modules. Recall that H ∗ ( A ) p := H ∗ ( M, M − M p ) . Using the approach developed in this section, we can easily reprove Zaslavsky’sresult [33] about the f -polynomials of hyperplane arrangements in M = R d . Corollary 4.5.
Let A be a central hyperplane arrangement in R d with intersection lattice L .Then the number of k -faces equals X p ∈ L,r ( p )= d − k dim( A ∗ ([ p, ])) where A ∗ ([ p, ]) is the canonical OS-algebra of [ p, ] with any field coefficients.Proof. Choose quasi-intersection poset to be intersection lattice of A . Every k -face is aregion of some S p with r ( p ) = d − k . Then the number of k -faces equals P p ∈ L,r ( p )= d − k dim( H ∗ ( N ( S p ) , N ( S p ) )) by Thom isomorphism. Since M p = R d − r ( p ) , thestructure map f ∗ p,q of H ∗ ( A ) is zero and the spectral sequence in Corollary 4.4 col-lapses on E -term because of the dimensional reason, so dim( H ∗ ( N ( S p ) , N ( S p ) )) =dim( A ∗ ([ p, ])) . (cid:3) Remark . Above corollary agrees with the original description of Zaslavsky [33]. Inthis view point, Theorem 4.2 essentially considers not only "regions (or chambers)" butalso the information of all "faces" of a manifold arrangement A , so it can be regardedas a topological generalization of Zaslavsky’s f -polynomial. .3. The relationship with mixed Hodge structure.
In this subsection, we discuss themixed Hodge structure of our model.Let R be a noetherian subring of C such that R ⊗ Q is a field. A Hodge R -complex K of weight n is a system [ K R , ( K C , F ) , α : K R ⊗ C ≃ K C ] where K R is a R -complex with finite type cohomology, K C is a C -complex with de-creasing filtration F and α is an isomorphism in D + ( C ) , satisfying: (1) the differentialof K C is strictly compatible with filtration F ; (2) the induced filtration F on H k ( K C ) defines a pure Hodge structure of weight n + k .A R -mixed Hodge complex (MHC) K is a system [ K R ; ( K R ⊗ Q , W ) , α : K R ⊗ Q ≃ K R ⊗ Q ; ( K C , W, F ) , β : ( K R ⊗ Q , W ) ⊗ C ≃ ( K C , W )] where K R is a R -complex with finite type cohomology, K R ⊗ Q is a R ⊗ Q -complex withincreasing filtration W and α is an isomorphism in derived category of R ⊗ Q -module, K C is a C -complex with increasing filtration W and decreasing filtration F and β isan isomorphism in derived category of filtrated C -module, satisfying: the n -th gradedpiece Gr Wn K = [ Gr Wn K R ⊗ Q , Gr Wn ( K C , F ) , Gr Wn β ] is a R ⊗ Q -Hodge complex of weight n for every n .There is a functorial MHC associated with any smooth complex algebraic variety U (see [34, Theorem 4.8]) , denoted by K ( U ) , which induces the canonical mixed Hodgestructure on H ∗ ( U ) . For more details about MHC and the definition of morphism ofMHC, we refer to [26].The model T ot ( A ∗ ( L , C )) will be a mixed Hodge complex if C is a cosheaf of mixedHodge complex, i.e., all f p,q : C p → C q are morphisms of mixed Hodge complex. Definition 4.2.
Assume C is a cosheaf of mixed Hodge complex. Namely C = [ C R ; ( C R ⊗ Q , W ) , α : C R ⊗ Q ≃ C R ⊗ Q ; ( C C , W, F ) , β : ( C R ⊗ Q , W ) ⊗ C ≃ ( C C , W )] where C R : p pR (resp. C R ⊗ Q , C C ) is a cosheaf of R (resp. R ⊗ Q , C ) module and f p,q : C p → C q is the morphism of mixed Hodge complex. Define T ot ( A ∗ ( L , C )) be asystem [ T ot ( A ∗ ( L , C R )); ( T ot ( A ∗ ( L , C R ⊗ Q )) , W ) , ˜ α ; ( T ot ( A ∗ ( L , C C )) , W, F ) , ˜ β ] where W m T ot ( A ∗ ( L , C R ⊗ Q )) = M p A ∗ ([0 , p ]) p ⊗ W m − r ( p ) C pR ⊗ Q F m T ot ( A ∗ ( L , C C )) = M p A ∗ ([0 , p ]) p ⊗ F m C p C and ˜ α, ˜ β is the isomorphism induced by α, β . Proposition 4.2. If C is a cosheaf of mixed Hodge complex, then T ot ( A ∗ ( L , C )) in above defi-nition is a well defined mixed Hodge complex.Proof. A direct calculation gives that the n -th graded piece Gr Wm T ot ( A ∗ ( L , C )) equals L p A ∗ ( L ) p ⊗ Gr Wm − r ( p ) C p with differential induced by δ on every C p (the contribution of ∂ to the differential vanishes since ∂ preserve W ), notice every element • ⊗ C ip have totaldegree i − r ( p ) , so A ∗ ( L ) p ⊗ Gr Wm − r ( p ) C p is a weight m Hodge complex for any p ∈ L . (cid:3) alculate the spectral sequence associated with weight filtration of T ot ( A ∗ ( L , C )) carefully, we have E − p,q = L s A ∗ ([0 , s ]) s ⊗ E − p + r ( s ) ,q ( C s ) , and the complex of E page · · · → E − p,q → E − p +1 ,q → · · · equals T ot ( A ∗ ( L , E ∗ ,q ( C ))) , where E ∗ ,q ( C ) s is the E page of weight filtration of every C s . Notice that E − p,q = Gr Wq H q − p ( K R ⊗ Q ) for every MHC K [26, Theorem 3.18], we have Corollary 4.6. Gr Wq H q − p ( T ot ( A ∗ ( L , C R ⊗ Q ))) = H − p ( T ot ( A ∗ ( L , E ∗ ,q ( C R ⊗ Q )))) The total complex
T ot ( A ∗ ( L , E ∗ ,q ( C R ⊗ Q ))) in the right side will be simpler if C satisfiessome "purity" condition, for example: Corollary 4.7.
If every H k ( C s ) is pure of weight k , then Gr Wq H q − p ( T ot ( A ∗ ( L , C R ⊗ Q ))) = H − p ( A ∗ ( L , H q ( C R ⊗ Q )) , ∂ ) . Proof.
In this case, E p,q ( C R ⊗ Q ) = 0 for any p = 0 and E ,q ( C R ⊗ Q ) = H q ( C R ⊗ Q ) . Thenwe can calculate H ∗ ( T ot ( A ∗ ( L , E ∗ ,q ( C R ⊗ Q )))) by "calculate cohomology twice", that is, H − p ( T ot ( A ∗ ( L , E ∗ ,q ( C R ⊗ Q )))) = H − p ( A ∗ ( L , H q ( C R ⊗ Q )) , ∂ ) as desired. (cid:3) Now, suppose that M and every N i ∈ A are complex smooth algebraic varieties. Weare going to construct a cosheaf of mixed Hodge complex as a substitute for C ( A ) inTheorem 4.2. Recall that C ( A ) p = C ∗ ( M, M − M p ) . One possible choice is the substitute K ( A ) p = Cone ( K ( M ) → K ( M − M p ))[ − for C ( A ) p , where K ( • ) is the associated MHCof any variety and Cone denotes the mixed cone of a map. This simple constructiondoes not behave very well since the quotient map
T ot ( A ∗ ( L , C ( A ))) → C ∗ ( M ( A )) inCorollary 4.3 is no longer a chain map of complexes if we replace C by K . We will usethe technique of mapping telescope (cone) to fix this problem.Recall that there is a mixed cone Cone ( φ ) for any morphism φ : K → L of MHC K, L .Roughly speaking,
Cone ( φ ) = K [1] ⊕ L with d ( x, y ) = ( dx, f ( x ) + dy ) and filtrations W [1] ⊕ W and F ⊕ F . Actually Cone ( φ ) is a MHC with these filtrations, we refer to [34,Definition 3.31] for strict definition about mixed cone.We define our mixed telescope cone by using mixed cone as follows. Definition 4.3.
Let K f → K f → · · · f n − → K n be a sequence of map of MHC and n ≥ .Construct a function ˜ f : L n − K i → L n K i as follows ˜ f ( x , x , . . . , x n − ) = ( x + f ( x ) , . . . , x n − + f n − ( x n − ) , f n − ( x n − )) . We define the mixed telescope cone of the sequence K f → K f → · · · f n − → K n as themixed cone Cone ( ˜ f ) , also denoted by T Cone ( K f → K f → · · · f n − → K n ) . This coincideswith the usual definition of mixed cone if n = 2 . Remark . It is easy to check there is a short exact sequence of MHC → T Cone ( K f → · · · f s − → K s ) → T Cone ( K f → · · · f n − → K n ) → T Cone ( K s f s → · · · f n − → K n ) → . If K f → · · · f n − → K n is the associated MHC of U n ⊆ · · · ⊆ U , then the long exactsequence induced by above short exact sequence is just the long exact sequence oftriple ( U , U s , U n ) . efinition 4.4. Define K ( A ) p = T Cone ( K ( M ) → · · · → K ( F ip M ) → · · · → K ( M − M p ))[ − where F ip M is the filtration in Definition 3.1. Proposition 4.3.
The K ( A ) in above definition is a cosheaf of mixed Hodge complex, and thereis a quotient map T ot ( A ∗ ( L , K ( A ))) → Cone ( K ( F r ( p )+1 p M ) → K ( F r ( p ) p M ))[ − which is a morphism of MHC and induces an isomorphism with mixed Hodge structure H ∗ ( T ot ( A ∗ ([ p, ∞ ) , K ( A )))) ∼ = H ∗ ( F r ( p )+1 p M, M − M p ) . In particular, for p = , H ∗ ( T ot ( A ∗ ( L , K ( A )))) ∼ = H ∗ ( M ( A )) with mixed Hodge structure.Proof. Firstly, we need to define map f p,q : K ( A ) p → K ( A ) q . The F iq M is always asubspace of F ip M for any p ≥ q by definition. Then these maps K ( F ip M ) → K ( F iq M ) induce map of telescope cone f p,q : K ( A ) p → K ( A ) q . It is obvious that K ( A ) is a cosheafof mixed Hodge complex with these f p,q .In the proof of Theorem 4.2, we use a filtration of C ( A ) such that E -term is sky-scraper cosheaf. Similarly, let F i K ( A ) p = ( T Cone ( K ( M ) → · · · → K ( F ∗ p M ) → · · · → K ( M − M p ))[ − , if i < r ( p ) T Cone ( K ( M ) → · · · → K ( F ip M ))[ − , if i ≥ r ( p ) . It is easy to check F i K ( A ) is a filtration of cosheaf ( f p,q preserves this filtration). Thereis no difference between the homologies of F i K ( A ) p and F i C ( A ) p , i.e., they are isomor-phic to H ∗ ( M, F ip M ) . So all processes in the proof of theorem 4.2 can still be carried outvery well for the filtration F i K ( A ) p . Notice that F r ( p ) K ( A ) q /F r ( p )+1 K ( A ) q ∼ = ( Cone ( K ( F r ( p )+1 p M ) → K ( F r ( p ) p M ))[ − if p = q if q > p. Then we have that H ∗ ( T ot ( A ∗ ([ p, ∞ ) , K ( A )))) ∼ = H ∗ ( F r ( p )+1 p M, M − M p ) with mixed Hodge structure. (cid:3) In particular, if M is projective, we know that H k ( A ) p = H k ( M, M − M p ) is pure ofweight k (actually, it equals a Tate twist of H ∗ ( M p ) ). Applying Corollary 4.7, we have Corollary 4.8.
Assume M is projective and using Q coefficients, we have Gr Wn H n − i ( M ( A )) = H − i ( A ∗ ( L , H n ( A )) , ∂ ) At the end of this subsection, we shall compare the weight filtration with the "column-wise" filtration (4.1) τ −∗ of T ot ( A ∗ ( L , K ( A ))) . Theorem 4.9.
Assume that M is projective and we use Q as coefficients. Then the "column-wise" filtration τ −∗ and the weight filtration W ∗ on T ot ( A ∗ ( L , K ( A ))) induce the same filtra-tion on cohomology group H ∗ ( T ot ( A ∗ ( L , K ( A )))) ∼ = H ∗ ( M ( A )) . roof. Assume C is a cosheaf of MHC and every H k ( C s ) is pure of weight k . We provea more general version that the "column-wise" filtration τ −∗ and the weight filtration W ∗ on T ot ( A ∗ ( L , C )) induce the same filtration on cohomology H ∗ ( T ot ( A ∗ ( L , C ))) .Recall that τ − i T ot ( A ∗ ( L , C )) = L p ∈ L ,r ( p ) ≤ i A ∗ ([0 , p ]) p ⊗ C p = T ot ( A ∗ ( { p ∈ L | r ( p ) ≤ i } , C )) . We need to show that Im( H ∗ ( τ − i T ot ( A ∗ ( L , C ))) → H ∗ ( T ot ( A ∗ ( L , C )))) equals W i H ∗ ( T ot ( A ∗ ( L , C ))) .Corollary 4.7 shows that(1) Gr Wk + j H k ( τ − i T ot ( A ∗ ( L , C ))) = H − j ( A ∗ ( { p ∈ L | r ( p ) ≤ i } , H k + j C ) , ∂ ) (2) Gr Wk + j H k ( T ot ( A ∗ ( L , C ))) = H − j ( A ∗ ( L , H k + j C ) , ∂ ) , so the inclusion map induces a surjective map on Gr Wk + j H k = W j /W j − for j = i andan isomorphism for j < i . Then it induces a surjective map on W i as well and therefore Im( W i H ∗ ( τ − i T ot ( A ∗ ( L , C ))) → H ∗ ( T ot ( A ∗ ( L , C )))) equals W i H ∗ ( T ot ( A ∗ ( L , C ))) .Now, we only need to check W i H ∗ ( τ − i T ot ( A ∗ ( L , C ))) equals H ∗ ( τ − i T ot ( A ∗ ( L , C ))) it-self. Observe that Gr Wk + j H k ( τ − i T ot ( A ∗ ( L , C ))) = 0 for j > i by equation (1), whichmeans that W j /W j − = 0 for j > i , so W i H ∗ ( τ − i T ot ( A ∗ ( L , C ))) equals H ∗ ( τ − i T ot ( A ∗ ( L , C ))) itself, which completes the proof. (cid:3) Remark . Unlike the weight filtration W ∗ , the filtration τ −∗ is also defined for ourmodel T ot ( A ∗ ( L , C ( A ))) with any coefficients where the manifolds in A need not to bevarieties.4.4. Inclusion map of sub-arrangements.
In this section, we will consider the sub-arrangements A| p = { M a i | r ( a i ) = 1 , a i ≤ p } . It is easy to see that every A| p is also amanifold arrangement with intersection lattice [0 , p ] . M ( A ) is obviously a subspace of M ( A| p ) , so there is the inclusion i : M ( A ) ֒ → M ( A| p ) . A natural question arises: whatis the induced map i ∗ : H ∗ ( M ( A| p )) → H ∗ ( M ( A )) under the isomorphism in Theorem 4.2? For an OS-algebra A ∗ ( L ) , we know that there is an inclusion map A ∗ ([ , p ]) → A ∗ ( L ) for every p ∈ L , which maps A ∗ ([ , p ]) q isomorphically onto A ∗ ( L ) q for every q ≤ p .Then we have an inclusion map j : A ∗ ([ , p ] , C ( A )) → A ∗ ( L, C ( A )) as double com-plexes. Proposition 4.4.
There is a commutative diagram H ∗ ( M ( A| p )) i ∗ −−−→ H ∗ ( M ( A )) x x H ∗ ( T ot ( A ∗ ([ , p ] , C ( A )))) j ∗ −−−→ H ∗ ( T ot ( A ∗ ( L, C ( A )))) where j ∗ is the induced map of cohomology by j , and every column arrow is an isomorphism asin Corollary 4.3.Proof. We see in the proof of Theorem 4.2 that the isomorphism H ∗ ( T ot ( A ∗ ( L, C ( A )))) → H ∗ ( M ( A )) is induced by the quotient A ∗ ( L, C ( A )) → C ∗ ( M ( A )) by mapping A ∗ ( L ) q ⊗C ( A ) q to zero for all q = and mapping C ( A ) = C ∗ ( M ) to C ∗ ( M ( A )) . Now it suffices o check that the following diagram is commutative: C ∗ ( M ( A| p )) −−−→ C ∗ ( M ( A )) x x A ∗ ([ , p ] , C ( A )) −−−→ A ∗ ( L, C ( A )) . Choose an element P x q ⊗ c q ∈ A ∗ ([ , p ] , C ( A )) where x q ∈ A ∗ ([ , p ]) q and c q ∈ C ( A ) q ,the image of this element in C ∗ ( M ( A )) is the image of x c under the quotient C ∗ ( M ) → C ∗ ( M ( A )) regardless of the ’path’ we choose. (cid:3)
5. P
RODUCT S TRUCTURE
In this section, we are going to study the product structure on the double complex A ∗ ([ p, ∞ ) , C ( A )) . We first discuss some general construction in subsection 5.1. Wewill see that if C is a monoidal cosheaf of DGA, then T ot ( A ∗ ( L , C )) is a differentialalgebra. Unfortunately, C ( A ) p = C ∗ ( M, M − M p ) is not a monoidal cosheaf underthe cup product because the cup product c ∪ c for c ∈ C ∗ ( M, M − M p ) and c ∈ C ∗ ( M, M − M q ) may not be contained in L s ∈ p ˚ ∨ q C ∗ ( M, M − M s ) , where C ∗ ( M, M − M p ) and C ∗ ( M, M − M q ) are regarded as sub-complexes of C ∗ ( M ) . We will modify C ( A ) into a monoidal cosheaf ˆ C ( A ) in subsection 5.2.5.1. Monoidal cosheaf of DGA.Definition 5.1.
Let C be a cosheaf of cochain complex. We call C a monoidal cosheafof DGA if C is a monoidal cosheaf with the monoidal product satisfying δ ( c c ) = δ ( c ) c + ( − i c δ ( c ) for all c ∈ C ip , c ∈ C jq , where δ is the differential of every cochain complex C p . Remark . For the monoidal product on C , we have C p · C p ⊂ C p since p ˚ ∨ p = { p } , soevery C p be a differential graded algebra. This is the reason why we use the name’monoidal cosheaf of DGA’.Let C be a monoidal cosheaf of DGA on locally geometric poset L . Then the asso-ciated double complex ( A ∗ ( L , C ) , ∂, δ ) has a natural product structure induced by theOS-algebra and the monoidal product of C , as we defined in Definition 2.7, that is(5.1) ( x ⊗ c ) · ( y ⊗ c ) = ( − deg( c ) r ( q ) X s ∈ p ˚ ∨ q ( i s x · i s y ) ⊗ j s ( c · c ) for x ∈ A ∗ ([ , p ]) p , y ∈ A ∗ ([ , q ]) q , c ∈ C p , c ∈ C q , where product " · " on the right side isthe product of OS-algebra A ∗ ([ , s ]) , j s is projection on C s and i s is the imbedding mapin Proposition 2.3(3). Furthermore, ∂ and δ are both derivation of algebra A ∗ ( L , C ) , weneed a lemma before prove this property. Lemma 5.1.
The following equations hold in A ∗ ( L , C ) X t ∈ q ˚ ∨ s x t ⊗ j t ( b · f p,q a ) = X k ∈ p ˚ ∨ s x λk ⊗ f k,λk ( j k ( b · a )) and X t ∈ q ˚ ∨ s x t ⊗ j t (( f p,q a ) · b ) = X k ∈ p ˚ ∨ s x λk ⊗ f k,λk ( j k ( a · b )) here p, q, s ∈ L , q ≤ p , x t ∈ A ∗ ([ , t ]) t for all t ∈ q ˚ ∨ s , a ∈ C p , b ∈ C s , λ = λ psqs be thecanonical map p ˚ ∨ s → q ˚ ∨ s Proof.
By a direct calculation, we have that b · f p,q ( a ) = f p ˚ ∨ s,q ˚ ∨ s ( b · a ) = X k ∈ p ˚ ∨ s f k,λk j k ( b · a )= X t ∈ q ˚ ∨ s X k ∈ λ − t f k,t j k ( b · a ) . Doing the projection on two sides to C t term, we have j t ( b · f p,q a ) = P k ∈ λ − t f k,t j k ( b · a ) .Then X t ∈ q ˚ ∨ s x t ⊗ j t ( b · f p,q a ) = X t ∈ q ˚ ∨ s X k ∈ λ − t x t ⊗ f k,t j k ( b · a )= X k ∈ p ˚ ∨ s x λk ⊗ f k,λk j k ( b · a ) as desired. The second equation follows in a similar way as above. (cid:3) Proposition 5.1.
Two differentials ∂ and δ with respect to the product (5.1) of A ∗ ( L, C ) satisfythe Leibniz laws ∂ ( αβ ) = ∂ ( α ) β + ( − i − r ( p ) α∂ ( β ) δ ( αβ ) = δ ( α ) β + ( − i − r ( p ) αδ ( β ) for all α ∈ A ∗ ([ , p ]) p ⊗ C ip and β ∈ A ∗ ([ , q ]) q ⊗ C jq .Proof. Write α = x ⊗ c and β = x ⊗ c where x ∈ A ∗ ([ , p ]) p , x ∈ A ∗ ([ , q ]) q , c ∈ C ip ,and c ∈ C jq . Let ∂ ( x ) = P u x p u and ∂ ( x ) = P v x q v such that p covers p u and q covers q v . Since the differential of the OS-algebra A ∗ ( L ) satisfies the Leibniz law, we have that ∂ ( x x ) = ∂ ( x ) x + ( − r ( p ) x ∂ ( x ) = P x p u x + ( − r ( p ) P x x q v . So ( − r ( q ) i ∂ ( αβ ) = X k ∈ p ˚ ∨ q ∂ (( i k x i k x ) ⊗ j k ( c c ))= X k ∈ p ˚ ∨ q ( X u i k x p u i k x ⊗ f k,λk j s ( c c ) + ( − r ( p ) X v i k x i k x q v ⊗ f k,λk j s ( c c )) where the first λ is λ pqp u q , the second is λ pqpq v . Using Lemma 5.1, the right side equals X u X t ∈ p u ˚ ∨ q i t x p u i t x ⊗ j t ( f p,p u ( c ) c ) + ( − r ( p ) X v X t ∈ p ˚ ∨ q v i t x i t x q v ⊗ j t ( c f q,q v ( c ))= ( − r ( q ) i ( X u x p u ⊗ f p,p u ( c ))( x ⊗ c ) + ( − r ( p )+ i ( r ( q ) − ( x ⊗ c )( X v x q v ⊗ f q,q v ( c ))= ( − r ( q ) i ∂ ( α ) β + ( − r ( p )+ r ( q ) i − i α∂ ( β ) rom which our first equation follows, and ( − r ( q ) i δ ( αβ ) equals X k ∈ p ˚ ∨ q δ (( i k x i k x ) ⊗ j k ( c c ))= ( − r ( p )+ r ( q ) X k ∈ p ˚ ∨ q ( i k x i k x ) ⊗ δj k ( c c )= ( − r ( p )+ r ( q ) X k ∈ p ˚ ∨ q ( i k x i k x ) ⊗ j k ( δ ( c ) c ) + ( − r ( p )+ r ( q )+ i X k ∈ p ˚ ∨ q ( i k x i k x ) ⊗ j k ( c δ ( c ))= ( − r ( p )+ r ( q ) i [( x ⊗ δ ( c ))( x ⊗ c ) + ( − r ( q )+ i ( x ⊗ c )( x ⊗ δ ( c ))]= ( − r ( q ) i δ ( x ⊗ c )( x ⊗ c ) + ( − r ( p )+ i + r ( q ) i ( x ⊗ c ) δ ( x ⊗ c )= ( − r ( q ) i δ ( α ) β + ( − r ( p )+ i + r ( q ) i αδ ( β ) which induces our second required equation. (cid:3) Corollary 5.1.
Let C be a monoidal cosheaf of DGA on a locally geometric poset L , then thetotal complex T ot ( A ∗ ( L , C )) is a differential algebra with the total differential δ + ∂ .Proof. It suffices to show that δ + ∂ satisfies the Leibniz law, that is, ( δ + ∂ )( αβ ) = ( δ + ∂ )( α ) β + ( − i − r ( p ) α ( δ + ∂ )( β ) . This immediately follows from the Leibniz laws of δ and ∂ in Proposition 5.1. (cid:3) Recall that the "column-wise" filtration of the double complex A ∗ ( L , C ) is τ − k T ot ( A ∗ ( L , C )) = M q with r ( q ) ≤ k A ∗ ( L , C ) q . If C is a monoidal cosheaf, this is a decreasing filtration of algebra by above discussing, satisfying the condition of [21, Theorem 2.14]. Thus we have Corollary 5.2.
There is a spectral sequence associated with filtration τ −∗ satisfying E − i,j = H − i ( A ∗ ([ p, ∞ ) , H j ( C )) , ∂ ) which converges to H ∗ ( T ot ( A ∗ ([ p, ∞ ) , C ))) as algebras, i.e. E −∗ , ∗∞ = Gr τ ∗ H ∗ ( T ot ( A ∗ ([ p, ∞ ) , C ))) as algebras. Our spectral sequence with the product structure may induce some good conditionsof degeneration.
Theorem 5.3.
Assume that the algebra H −∗ ( A ∗ ([ p, ∞ ) , H ∗ ( C ))) is generated by H and H − of the chain complex ( A ∗ ([ p, ∞ ) , H ∗ ( C )) , ∂ ) . Then the spectral sequence in Corollary 5.2 col-lapse at E ; namely, M i,j E − i,j ∼ = Gr τ ∗ H ∗ ( T ot ( A ∗ ([ p, ∞ ) , C ))) as algebras.Proof. The "column-wise" filtration F − k T ot ( A ∗ ([ p, ∞ ) , C )) = M q with r ( q ) ≤ k + r ( p ) A ∗ ([ p, ∞ ) , C ) q s an decreasing filtration, so the differential d maps E − i,j to E − i +2 ,j − . Then d ( E − ,j ) =0 and d ( E ,j ) = 0 since E − i,j = H − i ( A ∗ ([ p, ∞ ) , H j ( C ))) = 0 for i < . Since we haveassumed that H −∗ ( A ∗ ([ p, ∞ ) , H ∗ ( C ))) is generated by degree H and H − , this meansthat E i,j is generated by E , ∗ and E , ∗ . Furthermore, d ( E − i,j ) = 0 for all i, j since d satisfies the Leibniz law on E term, so E − i,j = E − i,j , and all d r for r ≥ are zero. (cid:3) Construction of monoidal cosheaf ˆ C ( A ) . Firstly, let us review some definitionabout presheaf of singular cochains.
Definition 5.2.
Let C ∗ ( X ) be those singular cochains f which are zero on all elements ofa suitable open cover of X (which may depends on f ), define ˆ C ∗ ( X ) = C ∗ ( X ) /C ∗ ( X ) .If ϕ : Y → X is a map of space, ϕ ∗ : C ∗ ( X ) → C ∗ ( Y ) maps C ∗ ( X ) to C ∗ ( Y ) , so itinduces a well defined map ˆ ϕ ∗ : ˆ C ∗ ( X ) → ˆ C ∗ ( Y ) . Let ˆ C ∗ ( X, A ) be the kernal of map ˆ C ∗ ( X ) → ˆ C ∗ ( A ) where A is a subspace of X . Literally, ˆ C ∗ ( X, A ) is the subset of ˆ C ∗ ( X ) ,each of whose elements can be represented by a cochain f which valuates zero on allelements of a suitable open cover of A . Remark . Let S ∗ : U → C ∗ ( U ) be the presheaf of singular cochains on X and ˆ S ∗ bethe associated sheaf. Theorem 6.2 of [5, Chapter I] tells us that ˆ C ∗ ( X ) = Γ ( X, ˆ S ∗ ) if X is paracompact. C ∗ ( X ) has zero cohomology by the discussion of subdivision, so thequotient map C ∗ ( X ) → ˆ C ∗ ( X ) is a quasi-isomorphism of complexes. Lemma 5.2.
Let
A, B be open subspaces of a metric space X . Then(1) The cup product on C ∗ ( X ) induces a cup product ∪ : ˆ C ∗ ( X, A ) ⊗ ˆ C ∗ ( X, B ) → ˆ C ∗ ( X, A ∪ B ) . (2) There is a short exact sequence → ˆ C ∗ ( X, A ∪ B ) → ˆ C ∗ ( X, A ) ⊕ ˆ C ∗ ( X, B ) → ˆ C ∗ ( X, A ∩ B ) → . (3) If { A i } consists of open subspaces of X such that A i ∪ A j = X for any different i, j ,then ˆ C ∗ ( X, \ i A i ) = M i ˆ C ∗ ( X, A i ) . Proof. (1) Let f (resp. f ) be a cochain of ˆ C ∗ ( X, A ) (resp. ˆ C ∗ ( X, B ) ) such that its valua-tion vanishes on all elements of open cover U (resp. U ) of A (resp. B ). Then f ∪ f isa cochain whose valuation vanishes on an open cover U ∪ U of A ∪ B , representingan element of ˆ C ∗ ( X, A ∪ B ) .(2) ˆ C ∗ ( X, A ) , ˆ C ∗ ( X, B ) are both subgroups of ˆ C ∗ ( X, A ∩ B ) . It is clear that ˆ C ∗ ( X, A ) ∩ ˆ C ∗ ( X, B ) = ˆ C ∗ ( X, A ∪ B ) . Thus it suffices to show ˆ C ∗ ( X, A ∩ B ) = ˆ C ∗ ( X, A ) + ˆ C ∗ ( X, B ) .Let f represent an element of ˆ C ∗ ( X, A ∩ B ) . Two elements f | A ∈ ˆ C ∗ ( A ) and ∈ ˆ C ∗ ( B ) give the same restriction on A ∩ B , so we can "glue" them together and get an element f A ∈ ˆ C ∗ ( A ∪ B ) satisfying that f A | A = f | A and f A | B = 0 since U ˆ C ∗ ( U ) is a sheafby Remark 12. Now let g ∈ ˆ C ∗ ( X ) such that g | A ∪ B = f A , then f − f | A represents anelement of ˆ C ∗ ( X, A ) and f | A represents an element of ˆ C ∗ ( X, B ) . These two elementsgive us the required decomposition.(3) Notice that A ∪ ( A ∩ A ∩ · · · ) = ( A ∪ A ) ∩ ( A ∪ A ) ∩ · · · = X ∩ X ∩ · · · = X .Then the required result follows by using (2) and an induction. (cid:3) ext let us return back to the manifold arrangement A with a quasi-intersectionposet L . Definition 5.3.
We define ˆ C ( A ) p = ˆ C ∗ ( M, M − M p ) Lemma 5.3. ˆ C ( A ) is a monoidal cosheaf.Proof. The map f p,q : ˆ C ( A ) p → ˆ C ( A ) q induced by inclusion make ˆ C ( A ) is a cosheaf. Thecup product ∪ maps ˆ C ( A ) p ⊗ ˆ C ( A ) q to ˆ C ∗ ( M, M − M p ∩ M q ) by Lemma 5.2(1), where M p ∩ M q = ⊔ s ∈ p ˚ ∨ q M s . Then ˆ C ∗ ( M, M − M p ∩ M q ) = L s ∈ p ˚ ∨ q ˆ C ( A ) s by Lemma 5.2(3). Thecondition b · f p,q ( a ) = f p ˚ ∨ s,q ˚ ∨ s ( b · a ) (resp. f p,q ( a ) · b = f p ˚ ∨ s,q ˚ ∨ s ( a · b ) ) of monoidal cosheafis trivial in this case since each f ∗ , ∗ is an inclusion map and each side of equation isrepresented by b ∪ a (resp. a ∪ b ). (cid:3) Remark . As two monoidal cosheaves of cohomology groups, H ∗ ( ˆ C ( A )) and H ∗ ( C ( A )) have no any difference by the discussion of subdivision. They both are isomorphic tothe monoidal cosheaf H ∗ ( A ) p = H ∗ ( M, M − M p ) . Notation.
For any manifold arrangement A with a quasi-intersection poset L , thedouble complex A ∗ ( L , ˆ C ( A )) is an algebra as discussed in last subsection, abbreviate itas A ∗ ( A ) , which is called the global OS-algebra associated with A .We are going to prove a similar result of Theorem 4.2 in the sense of algebras. Foreach p ∈ L , given a similar filtration of ˆ C ( A ) as F i ˆ C ( A ) p = ˆ C ∗ ( M, F ip M ) . This is a similar "hat" version of the filtration F i C ( A ) p appeared in Definition 3.2. Wecan calculate the cohomology algebra of T ot ( A ∗ ([ p, ∞ ) , ˆ C ( A ))) by this filtration. Theorem 5.4. H ∗ ( T ot ( A ∗ ([ p, ∞ ) , ˆ C ( A )))) and H ∗ ( N ( S p ) , N ( S p ) ) are isomorphic as alge-bras. In particular, for p = , H ∗ ( T ot ( A ∗ ( A ))) and H ∗ ( M ( A )) are isomorphic as algebras.Proof. There is no difference between the homologies of F i ˆ C ( A ) p and F i C ( A ) p by sub-division. So all processes in the proof of theorem 4.2 can still be carried out verywell for the filtration F i ˆ C ( A ) p . Then we first have that H ∗ ( T ot ( A ∗ ([ p, ∞ ) , ˆ C ( A )))) ∼ = H ∗ ( N ( S p ) , N ( S p ) ) as modules. This isomorphism is induced by a quotient map ofalgebras, so it is also an isomorphism of algebras. (cid:3) Product structure on our spectral sequence.
Now applying Corollary 5.2, wehave
Corollary 5.5.
There is a spectral sequence associated with filtration τ −∗ of the double complex A ∗ ([ p, ∞ ) , ˆ C ( A )) such that E − i,j = A ∗ ([ p, ∞ ) , H j ( A )) − i with d = ∂ , and E − i,j = H − i ( A ∗ ([ p, ∞ ) , H j ( A )) , ∂ ) , which converges to H ∗ ( N ( S p ) , N ( S p ) ) as algebras. In particular, for p = , there exists aspectral sequence with E − i,j = H − i ( A ∗ ( L , H j ( A )) , ∂ ) , hich converges to H ∗ ( M ( A )) as algebras, i.e. Gr τ ∗ H ∗ ( M ( A )) ∼ = H −∗ ( A ∗ ( L , H ∗ ( A )) , ∂ ) asalgebras.Remark . Note that H ∗ ( A ) is a monoidal cosheaf on L , and the product structure on E is given by the ’global’ OS-algebra A ∗ ([ p, ∞ ) , H ∗ ( C ( A ))) . Example . Although the double complex in Theorem 5.4 may be very complicated,the E page of above spectral sequence can be simple in some cases. Actually we canwrite down it explicitly, see Theorem 6.4 as an example. In particular, this spectralsequence degenerates at E page for the case of complex projective varieties, see fol-lowing subsection.5.4. Arrangements of subvariety.
In this subsection, we assume that M is a complexprojective smooth variety and each N i ∈ A is also a smooth subvariety. Theorem 5.6.
Using Q as coefficients, we have H ∗ ( M ( A )) ∼ = H −∗ ( A ∗ ( L , H ∗ ( A )) , ∂ ) as algebras, where the product of ring in the above right side is given by Definition 2.7.Proof. Deligne proved in [10] that the cohomology ring of any algebraic variety is iso-morphic to the associated graded ring with respect to the weight filtration. Actually,this isomorphism can be defined by Deligne splitting I p,q := F p ∩ W p + q ∩ ( F q ∩ W p + q + X j ≥ F q − j +1 ∩ W p + q − j ) see [26, Lemma-Definition 3.4] for details. Mapping each L p + q = k I p,q to Gr Wk by pro-jection (notice that the cup product is with respect to F, W ), we see that I p,q ∪ I s,t ⊆ I p + s,q + t , so this map preserves the product and is an natural isomorphism λ : H ∗ ( U ) → L k Gr Wk H ∗ ( U ) as algebras.Compare Corollaries 4.8 and 5.5, the weight filtration and τ −∗ have the same E page,so the spectral sequence in Corollary 5.5 also degenerates on E , which means that Gr τ ∗ H ∗ ( M ( A )) ∼ = E ∗ , ∗ as algebras. Meanwhile, Gr τ ∗ H ∗ ( M ( A )) = Gr W ∗ H ∗ ( M ( A )) byTheorem 4.9, so H −∗ ( A ∗ ( L , H ∗ ( A )) ∼ = E ∗ , ∗ ∼ = Gr W ∗ H ∗ ( M ( A )) ∼ = H ∗ ( M ( A )) as algebras(probably with different mixed Hodge structures). (cid:3)
6. A
PPLICATION TO CHROMATIC CONFIGURATION SPACES
Chromatic configuration space.
In the classical vertex coloring problem of a graph G , a usual way we use is to color the vertices of G with m colors from [ m ] = { , ..., m } ,so that adjacent vertices would receive different colors, a so-called proper m -coloring.It is well-known that the number χ G ( m ) of proper m -colorings is a polynomial of m ,called the chromatic polynomial of G . In this section, we consider the problem, but wewill use a manifold M as a color set to color the vertices of G , and the resulting col-orings will also form a manifold, called the chromatic configuration space of M on G . Definition 6.1.
Let G be a simple graph with vertex set [ n ] = { , ..., n } and M be asmooth manifold without boundary. Then the chromatic configuration space of M on G consists of all the proper colorings of G with all points of M as colors F ( M, G ) = { ( x , ..., x n ) ∈ M n | ( i, j ) ∈ E ( G ) ⇒ x i = x j } here E ( G ) denotes the set of all edges of G .This generalizes the concept of the classical configuration space F ( M, n ) = { ( x , ..., x n ) ∈ M n | i = j ⇒ x i = x j } . Actually, F ( M, G ) = F ( M, n ) when G is a complete graph. In the viewpoint of config-uration spaces, the definition of F ( M, G ) first appeared in the work of Eastwood andHuggett [13], where F ( M, G ) was called the generalized configuration space, and thecase in which M is a Riemann surface was studied in [2]. In addition, Dupont in [12]also studied the hypersurface arrangements. In this section, we will study the moregeneral case of F ( M, G ) by our approach about manifold arrangements.Following the assumption and the notion in Definition 6.1, by L G we denote theassociated geometric lattice of G , also see Example 2.1. It is well-known that L G is ageometric lattice (the matroid associated with this lattice is also known as the cyclematroid of G ), the partition induced by a single edge is an atom of L G . By and wealso denote the minimum and maximum element of L G , respectively. Definition 6.2.
For x = ( x , ..., x n ) ∈ M n , define G ( x ) to be the spanning subgraphgiven by those edges ( i, j ) in E ( G ) with x i = x j . By L G ( x ) we denote the element of L G induced by the subgraph G ( x ) . For p ∈ L G , define the L G -indexed ’diagonal’ of M n as ∆ p = { x ∈ M n | L G ( x ) ≥ p } . Theorem 6.1.
The set of diagonal A G = { ∆ a | a ∈ Atom ( L G ) } is a manifold arrangement in M n such that each ∆ a is closed in M n , and the intersection lattice of A G is just L G . Choosequasi-intersection poset to be intersection lattice. In particular, for p ∈ L G , ( M n ) p equals thediagonal ∆ p defined as above.Proof. In order to show that A G = { ∆ a | a ∈ Atom ( L G ) } is a manifold arrangement, itsuffices to check that A G is locally diffeomorphic to a subspace arrangement. In fact,choose x = ( x , x , ..., x n ) ∈ M n and its sufficient small neighborhood U = U × U ×· · · × U n with x i = x j ⇒ U i ∩ U j = ∅ and x i = x j ⇒ U i = U j , such that each U i isdiffeomorphic to some R -linear space W i . Clearly, the intersection ∆ a ∩ U is some diag-onal subspace (i.e., the subspace with some coordinates being equal) of Q U i , which isdiffeomorphic to some diagonal subspace of W = Q W i . Since each diagonal subspaceof W is linear, the diffeomorphism between U and W maps { ∆ a ∩ U } onto a subspacearrangement in W .It is obvious that each diagonal ∆ a is closed. Since ∆ p ∩ ∆ q = { x ∈ M n | L G ( x ) ≥ p and L G ( x ) ≥ q } = { x ∈ M n | L G ( x ) ≥ p ∨ q } = ∆ p ∨ q , we see that the intersection latticeof arrangements ∆ a , a ∈ Atom ( L G ) is just L G and ( M n ) p = ∆ p . (cid:3) Consider the monoidal cosheaf ˆ C ( A G ) and the model A ∗ ( A G ) = A ∗ ( L G , ˆ C ( A G )) asso-ciated with arrangement A G . Theorem 6.2. H ∗ ( F ( M, G )) is isomorphic to H ∗ ( T ot ( A ∗ ( A G ))) as algebras. In particular,there is a spectral sequence with E − i,j = A ∗ ( L G , H j ( A G )) − i with d = ∂, which converges to H ∗ ( F ( M, G )) as algebras.Proof. Applying theorem 5.4. (cid:3) sing a field as coefficients, the dimension of E − i,j term would be easy to calculateas follows: dim E − i,j = X p with r ( p )= i dim A ∗ ( L G ) p dim H j − r ( p ) m (∆ p ) by definition. This formula will be more elegant if we consider a polynomial with twovariables P M,G ( s, t ) = X i,j dim E − i,j s − i t j . Lemma 6.1.
Let M be a m -dimensional manifold without boundary, and G be a simple graphwith n vertexes. Then P M,G ( s, t ) = ( − n s − n t mn χ G ( − P ( M, t ) st − m ) where χ G is the chromatic polynomial of G , and P ( M, t ) = P i dim H i ( M ) t i is the Poincarépolynomial of M .Proof. It is well-known that dim( A ∗ ( L G ) p ) = ( − r ( p ) µ (0 , p ) where µ is the Möbius func-tion on L G and notice that dim H j ( M n , M n − ∆ p ) = dim H j − r ( p ) m (∆ p ) by Thom isomor-phism. Then we have P M,G ( s, t ) = X i,j dim E − i,j s − i t j = X p ∈ L G ,j dim A ∗ ( L G ) p dim H j − r ( p ) m (∆ p ) s − r ( p ) t j = X p ∈ L G ( − r ( p ) µ (0 , p ) s − r ( p ) P ( M, t ) n − r ( p ) t r ( p ) m . Moreover, the required equation follows from the following known result for the chro-matic polynomial χ G ( t ) = X p ∈ L G µ (0 , p ) t n − r ( p ) . (cid:3) Remark . The "Deletion–contraction" formula χ G = χ G − e − χ G/e of chromatic poly-nomial induces a "Deletion–contraction" formula of P M,G . It is easy to check that P M,G = P M,G − e + s − t m P M,G/e . Remark . We can also consider the 2-variable polynomial of E ∞ term for every man-ifold arrangements A , i.e., P ∞ , A ( s, t ) = P i,j dim( E − i,j ∞ ) s − i t j , which is hard to calculatein general. This polynomial is a natural invariant of A that contains much more infor-mation than the Poincaré polynomial of M ( A ) .6.2. More explicit result about H ∗ ( F ( M, G )) . For some special case, the spectral se-quence in Section 4 is simpler.The first case is that M be a complex projective smooth variety. Combine Corollary4.8, Theorem 5.6 and last subsection, we have orollary 6.3. Assume M is a complex projective smooth variety and we use Q as coefficients.Then the mix Hodge structure of H ∗ ( F ( M, G )) satisfies Gr Wk + i H k ( F ( M, G )) ∼ = H − i ( A ∗ ( L G , H k + i ( A G )) , ∂ ) and there is the following isomorphism H ∗ ( F ( M, G )) ∼ = H −∗ ( A ∗ ( L G , H ∗ ( A G )) , ∂ ) as algebras. For a general smooth manifold M , the spectral sequence in Theorem 6.2 may havenon-trivial higher differential. This spectral sequence will be simple if the diagonalcohomology class of M is zero. Recall (see Milnor’s book [22]) that the diagonal co-homology class of M is the image of the Thom class under the map H ∗ ( M , M − ∆( M )) → H ∗ ( M ) , where ∆( M ) is the diagonal of M . From now on, we always use Z -coefficients, and assume that the diagonal cohomology class of M vanishes. Thisis the case if M = M ′ × R where M ′ is another manifold.Under the above assumption, the monoidal cosheaf H ∗ ( A G ) becomes simpler, andit is completely only determined by G and H ∗ ( M ) . Firstly, observe that H ∗ ( A G ) p = H ∗ ( M n , M n − ∆ p ) = H ∗ (∆ p )[ − mr ( p )] (let dim( M ) = m ) by Thom isomorphism, soevery element of H i ( A G ) p can be represented by an element x p ∈ H i − mr ( p ) (∆ p ) . Theorem 6.4.
Assume that the diagonal cohomology class of M is zero and we use Z ascoefficients and G is a simple graph with vertex set [ n ] . Let E ∗ ( M, G ) be the exterior al-gebra over ring H ∗ ( M ) ⊗ n , generated by elements e ij where ( ij ) is edge of G . The E page A ∗ ( L G , H ∗ ( A G )) has zero differential ( ∂ = 0 ) and is isomorphic to E ∗ ( M, G ) /I as algebraswhere I is the ideal generated by elements (1 i − ⊗ x ⊗ n − i − − j − ⊗ x ⊗ n − j − ) e ij (1) X s e i i . . . \ e i s i s +1 . . . e i k i (2) for all edges ( i, j ) and all cycles ( i i , i i , . . . , i k i ) . The proof will be completed in appendix.
Remark . These elements in (1) of Theorem 6.4 also appeared in Cohen and Taylor’smodel when G is complete graph, see [8, page 111]. Elements in (2) of Theorem 6.4appeared in the definition of OS-algebra. Of course, there also exists a similar resultfor the structure of the monoidal cosheaf H ∗ ( A G ) with Z coefficients if we considerthe orientation of the Thom class carefully. Here we use Z as coefficients only for asimplicity since it is enough to illustrate the application of our approach in this section. Corollary 6.5.
Assume that the diagonal cohomology class of M is zero and using Z coeffi-cients. Let G be a simple graph with vertex set [ n ] . Then there exists a filtration of H ∗ ( F ( M, G )) such that Gr ( H ∗ ( F ( M, G ))) is isomorphic to A ∗ ( L G , H ∗ ( A G )) as algebras.Proof. The differential ∂ of the global OS-algebra A ∗ ( L G , H ∗ ( A G )) is zero by Theorem6.4, so E − i,j = E − i,j = A ∗ ( L G , H j ( A G )) − i . ow we only need to show that this algebra is generated by elements of E − ,j and E ,j , by making use of the degeneration condition in Theorem 5.3. This is obvious byTheorem 6.4. (cid:3) Example . Assume that the diagonal cohomology class of M is zero and using Z co-efficients. Let G be the cycle graph C n of length n , then Gr ( H ∗ ( F ( M, G ))) is isomorphicto the exterior algebra over ring H ∗ ( M ) ⊗ n , generated by e , e , . . . , e n , with relations (1 i − ⊗ x ⊗ n − i ) e i = (1 i ⊗ x ⊗ n − i − ) e i for i < n (1 n − ⊗ x ) e n = ( x ⊗ n − ) e n X s e . . . b e s . . . e n = 0 . Consider the Poincaré polynomial, we have
Corollary 6.6.
With the same assumption as in Corollary 6.5. Then P ( F ( M, G )) = ( − n t n ( m − χ G ( − P ( M ) t − m ) where P ( − ) denote the Poincaré polynomial of Z -cohomology with a variable t , dim M = m ,and χ G ( t ) is the chromatic polynomial of G .Proof. In the proof of Corollary 6.5, the spectral sequence collapses on E term, so thePoincaré polynomial of F ( M, G ) equals P M,G ( t, t ) , where P M,G is the two-variable poly-nomial of E term which is defined in Lemma 6.1. The required equation is a directresult of Lemma 6.1. (cid:3) Remark . With the same assumption of the above Corollary, P ( F ( M, G )) satisfies asimilar "Deletion–contraction" formula P ( F ( M, G )) = P ( F ( M, G − e )) + t m − P ( F ( M, G/e )) by the "Deletion–contraction" formula of chromatic polynomial.In a more special condition, we can overcome the gap between Gr ( H ∗ ( F ( M, G ))) and H ∗ ( F ( M, G )) by a simple method of "counting degree". Theorem 6.7.
Using Z coefficients, assume dim M = m and the diagonal cohomology classof M is zero and H i ( M ) = 0 for all i ≥ ( m − / . Then H ∗ ( F ( M, G )) is isomorphic to A ∗ ( L G , H ∗ ( A G )) as algebras.Proof. Firstly, observe that if G is a disjoint union of some connected graphs G i , then F ( M, G ) = Q F ( M, G i ) and A ∗ ( A G ) = L A ∗ ( A G i ) , so it suffices to prove the case that G is connected.We perform our work by induction on the number of edges. If G has no edge, thenthe result is obvious. Now we assume that the result is true for all connected G that | E ( G ) | < s . Consider the case in which G is connected with | E ( G ) | = s .Corollary 6.5 says that there is an isomorphism η : A ∗ ( L G , H ∗ ( A G )) → H ∗ ( F ( M, G )) as modules, so we need to show this is also an isomorphism as algebras.Given two elements x p ∈ A ∗ ( L G , H ∗ ( A G )) p and x q ∈ A ∗ ( L G , H ∗ ( A G )) q , where p, q ∈ L G with r ( p ) = i and r ( q ) = j . If p ∨ q < , since we have assumed G is connected, his means that p ∨ q must have two or more components, so the subgraph G | p ∨ q must have less edges than G . Consider the arrangement A G | p ∨ q associated with thespace F ( M, G | p ∨ q ) , we know that this arrangement is a sub-arrangement of A G asdiscussed in Section 4.4. Then Theorem 4.4 and Corollary 6.5 give us a commutativediagram H ∗ ( F ( M, G | p ∨ q )) i ∗ −−−→ H ∗ ( F ( M, G )) θ x η x A ∗ ([0 , p ∨ q ] , H ∗ ( A G | p ∨ q )) j ∗ −−−→ A ∗ ( L G , H ∗ ( A G )) where η, θ are two module-isomorphisms (note that θ is also an algebra-isomorphismby induction hypothesis), j ∗ is a natural inclusion of algebra, and i ∗ is induced bythe inclusion F ( M, G ) ֒ → F ( M, G | p ∨ q ) . It is easy to see that those two elements x p , x q and their product are located in sub-algebra A ∗ ([0 , p ∨ q ] , H ∗ ( A G | p ∨ q )) since θ, i ∗ preserveproduct, so η is also compatible with the product of those two elements by the diagram.Now assume that p ∨ q = . We know that η induces an algebra-isomorphism: A ∗ ( L G , H ∗ ( A G )) → Gr ( H ∗ ( F ( M, G ))) . Then we can write the image of the product as η ( x p x q ) = η ( x p ) ∪ η ( x q ) + X i η ( x p i ) for some x p i ∈ A ∗ ( L G , H ∗ ( A G )) p i , p i < . We see that r ( p ) + r ( q ) ≥ r ( ) = n − since G is connected, η ( x p x q ) should have degree at least ( n − m − in H ∗ ( F ( M, G )) .Let us observe the degree of elements x p i with p i < . Write k = r ( p i ) and x p i = a p i ⊗ c p i where a p i ∈ A ∗ ( L G ) p i and c p i ∈ H ∗ ( A G ) p i = H ∗ ( M n , M n − ∆ p i ) . Then a p i hasdegree − k , and c p i equals the product of a Thom class u p i with deg u p i = mk and somenonzero elements of degree less than ( n − k )( m − / by the assumption of this theorem.Thus, deg( a p i ⊗ c p i ) < ( m − k + ( n − k )( m − / m − n + k ) / , which implies that k ≤ n − , so deg x p i < ( m − n − , a contradiction of deg η ( x p x q ) ≥ ( n − m − .Then this forces these items x p i to be zero. Therefore, η ( x p x q ) = η ( x p ) ∪ η ( x q ) . (cid:3) Remark . If M = R m , then Theorem 6.7 agrees with the classical result of H ∗ ( R m , n ) given by F. Cohen. In addition, Theorem 6.7 also shows a "standard" process how toapply our main result: (i) Determine the structure of monoidal cosheaf H ∗ ( A ) . (ii)Calculate the E -term H − j ( A ∗ ( L, H i ( A )) , ∂ ) and check the condition of degeneration(iii) Observe the gap between H ∗ ( M ( A )) and Gr ( H ∗ ( M ( A ))) . Using this process, wecan also easily reprove the classical result of Orlik-Solomon for complex hyperplanearrangements. 7. A PPENDIX
In this section, we review some known result of spectral sequence of filtrated differ-ential algebra, see [21]. For definition of subgraph G | p ∨ q , see Example 2.1 .1. Spectral sequence of filtrated differential algebra.
Let A = L p A p be a gradedmodule, and ... ⊂ F p A p + q ⊂ F p − A p + q ⊂ ... be an decreasing filtration with the differ-ential d such that d ( F p A p + q ) ⊂ F p A p + q +1 . Define Z p,qr = F p A p + q ∩ d − ( F p + r A p + q ) B p,qr = F p A p + q ∩ d ( F p − r A p + q − ) Z p,q ∞ = F p A p + q ∩ ker( d ) B p,q ∞ = F p A p + q ∩ im ( d ) E p,qr = Z p,qr / ( Z p +1 ,q − r − + B p,qr − ) Proposition 7.1.
The differential d which maps Z p,qr to Z p + r,q − r +1 r induces the differential d r : E p,qr → E p + r,q − r +1 r of the associated spectral sequence, such that H ∗ ( E ∗ , ∗ r , d r ) = E ∗ , ∗ r +1 E p,q = H p + q ( F p A/F p +1 A ) E p,q ∞ = F p H p + q ( A, d ) /F p +1 H p + q ( A, d ) where F p H ∗ ( A, d ) = Im ( H ∗ ( F p A, d ) → H ∗ ( A, d )) . Furthermore, if A is also an algebra, then we have Proposition 7.2.
Suppose that ( A, d, F ∗ A ) is a decreasing filtered differential graded algebrawith product A ⊗ A → A satisfying F p A · F q A ⊂ F p + q A Then there is an induced product on E ∗ , ∗ r satisfying E p,qr · E s,tr ⊂ E p + s,q + tr and d r ( x · y ) = d r ( x ) · y + ( − p + q x · d r ( y ) where x ∈ E p,qr and y ∈ E s,tr . If the filtration F ∗ A is bounded, then the spectral sequence ( E ∗ , ∗ r , d r ) converges to H ( A, d ) as algebras, i.e., L p,q E p,q ∞ is isomorphic to the associated gradedalgebra Gr ( H ( A, d )) = L p,q F p H p + q ( A, d ) /F p +1 H p + q ( A, d ) and this isomorphism obeys thebigrading ( p, q ) .Remark . Gr ( H ∗ ( A, d )) determines H ∗ ( A, d ) up to the extension problem. If we usea field as coefficients or E ∞ is free with Z coefficients, the extension problem is triv-ial, i.e., H ∗ ( A, d ) ∼ = Gr ( H ∗ ( A, d )) as modules. This isomorphism will also obey theproduct of A in some special case. For example, if there exists an integer k suchthat x ∈ F l H ∗ ( A, d ) ⇔ deg x ≤ kl for any homogeneous element x , then H ∗ ( A, d ) ∼ = Gr ( H ∗ ( A, d )) as algebras by dimensional reason. .2. Proof of Theorem 6.4.
In this subsection, we give the proof of Theorem 6.4. Firstwe prove a lemma.
Lemma 7.1.
Assume that V , V are two linear subspaces of R n such that V ∩ V = 0 , dim( V ) = i and dim( V ) = n − i . By µ , µ , µ we denote the fundamental classes of H n − i ( R n , R n − V ) , H i ( R n , R n − V ) , H n ( R n , R n − , respectively. Then µ ∪ µ = µ .Proof. Write R n = R × R × · · · × R n be the product of n -copies of R . Without the loss ofgenerality, assume that V = R × · · · × R i × and V = 0 × R i +1 × · · · × R n . Let e i be thefundamental class of H ( R i , R i − . Then we can write µ = 1 × · · · × × e i +1 × · · · × e n , µ = e × · · · × e i × × · · · × , so µ ∪ µ = e × e × · · · × e n , which is just the fundamentalclass of H n ( R n , R n − . (cid:3) Proof of Theorem 6.4.
Let u p be the Thom class of H ∗ ( M, M − ∆ p ) for any p ∈ L G . Assumethat p, q are independent. Then r ( p ) + r ( q ) = r ( p ∨ q ) , which means that dim ∆ p +dim ∆ q = dim M n + dim ∆ p ∨ q . Let F x be the fiber at x of the normal bundle N (∆ p ∨ q ) (here we do not distinguish the normal bundle and the tubular neighborhood). We canchoose suitable local charts and tubular neighborhoods such that F x ∩ ∆ p and F x ∩ ∆ p arelinear subspaces of F x , denoted by V , V , respectively. Then we know that V ∩ V = 0 and V ⊕ V = F x since p, q are independent. Moreover, N (∆ p ) ∩ F x = F x − V and N (∆ q ) ∩ F x = F x − V , so u p | ( F x ,F x − V ) is the fundamental class of H r ( p ) m ( F x , F x − V ) and u q | ( F x ,F x − V ) is the fundamental class of H r ( q ) m ( F x , F x − V ) . By Lemma 7.1, ( u p ∪ u q ) | ( F x ,F x − is the fundamental class of H r ( p ∨ q ) m ( F x , F x − for every x ∈ ∆ p ∨ q . Thus, u p ∪ u q must be the Thom class u p ∨ q of H ∗ ( M n , M n − ∆ p ∨ q ) by uniqueness of Thom class.Now we want to check that u p ∪ u q = 0 if p, q are dependent. Firstly, assume that p isan atom. Then we know that ( M n , M n − ∆ p ) = ( M , M − ∆( M )) × M n − , so u p is justthe product of the Thom class u of ( M , M − ∆( M )) with the unit of H ∗ ( M n − ) , and u = u · w m ( T ( M )) by Milnor’s book [22], where w m ( T ( M )) is the top Stiefel-Whitneyclass of the tangent bundle T ( M ) . Furthermore, u = 0 since w m ( T ( M )) is the imageof the diagonal cohomology class, so u p is also zero for the atom p . Thus, for every p , u p = 0 since p is always a join of some independent atoms, and so u p ∪ u q = 0 for p ≤ q since u q can be expressed as u p ∪ u s ∪ u s · · · for some independent atoms s i . For anydependent p, q , let q = s ∨ s · · · for some independent atoms s i , then there exists some k such that p ∨ s ∨ · · · ∨ s k ≥ s k +1 , so u p ∪ u q = u p ∨ s ∨···∨ s k ∪ u s k +1 ∪ · · · , which must beequal to zero by the above discussion.Now, Let ψ p,q : H ∗ (∆ p ) → H ∗ (∆ q ) be the map induced by inclusion for p ≤ q . Noticethat every H ∗ ( C ( A ) p ) = H ∗ ( M n , M n − ∆ p ) is an H ∗ ( M n ) -algebra such that u p ∪ x = u p · ψ ,p ( x ) for x ∈ H ∗ ( M n ) , it is not difficult to check that ( u p · x p ) ∪ ( u q · x q ) = u p ∨ q · ( ψ p,p ∨ q ( x p ) ∪ ψ q,p ∨ q ( x q )) . Then we can define E ∗ ( M, G ) /I → A ∗ ( L G , H ∗ ( A G )) by map e ij · x to e ij ⊗ ψ ,p ( x ) (recall that OS-algebra A ∗ ( L G ) is the exterior algebra generated by e ij with relations P s e i i . . . \ e i s i s +1 . . . e i k i = 0 for all cycle), it’s a well defined injectivemap of algebra by above discussion, it’s an isomorphism by checking the dimensionof each side.The complex ( A ∗ ( L G , H ∗ ( A G )) , ∂ ) have zero differential by checking that f ∗ p,q = 0 forall p ≥ q . We only need to check the case that p covers q by the functorial property of f ∗ . Assume that p = q ∨ s for an atom s . Then we have that f ∗ p,q ( u p ) = f ∗ p,q ( u q ∪ u s ) = q ∪ f ∗ s, ( u s ) , which is zero since f ∗ s, ( u s ) is the product of the diagonal cohomology classwith some unit. (cid:3) R EFERENCES [1] Vladimir Baranovsky and Radmila Sazdanovic. Graph homology and graph configuration spaces.
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