The homology of real subspace arrangements

Transcription

1 Journal o Topology Advance Access published October 14, 2010 Journal o Topology (2010) Page 1 o 33 C 2010 London Mathematical Society doi: /jtopol/jtq027 The homology o real subspace arrangements Eric M. Rains Abstract Associated to any subspace arrangement is a De Concini Procesi model, a certain smooth compactiication o its complement, which in the case o the braid arrangement produces the Deligne Mumord compactiication o the moduli space o genus 0 curves with marked points. In the present work, we calculate the integral homology o real De Concini Procesi models, extending earlier work o Etingo, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology o the real locus o the moduli space. To be precise, we show that the integral homology o a real De Concini Procesi model is isomorphic modulo its 2-torsion to a sum o cohomology groups o subposets o the intersection lattice o the arrangement. As part o the proo, we construct a large amily o natural maps between De Concini Procesi models (generalizing the operad structure o moduli space), and determine the induced action on poset cohomology. In particular, this determines the ring structure o the cohomology o De Concini Procesi models (modulo 2-torsion). Contents 1. Introduction De Concini Procesi models Poset homology and operad maps Blow-ups and homology A cell structure or Y adic homology Operadicity Further directions Reerences Introduction The main result o [7] was the determination o the cohomology ring structure o the real locus M 0,n (R) o the moduli space o stable genus 0 curves with n marked points. It was shown there that the cohomology o M 0,n (R) could be expressed in terms o the homology o intervals o a certain poset, namely the poset o partitions o {1, 2,...,n 1} in which each part is o odd size. This was, o course, quite reminiscent o the corresponding result (see, or example, [15]) or the cohomology o the coniguration space o n distinct points in P 1 (C), which is expressed in terms o the homology o the poset o all partitions o {1, 2,...,n 1}. As the latter result generalizes to arbitrary subspace arrangements [5, 10, 16, 19], it was natural to look or a corresponding generalization o [7]. In our context, the appearance o subspace arrangements comes rom an interpretation o M 0,n (real or complex) as a special case o a construction o De Concini and Procesi [4] Received 10 December Mathematics Subject Classiication 14N20 (primary), 14F25 (secondary). The author was supported in part by NSF rant No. DMS

2 Page 2 o 33 ERIC M. RAINS o a wonderul compactiication associated to an arbitrary subspace arrangement (in this case, the braid arrangement o type A n 2, as remarked in [4, p. 483]). When the spaces in the arrangement are real, the De Concini Procesi model is a smooth real projective variety, and thus gives rise to a smooth maniold. In the present note, we determine the (integral) homology groups o these maniolds. More precisely, the homology o these maniolds consists o a large amount o 2-torsion (analogous to the homology o the spaces RP n ), which is implicitly determined (via the mod 2 homology) in Section 6 below, but i we quotient out by this 2-torsion, the result can then be expressed (canonically) as a direct sum o cohomology groups o certain simplicial complexes described in Section 3. These simplicial complexes can in turn be subdivided to obtain the order complexes o certain subposets o the lattice o subspaces in the arrangement, the primary constraint (2-divisibility) on the subspaces being that they decompose as transverse intersections o even-codimensional subspaces. The precise statement o the main theorem is given as Theorem 3.7 below. As might be guessed rom the act that we switched to considering homology rather than cohomology as in [7], our techniques are somewhat dierent; in particular, we do not have the luxury o an explicit presentation or the cohomology algebra. (Similarly, while [7] made some use o the act that M 0,n (R) isak(π, 1), this ails to hold or arbitrary real De Concini Procesi models (or example, RP n ).) The approach o [7] o obtaining inormation about 2-adic (co)homology using the mod 2 (co)homology does play a signiicant role, however (see Section 6 below). Moreover, one o the main tools o [7] was the act that the moduli spaces orm an operad, giving rise to a large collection o natural maps that were used there to distinguish cohomology classes; one o our main tools (even more so than in [7]) is the observation that these operad maps can be deined or general De Concini Procesi models. In particular, this includes diagonal morphisms; thus, although we work exclusively with homology, we still eectively (i somewhat implicitly) determine the ring structure on cohomology. Our primary remaining tool, which was not used in [7], is a certain long exact sequence associated to blow-ups o real varieties. Since every De Concini Procesi model is a blow-up o a simpler De Concini Procesi model, this allows us to reduce a signiicant portion o the main theorem (localized away rom the prime 2) to the (trivial) case o products o projective spaces. In particular, this allows us to resolve a conjecture o [7] by showing that the (co)homology o M 0,n (R) has no odd torsion. This holds or arbitrary Coxeter arrangements (see [11]) but again not or general De Concini Procesi models. (In act, even i one restricts one s attention to hyperplane arrangements, one can arrange or the homology o an arbitrary inite simplicial complex to appear as a graded piece o the homology o some De Concini Procesi model; see the remark ollowing the statement o the main theorem.) The plan o the paper is as ollows. In Section 2, we recall De Concini and Procesi s construction, and introduce the associated operad maps (together with the main composition law that they satisy); we also show how these maps give rise to a natural grading o the homology groups by the lattice o subspaces in the arrangement. In Section 3, we introduce the corresponding combinatorial data, in particular allowing us to state the main theorem (Theorem 3.7). The inal set-up section, Section 4, establishes the long exact sequence associated to a real blow-up. Corollary 4.5 o this section discusses the special case o the blow-up long exact sequence associated to De Concini Procesi models, and in particular shows how this sequence interacts with the natural grading by subspaces. The proo o the main theorem spans Sections 5 7. In Section 5, we construct a amily o natural cell structures on the De Concini Procesi model, which allow us to construct the isomorphism o the main theorem via a chain map. It is then airly straightorward to show that the chain map respects the blow-up long exact sequence, and thus by induction that it induces an isomorphism on homology with coeicients in Z[1/2]. This argument ails to control the 2 k -torsion, however, which is the subject o Section 6; there, we show that the chain map gives an isomorphism (modulo 2-torsion) over Z/4Z, which then implies the isomorphism property

3 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 3 o 33 over the 2-adic integers. In the process, we give an explicit basis o the mod 2 homology consistent with the natural grading (its primary distinction rom the cohomology basis o [18]), which together with the main theorem determines the 2-torsion o the homology groups. The proo o the main theorem is inished in Section 7, where we show that the action o the operad maps on homology agrees with the action described combinatorially in Section 3. This in particular determines the ring structure on cohomology. Finally, in Section 8, we discuss some possibilities or urther generalizations; see especially Conjecture 1, which considers the case o R-rational (that is, closed under conjugation) arrangements. Also o some interest is Theorem 8.1, which gives an interpretation o the cohomology o the ull poset o 2-divisible subspaces (when that poset has no maximal element) in terms o the homology, twisted by a certain shea, o the De Concini Procesi model. Notational convention. Unless otherwise stated, all modules are supermodules, with corresponding conventions or tensor products. 2. De Concini Procesi models We will need to recall (and extend) some notions rom [4]. Note that our deinition is slightly more general than theirs, but only in that their construction is not closed under taking products (see Lemma 2.3 and surrounding remarks). The considerations o unctoriality and operadicity are new, and in particular the construction o a natural grading by subspaces on the (co)homology groups (see the remark ollowing Corollary 2.8). A subspace arrangement in a inite-dimensional vector space V is a inite collection o subspaces o V. Note that this induces by duality a corresponding collection o subspaces o V, but it will be convenient to use the dual notation. iven a subspace arrangement, we deine C to be the lattice generated by ; that is, the set o all sums o subsets o. (By convention, this includes the empty sum; that is, 0 C.) Note that C is indeed a lattice with respect to inclusion, but the meet operation in this lattice is not simply the intersection o subspaces. iven U C,a-decomposition o U is a collection o nonempty subspaces U i C such that U = U i 1 i k and or every such that U, U i or exactly one i. Note that every element o is -indecomposable, and the notion o decomposition depends only on the collection o -indecomposable subspaces. In particular, i we let denote that collection, then C = C and an element is -indecomposable i and only i it is -indecomposable. A building set is a subspace arrangement such that = ; we have thus seen that every arrangement induces a building set. I is a building set in V,and is a building set in V, then a morphism rom to is a linear transormation : V V such that () or all. We thus obtain or every ield K a category Build(K) o building sets o vector spaces over K. There are two important special cases o morphisms. First, i are both building sets in V, then the identity map on V induces a morphism ι :. Second, we have morphisms such that = { () : }, (2.1) = () :={C : C C (C) }. (2.2)

4 Page 4 o 33 ERIC M. RAINS Since, or a general morphism, { () : }, (2.3) (), (2.4) { () : } = { () : ()}, (2.5) it ollows that any morphism can be decomposed as ι ι with satisying (2.1) and (2.2). Both types o morphisms can be urther decomposed; indeed, it suices to take morphisms with dim(ker ) =1and = (), and morphisms ι such that = 1. The only nontrivial thing to prove is the ollowing. Lemma 2.1. iven any two building sets on the same space, there exists an element \ such that \{} is a building set. Proo. 2.5(1)]. Let be a minimal element o not in, and ollow the proo o [4, Proposition De Concini and Procesi associate a smooth projective variety to a building set as ollows. Let A be the aine variety A = V \. Then or each we have a map A P, where P = P(V/ ), and we thus have a map A P. Then Y is the closure o the image o this map. A more local description can be given as ollows: let ρ : Y P be the natural map. Then Y is the locus o points x P such that or every pair H,(ρ (x),ρ H (x)) is in the (closed) graph o the projection P P H. I dim() = 1 (that is, i is a hyperplane), then P is a single point, so we ind that removing all hyperplanes rom a building set has no eect on the corresponding variety. Conversely, adjoining a hyperplane to a building set has no eect, so long as the result is still a building set. A useul criterion or this is the ollowing. Lemma 2.2. Suppose v V is a nonzero vector such that or any C C containing v there exists such that v C. Then { v } is a building set. Proo. Let := { v }, and consider C C which is -indecomposable. We need to show that C. I v/ C, then C C.A-decomposition o C would then also decompose C as an element o C ; it ollows that C. I v C, but still C C, then by assumption, there exists with v C. In particular, is contained in some component o the -decomposition o C, and thus v is contained in that component. It ollows that the -decomposition o C is the same as its -decomposition, and thus C. Finally, we have the case v C, C/ C. In particular, we can write C = D + v or some D C not containing v. We claim that the -decomposition o C is obtained by adjoining v to the -decomposition o D. Suppose otherwise that there exists such that C but D. But then C = + D C, a contradiction. It ollows that C = v.

5 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 5 o 33 In particular, i v is such that i v C C, then C, then { v } is a building set. Over an ininite ield, there exists such a vector in any, and we may thus repeatedly adjoin hyperplanes to without aecting the variety, until each is the span o dim() + 1 hyperplanes in the collection, and those hyperplanes are in general position. Letting H denote this collection o hyperplanes, we ind H = H, and thus Y = Y H.In other words, every De Concini Procesi model is isomorphic to the De Concini Procesi model o a hyperplane arrangement. Thus by considering general building sets, we do not in act add any more generality than i we merely considered the hyperplane case; they do, however, orm a useul tool. We note that the construction o Y attributed to De Concini and Procesi above is not quite the construction they give. To be precise, they also include the map A P(V ), or equivalently in our notation assume that V, in which case A embeds in Y ; it will be notationally convenient to allow the slightly more general case. Our case reduces easily to the case V, as ollows. iven W V, deine the restriction W by W = { W }, a building set in W = V/W. Since (V/W )/ = V/ or W, we immediately obtain the ollowing, using the local description o Y.Theroot o a building set is the maximal element root() o the lattice C. Lemma 2.3. For any W V, there exists a natural map Y Y W.I k is the decomposition o root(), then Y is an isomorphism. In particular, dim Y = 1 i k Y i 1 i k (dim( i ) 1) = dim(root()) k. Remark. In particular, each o the actors contains the appropriate ambient space. Also, or topological purposes we note that Y in our notation is homotopic to the variety Y (the closure with a actor V added to the map) discussed in [4]. Proposition 2.4. The construction Y deines a unctor rom Build(K) to the category o smooth projective varieties over K. Proo. Let be a morphism rom to. To speciy the associated map Y : Y Y, it suices to speciy ρ Y or each. We simply take ρ Y = ρ () P(), where P() :P(V/( ()) ) P(V / ) is the natural morphism, which is well deined (and injective) since ( ()) = 1 ( ). The local conditions are then straightorward to veriy, as is the act that Y respects composition o morphisms. Remark. The deining maps ρ are associated in this way to the morphisms ι : {}. Similarly, the diagonal map Y Y Y is associated to the diagonal map Δ : V V V ; more precisely, the diagonal map is the composition Y Y Δ Y Y Y.

6 Page 6 o 33 ERIC M. RAINS In act, Y satisies a more general version o unctoriality, in that it has a sort o operadic structure. Let and be building sets in V and V, respectively. A weak morphism : is a linear transormation : V V such that () {0} or all. Theorem 2.5. (the operad map) iven any weak morphism :, there exists a natural morphism φ : Y ker( ) Y Y. These maps satisy the composition law, which states that given any two weak morphisms :,g:, the diagram Y ker g Y ker( ) Y φ g im() 1 1 φ Y ker g Y φ g Y ker g Y commutes, where g im() is the induced weak morphism φ g Y g im() : ker( ) ker( g ). Proo. We need to speciy ρ φ or each.i ker( ), then we simply set ρ φ = ρ, projecting rom Y ker( ). Otherwise, we compose ρ () (projecting rom Y ) with the induced map P() :P(V/( ()) ) P(V / ) as beore. Remark 1. Note that i is a linear transormation such that ker( ) C, then { () : () 0} is a building set in V, called the induced building set, and denoted by ( ). We will denote the corresponding weak morphism rom ( )to by τ(), and say that such a weak morphism is purely operadic. Note that the corresponding operad map is injective. For C, we denote by φ C the operad map associated to τ(i C ) where i C : C V is the inclusion map. Remark 2. In general, any weak morphism can be actored as a product o a morphism and a purely operadic weak morphism. Indeed, given a general weak morphism :,let C =root( ker ). Then we may actor the linear transormation as i C g. But then as a weak morphism, = τ(i C ) g. The composition law in this case simpliies, and thus we ind as one might expect. φ = φ C (1 Y g ) Remark 3. It will be helpul to note the orms the composition law takes when one o the maps is a morphism. I is a morphism, then φ g = φ g (1 φ ) while i g is a morphism, then φ g φ = φ g (φ g im() 1). O course, i both are morphisms, then the composition law is simply φ g = φ g φ.

7 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 7 o 33 Example. As an example, consider the braid arrangement A n 1 = { e i e j :1 i<j n}. The indecomposable subspaces in this arrangement are those o the orm e i e j : i, j S or some subset S {1, 2,...,n} with S 2, and general subspaces are associated to partitions o {1, 2,...,n}. The associated De Concini Procesi model is thus constructed as a subvariety o a product o projective spaces o the orm P S 2 ; S {1,2,...,n} to be precise, it is the intersection o the graphs o the maps P(S) P(T )ort S given by omitting those coordinates not in T. There is a natural morphism rom the moduli space M 0,n+1 (with one marked point singled out as special) to this De Concini Procesi model, as ollows. For each subset S {1, 2,...,n}, we map the moduli space to P(S) by irst orgetting all points with labels outside S, collapsing components as necessary to preserve stability, then urther collapsing all components not containing the special point. We thus obtain a copy o P 1 together with a unction rom S to P 1 that avoids the special point. Equivalently, taking the special point to, we obtain a point in the aine space A S, with at least two distinct coordinates. The urther isomorphisms o P 1 quotient this aine space by the diagonal subspace, then by scalar multiplication, and thus we obtain a point in P S 2 associated to each stable curve. The local conditions are clearly satisied, so this collection o morphisms to projective space induce a morphism to Y An 1.By considering its action on the open set M 0,n+1, we ind that this morphism is birational; it is also straightorward to veriy that it is bijective, and thus an isomorphism. The moduli spaces (or, rather, their real or complex loci) orm a topological operad in the usual sense, which is a special case o the above generalized operad structure. To be precise, or any composition α 1,...,α k o n with nonzero parts, there is an associated weak morphism α : A k 1 A n 1 which maps e 1 to the sum o the irst α 1 basis vectors, e 2 to the sum o the next α 2 basis vectors, etc. The kernel o α is then a subspace in the arrangement, to wit the subspace associated to the set partition {1, 2,...,α 1 }, {α 1 +1,...,α 1 + α 2 },..., and (A n 1 )=A k 1. In particular, α is purely operadic, and the associated operad map is the usual operad map M 0,α M 0,αk +1 M 0,k+1 M 0,n+1 obtained by gluing the special points o the irst k curves to corresponding nonspecial points o the last curve. (Similarly, the orget a point map is associated to a morphism A n A n 1.) Moreover, the usual operad axiom becomes just a special case o the general composition law. Note, however, that the cyclic operad structure o the moduli space does not seem to be compatible with its interpretation as a De Concini Procesi model, as it does not respect the role o the special point. Proposition 2.6. Let : be a weak morphism with ( ). Then φ is injective. Proo. Simply observe that i ( )=, then the map ρ on the codomain is the composition o the map ρ on the domain with an injection.

8 Page 8 o 33 ERIC M. RAINS Proposition 2.7. Let : be a surjective (weak) morphism. Then φ is surjective, and is birational i and only i root() = (root( )) and both roots have the same number o components. Proo. The image o the restriction : A A is dense i : V V is surjective, and thus the corresponding map o projective varieties must be surjective; birationality then ollows by comparing dimensions. Remark. In particular, this applies to any morphism o the orm ι :. Corollary 2.8. iven a building set and a space C C, the natural surjection π C := φ ι : Y Y C has a natural homotopy class π [ 1] C o splittings. Moreover, these maps satisy the identities (up to homotopy) π C D π C = π C D, (2.6) π [ 1] C π[ 1] C D π C D, (2.7) π [ 1] C D π[ 1] C D. (2.8) π C π [ 1] D Proo. The splitting maps arise rom by choosing a point in Y /C. Now, the composition law implies that φ C : Y C Y /C Y π C φ D = φ C D (π C D (φ π (C+D)/D )) : Y D Y /D Y C, where is the natural morphism :(/D) (C+D)/D ( C )/(C D). I C = D, then the right-hand side is just the projection Y C Y /C Y C and we thus obtain the desired splitting. More generally, i we choose a point in Y /D,we obtain the second identity. The other two identities ollow similarly rom the composition law. Remark. In particular, the associated retractions satisy (π [ 1] C π C ) (π [ 1] D π D ) π[ 1] C D π C D, and thus commute up to homotopy. It ollows that any associated (co)homology group is graded by C ; that is, splits as a direct sum indexed by C.Tobeprecise,oranyC C,let H (Y )[C] denote the subspace consisting o homology classes ixed by π [ 1] C π C and annihilated by π [ 1] D π D or D C. (Equivalently, these are the classes in the image o π [ 1] C which are annihilated by π D or D C.) Since these are retractions, so idempotents on (co)homology,

9 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 9 o 33 and commute, we ind that and more generally H (Y )= C C H (Y )[C], (π [ 1] C π C )H (Y )= D C H (Y )[D]. Proposition 2.9. I V /, then Y V is a projective space bundle over Y. Proo. Let 1,..., m be the maximal elements o ; note that by adjoining suitable hyperplanes, we can orce i i = V without changing Y. We thus need to show that the set o points in P(V ) compatible with a given point in Y orm a projective space. But the compatibility condition is simply that the projection to each P i, i deined, has the correct value. I we choose representatives p i V/ i \ 0, then a point in V \ 0 is compatible i and only i it is o the orm i α ip i ; it ollows that the preimage is P m 1 as desired. Proposition I = {}, and is not a maximal element o, then Y is the blow-up o Y along the image d o the injective map φ : Y Y / Y. The exceptional divisor is the image d o the injective map φ : Y Y / Y. Proo. The composition law gives φ ι φ = φ (φ ι 1), and thus φ ι (d )=d. Since dim(d )=dim(y ) 1 > dim(d )andd is a projective space bundle over d, it remains only to check that φ ι is injective on the complement o d,which is immediate. Remark. As in [4] (which used two special cases o this proposition), this immediately gives an inductive proo that Y is a smooth, irreducible variety, since by the above proposition and Lemma 2.1, it can be obtained rom a product o projective spaces by a sequence o blow-ups; it also ollows that Y (R) is a smooth, connected maniold or Build(R). Also note the consequence that the normal bundle o d is trivial i is minimal in, since then d is a product bundle. Note that this construction o Y as an iterated blow-up can almost certainly be generalized. Indeed, Keel s construction [13] om 0,n+1 as a blow-up o (P 1 ) n 1 is not o the above orm. In general, or any category C, we deine a universal operad in C to be a unctor F : Build C that also associates a morphism φ : F ( ker( ) ) F ( ) to every weak morphism, satisying the composition law φ g φ 1 = φ g φ g im() 1.

10 Page 10 o 33 ERIC M. RAINS A natural transormation between universal operads will be said to be operadic i it is compatible with the composition law in the obvious way. A universal cooperad in C is simply a universal operad in C op. Remark. A universal operad in a tensor category, equipped with natural isomorphisms F ( ) F () F ( ) (compatible with symmetry and associativity), induces (by restriction to the braid arrangements) an operad in the usual sense; this is not true or a general universal operad, however. Also, given a topological universal operad, we may take the homology o F (with appropriate coeicients), and thus obtain a universal operad in the appropriate category o modules (or, taking cohomology, a universal cooperad o rings, assuming the topological universal operad respects products). 3. Poset homology and operad maps In this section, and until urther notice, we restrict our attention to the case Build(R). In this case, there is a signiicant dierence between odd- and even-dimensional elements o : ordim() > 1, P (R) is orientable i and only i dim() is even. This suggests that evendimensional elements will have particular signiicance in the homology o Y (R). More generally, let Π (m) be the subposet o C consisting o elements A that can be written as direct sums o elements with dim() a multiple o m. (In the case o the braid arrangement = A n 1,Π (m) is the poset o partitions o {1, 2,...,n} into parts all o size congruent to 1 modulo m.) For the remainder o this section, we ix a choice o m; ater this section, we will in act take m = 2 almost exclusively. For any element A Π (m) and any commutative ring R, we deine H ([0,A]; R) tobethe homology o the chain complex C ([0,A]; R) inwhichc k+1 ([0,A]; R) is the ree R-module spanned by chains (0 <A 1 <...<A k <A) in Π (m), and the boundary map is deined by (0 <A 1 <...<A k <A)= ( 1) i (0 <A 1 <...<Âi <...<A k <A). i (Aside rom a shit in degree, this is the reduced homology o the order complex o the open interval (0,A).) By convention, C ([0, 0]; R) = 0 except in degree 0, where it is spanned by the single chain (0). It will be convenient to consider an alternate chain complex giving the same homology, deined in terms o -orests (these are called nested sets in [4], but we eel orest is more evocative o the relevant combinatorics, especially in the braid case). Deinition 1. A -orest is a subset F such that every collection o pairwise incomparable elements o F orms a decomposition; the root o F is the space root(f ):= F C. (Note that this subspace will not be an element o F in general, but merely a direct sum o elements o F.) iven F,thechild o in F is the space child F () := H. H F H A orest F is said to be m-divisible i every element o F has dimension a multiple o m.

11 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 11 o 33 Remark. There is a natural bijection between A n 1 -orests and orests in the usual sense in which the leaves are labeled 1,...,n. iven such a orest, i one labels each internal node by the set o its descendant leaves, the resulting collection o subsets orms an A n 1 -orest. The name orest is justiied by the ollowing lemma. Lemma 3.1. Let F be a -orest. For any element F, the set o elements o F containing orms a chain. Proo. Indeed, i H 1, H 2 F are incomparable elements, then by deinition they orm a decomposition; in particular, cannot be contained in both. In other words, F has a natural structure o a orest with nodes labeled by elements o, compatible with inclusion; the root in our sense is simply the direct sum o the labels o the roots. Now, or A Π (m), deine a chain complex C (A) as ollows. The R-module C k (A) is spanned by ordered m-divisible orests F with root A and k nodes, but with dierent orderings identiied, up to the obvious sign actor. For, we deine F = { ( 1) i 1 F \, = F i, 0, / F ; the boundary map is then given by the sum o with ranging over proper subspaces o components o A. We will also need a concatenation operation (F 1,F 2,...,F n )=(, F 1,F 2,...,F n ), or 0 i the result is not a orest. The ollowing proo is adapted rom [17, Section 2.6], which essentially considered the case Π (1) A n with A = V. Theorem 3.2. For all A Π (m), there is a canonical isomorphism H ([0,A]) = H (A). Proo. I A = 0, then the result is immediate (in both cases, H 0 = Z and all other homology groups are trivial); we may thus assume A 0, and thus both complexes are trivial in degree 0. Now, given a chain 0 <A 1 <A 2 <...<A k <A, consider the (partially closed) simplex o numbers 0 τ k <...<τ 1 <τ 0 = 1. The remainder o the closure o this simplex is naturally identiied with the disjoint union o simplices corresponding to chains with steps removed (i τ k = τ k 1, then remove A k ); as a result, we can glue together all o the simplices to obtain a geometric simplicial complex Σ. The result is simply the order complex o (0,A], so its local homology at the point (0 <A)is H (Σ, Σ \{(0 <A)}) =H +1 ([0,A]). Similarly, given a orest F, consider labelings τ o the nodes, subject to the condition that the labels sum to 1, are nonnegative, and the labels o all nonroots are positive. Again, i we include labelings o suborests (with the same root), with the convention that τ =0 or removed nodes, we obtain a closed simplex, and a geometric simplicial complex Σ.Ix is the

12 Page 12 o 33 ERIC M. RAINS centroid o the simplex corresponding to the orest A, the local homology at x is H (Σ, Σ \{x}) =H +1 (A). (Taking local homology with respect to x is equivalent to restricting the chain complex to those simplices containing the open simplex that contains x.) The theorem will ollow i we can establish a pointed homeomorphism Σ = Σ. Deine a (discontinuous) unction ρ :(0, 1) Π (m) by ρ(t) =A l whenever τ l <t<τ l 1 (with the convention A k+1 = A, τ k+1 = 0). Now, i F is the orest consisting o all components o the A k, we initially label a node by the dierence between the lim sup and the lim in o those t or which is a component o ρ(t). The only way such a label can be 0 is i τ 0 =0and is a component o A but not A 1 ; in particular nonroot nodes have a nonzero label. Since the sum o the labels is uniormly bounded away rom 0 and, we may rescale to make the sum 1, and obtain the desired labeling τ. To invert, we rescale so that the highest-weight path rom the root has weight 1, and can then immediately recover the labeled chain. I we begin with the chain (0 <A), we ind ρ A. The corresponding orest is just the set o components o A, each labeled (ater normalization) by 1/l (i A has l components). But this is indeed the centroid o the simplex corresponding to that orest. Remark. The general case m = 1 was established by dierent means in [8]. Similarly, the case m>1, = A n 1 was established in [6]. Under this isomorphism, each simplex o Σ is identiied with a union o simplices o Σ, and thus the above isomorphism induces a chain map. Corollary 3.3. Deine a map σ : C (A) C ([0,A]) by σ(f ):= σ(π)(0 <F π(1) <F π(1) + F π(2) <...<F π(1) F π(k 1) <A), π where the sum is over permutations o the nodes such that i F π(i) F π(j), then i j. Then σ is a chain map inducing an isomorphism on homology. A chain o the orm will be called a orest chain. (0 <F π(1) <F π(1) + F π(2) <...<F π(1) F π(k 1) <A) Lemma 3.4. A linear combination o orest chains in C ([0,A]) is in the image o σ i and only i its boundary is also a linear combination o orest chains. Proo. This ollows rom geometric considerations, but can also be shown directly as ollows. Suppose we order F in such a way that F i F j implies i j, and suppose F l, F l+1 are incomparable. Then the nonorest chain (0 <F 1 <...<F F l 1 <F F l+1 <...<A) occurs in the dierential o precisely two orest chains, namely and (0 <F 1 <...<F F l 1 <F F l 1 + F l <F F l+1 <...<A) (0 <F 1 <...<F F l 1 <F F l 1 + F l+1 <F F l+1 <...<A).

13 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 13 o 33 Thus the coeicient o this nonorest chain in the dierential is 0 i and only i the coeicients o the orest chains are negatives o each other. Since any two such orderings o F can be connected by a sequence o such transpositions, the claim ollows. Remark. In particular, a chain map between two complexes C ([0,A]) that takes orest chains to orest chains pulls back to a chain map on the associated orest complexes. Now, deine the Whitney homology W (m) () = A Π (m) H ([0,A]), a module graded by both the degree in homology and by the poset Π (m). This, o course, can be computed as the homology o the corresponding sum o poset or orest complexes; thus deine, or instance, C W () = A C C W ([0,A]), and similarly or orests. Theorem 3.5. The Whitney homology extends to a cooperad such that the maps φ are homogeneous and respect the poset grading: in particular φ Proo. φ (W (m) ( )[A]) = W (m) ( ker( ) )[(A ker ) (A)]; vanishes unless (A ker ) (A) Π (m) ker( ). Let : be a weak morphism. We deine a chain map φ : C W ( ) C W ( ker( )) C W () on chains 0 <...<A i <...<A k = A in Π (m) as ollows. I A ker is not in the chain, we set φ = 0. Otherwise, let l be the index o A ker in the chain, and deine φ (0 <...<A i <...<A)=(0<...<A i ker <...<A l ker = A ker ) (0 = (A l ) <...< (A l+i ) <...< (A)) C W l ( ker( )) C W k l(). Since deleting a step cannot introduce A ker to the chain, the dierential cannot leave the set o bad chains (that is, in the kernel o φ ). And the only way to produce a bad chain by deleting a node is to delete A ker ; but the corresponding term is missing rom the dierential on the image space. Thereore φ is indeed a chain map. Moreover, it is easily seen to satisy the composition law. Finally, we can extend it to a chain map by composing with the shule product obtaining the map on homology as required. φ : C W ( ) C W ( ker( ) ) C W ( ker( )) C W () C W ( ker( ) ), Remark 1. Since the Whitney homology is a direct sum, it is equivalent to consider only the induced maps φ : H ([0,A] ) H ([0,A ker (A)] ker( ) )

14 Page 14 o 33 ERIC M. RAINS where A ker (A) Π (m) ker, and similarly or the chain maps. Remark 2. The case o the diagonal morphism Δ : is o particular interest: { φ 0 <...<A i + B i <...A+ B i A B =0, Δ(0 <...<(A i,b i ) <...<(A, B)) = 0 otherwise. Composing with the shule product induces a ring structure on the Whitney homology, graded by Π (m). Since φ takes orest chains to orest chains, we conclude the ollowing. Corollary 3.6. For each weak morphism :, there is a chain map φ : C W producing a commutative diagram ( ) C W ( ker( ) ) C W ( ) σ C W ( ) φ C W ( ker( ) ) σ φ C W ( ker( ) ). Thus, in addition to the cooperad structure on the Whitney homology itsel, we also obtain two cooperads o chain complexes, and an operadic homotopy between them. It is unclear, however, how to deine the cooperad structure on the orest complex without passing through the poset complex. It turns out that the relation to De Concini Procesi models is more convenient in terms o poset and orest cohomology. O course, the cooperads o homology chain complexes immediately dualize to operads on the cohomology chain complexes, and thus give rise to an operad on the Whitney cohomology (deined in the obvious way). Even this is not quite the right structure, however. Note that the induced chain maps (φ ) : C ([0,B C] ker( ) ) C ([0,A] ) are 0 unless B = A ker, C = (A); in other words, we must have a short exact sequence 0 B A C 0. In particular, dim(a) = dim(b) + dim(c), and we may thus shit the degrees o the complex by this dimension without aecting homogeneity o (φ ). To be precise, we will use the chain complex C dim A ([0,A] ), which becomes a homology complex since the dierential now decreases the degree. A more subtle correction is a certain twisting o the operad structure. For any real vector space V, let or(v ):=H dim(v )(V,V \{0}) be the corresponding orientation module; note in particular the canonical isomorphisms and when dim(v ) is even, or(v ) = H dim(v ) 1 (S(V )), or(v ) = H dim(v ) 1 (P(V )).

15 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 15 o 33 In any event, every short exact sequence 0 V W X 0 induces a canonical isomorphism or(v ) or(x) or(w ). Thus every nonzero φ induces an isomorphism or(a) = or(b C), and we obtain an operad map (φ ) : C dim(b C) ([0,B C] ker( ) ) or(b C) C dim(a) ([0,A] ) or(a), as well as associated maps on (degree-reversed) cohomology. We may now state our main theorem. Recall that the notation [A] reers to the A-graded piece o the homology; see the remark ollowing Corollary 2.8. Also, we identiy the real algebraic variety Y with its real locus Y (R), whenever this can be done without causing conusion. Theorem 3.7. Let be a real building set, and A Π.IA/ Π (2), then 2H (Y )[A] =0; otherwise, there is a natural, operadic, isomorphism H dim(a) (2) ([0,A] ) or(a) 2H (Y )[A]. Remark 1. A similar result (minus the operad structure, and requiring some additional hypotheses or naturality) was already known [10] (seealso[5, 16, 19] or the ring structure o cohomology) or the complement o a real subspace arrangement, namely an isomorphism o its homology with H dim(a) (1) ([0,A] ) or(a). A Π (1) One curious consequence o this similarity is that i is obtained rom a complex building set by viewing each space as a real space o twice the dimension, then (since every subspace now has even dimension) there is an isomorphism 2H (Y ) = H (V ). In particular, the homology o Y (R) can be arbitrarily complicated (even or hyperplane arrangements, by the remark ollowing Lemma 2.2), since the same holds or the complements o complex subspace arrangements. Indeed, any inite simplicial complex is homeomorphic to the order complex o a inite atomic lattice (its lattice o aces), and thus, reversing inequalities, to the order complex o a inite coatomic lattice. But any inite coatomic lattice can be represented as a lattice o subspaces o the space o complex-valued unctions on the coatoms (each element corresponds to the space o unctions vanishing on the coatoms bounding it). Remark 2. Dually, there is a natural isomorphism rom the cohomology (modulo 2-torsion) o Y (R) to the (suitably twisted) Whitney homology o Π (2), and operadicity (applied to the diagonal map) implies that this is an isomorphism o (poset-graded) rings. The main theorem has an important consequence or the moduli space o stable genus 0 curves. Corollary 3.8 [7, Conjecture 2.13]. The groups 2H (M 0,n (R); Z) and 2H (M 0,n (R); Z) are ree. Proo. As remarked above, we have an isomorphism M 0,n+1 = Y An 1 o algebraic varieties, and thus o their respective real loci. It ollows that 2H (M 0,n (R), Z) is isomorphic to a direct

16 Page 16 o 33 ERIC M. RAINS sum o cohomology groups o subposets o the poset o set partitions with all parts odd. But this poset is Cohen Macaulay, and thus its cohomology is ree (and supported in the appropriate degree). The remaining claim ollows rom the universal coeicient theorem. Remark 1. The same argument applies to an arbitrary Coxeter arrangement (see [11]) again because the relevant posets are Cohen Macaulay. Remark 2. Note, however, that the explicit presentation o the ring H (M 0,n (R); Z)/H (M 0,n (R); Z)[2] given in [7] does not ollow rom the present methods. It is, o course, ar too much to hope or such a presentation or completely general building sets (or even or general hyperplane arrangements, by the discussion ollowing Lemma 2.2); or instance, the cohomology ring need not be generated in degree 1 in that case. 4. Blow-ups and homology The construction o Y via repeated blow-ups turns out to have extremely useul consequences in homology. As we are interested in the topological consequences o this, it will be useul to have a more topological (or, more precisely, dierentiable rather than algebraic) version o blowing-up. In addition to real and complex blow-ups, corresponding to algebraic blowing-up o real or complex varieties, there is a third spherical blow-up that it will be useul to consider. (For an early application o such blow-ups to subspace arrangements, see [2, 5].) In act, the spherical blow-up is in some sense universal; the other blow-ups can be constructed as quotients o the spherical blow-up. The basic idea o the spherical blow-up is to replace a submaniold Y by the sphere bundle N Y (X)/R +. That is, the blow-up is a new space X equipped with a continuous (smooth) map to X which is an isomorphism outside Y, and such that the preimage o a given point in Y is identiied with the space o unit normal vectors to Y at that point. The result is homotopic to the complement o Y in X, but has the merit o being compact and almost smooth; the ailure to be smooth being that i X is a compact maniold, then X is a compact maniold with boundary (the preimage o Y ). Similarly, i X is a maniold with boundary, then X can have corners. Recall that a smooth maniold with corners is a (paracompact, Hausdor) space X with a covering U i by open sets, each homeomorphic to a space R pi (R 0 ) qi, and in such a way that the compatibility maps are C. We extend this to pairs (X, Y ) by insisting that each Y U i either be empty or an intersection o coordinate hyperplanes. (This is essentially just a condition that Y meets the boundary and corners o X transversely.) In particular, given a pair, we may associate a normal bundle N Y (X), which on a patch U i Y is the quotient o the cone R pi (R 0 ) qi by the subspace U i Y. The spherical blow-up o the pair (X, Y ) is then deined as ollows. I Y is empty, then the blow-up o (X, Y ) is itsel; i X is a cone bundle over Y, then the blow-up is the pair ( X,Ỹ ), where X is the closure o the subset X Y X X/R +, and Ỹ = X/R + is the preimage o Y in X. In general, any pair (X, Y ) looks locally like one o the above two examples, and the above constructions are suiciently compatible to give a global construction. In particular, note that ( X,Ỹ ) is a smooth pair with corners, and the induced map X Ỹ X Y is a dieomorphism.

17 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 17 o 33 For instance, i X R n and Y X consists o a single point, then there is a natural projection X \ Y S n 1,and X is the closure o X \ Y in R n S n 1. The subset Ỹ can then be viewed as the set o all unit vectors pointing toward X rom Y. Since Ỹ = N Y (X)/R +, one can deine other blow-ups as push-orwards o appropriate surjections rom Ỹ. In particular, the real blow-up corresponds to the map N Y (X)/R + N Y (X)/R (valid so long as N Y (X) is a vector bundle, or example, i Y is disjoint rom the corners o X), while the complex blow-up corresponds to a map N Y (X)/R + N Y (X)/C, assuming such a complex structure exists. Both o these, unlike the spherical blow-up, preserve smoothness; however, the spherical blow-up is a useul technical tool, both because it maps to these cases and because it has the homotopy type o X\Y.(Seealso[9], which considers the analog o Y (R) replacing real blow-ups with spherical blow-ups.) It turns out that in general, there is a long exact sequence relating the homology o X, the homology o the blow-up, and the homology o the mapping cone o the projection π : π 1 (Y ) Y. For a continuous map : X Y, recall that the mapping cone M is deined as the quotient o cone(x) Y by identiying each point in X with its image in Y. (By convention, i X =, then cone(x) consists o a single point.) Note that commutative diagrams o the orm A B g h C k D induce continuous maps between mapping cones; moreover, there is a natural homeomorphism M (g,h):m M k = M(,k):Mg M h. iven a map o pairs :(A, B) (C, D), deine H () :=H (M :A C,M :B D ). Note that i B = D =, then this is the reduced homology o M :A C ;i is also an inclusion map, then H () = H (C, A). Lemma 4.1. exact sequence Let :(A, B) (C, D) be a continuous map o pairs. Then there is a long H (A, B) H (C, D) H () H 1 (A, B) unctorial in the sense that, or any commutative square the corresponding diagram (A, B) (C, D) (E,F) g (, H), H (A, B) H (C, D) H () H 1 (A, B) H (E,F) commutes. g H (, H) H (g) H 1 (E,F)

18 Page 18 o 33 ERIC M. RAINS Proo. H (A) I B = D =, A, C, then we have the commutative diagram H (C) H ( : A C) H 1(A) H +1(M,C) H (C) H (M ) H (M,C) which gives rise to a reduced homology version o the desired sequence. For the general case, let M A,B denote the mapping cone o the inclusion B A, and observe that the mapping cone o : M A,B M C,D is homeomorphic to the mapping cone o the inclusion M :B D M :A C. Thus we have the commutative diagram H (A, B) H (C, D) H ( :(A, B) (C, D)) H 1 (A, B) H (M A,B ) H (M C,D ) H ( : M A,B M C,D ) H 1 (M A,B ) Remark. More generally, given two maps :(A, B) (C, D), g :(C, D) (E,F), there is a long exact sequence H () (1,g) H (g ) (,1) H (g) H 1 () which gives the above result when (A, B) =(, ), and when B = D = E = F = gives the reduced homology version. Corollary 4.2. Suppose the map :(A, B) (C, D) induces an isomorphism on relative homology. Then there is a unctorial long exact sequence H (A) Proo. H (C) H ( : B D) H 1 (A) It suices to show that the inclusion maps induce an isomorphism H ( : B D) = H ( : A C). From the long exact sequence o relative homology, this is equivalent to the vanishing o the homology group H ( :(A, B) (C, D)) which by the mapping cone long exact sequence is equivalent to the claim that : H (A, B) H (C, D) is an isomorphism. In the case o a blow-up, we have the ollowing. Theorem 4.3. Let (X, Y ) be a smooth maniold pair with corners, and let ( X,Ỹ ) be a corresponding (spherical, real, complex) blow-up. Then there is a long exact sequence H ( X) H (X) H (Ỹ Y ) H 1( X) Moreover, the map H (X) H (Ỹ Y ) actors through the corresponding map H (X) H ((N Y (X)/R + ) Y ).

19 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 19 o 33 Proo. It suices to show that the natural map π : H ( X,Ỹ ) H (X, Y ) is an isomorphism; indeed, then the long exact sequence ollows rom Corollary 4.2, and the act that the map actors ollows rom unctoriality. Now, i U is an open subset o a patch, then π : H (π 1 (U),π 1 (U Y )) H (U, U Y )is an isomorphism, since π 1 (U Y )andu Y are both deormation retracts o neighborhoods. It then ollows rom Mayer Vietoris that this holds whenever U is a inite union o such subsets, and then by taking a direct limit, that it holds in general. The act that the map actors through the spherical case is surprisingly powerul, and in particular gives short exact sequences in many cases. Corollary 4.4. Let (X, Y ) and ( X,Ỹ ) be as above. In the case o a complex blow-up, the connecting map H (X; R) H (Ỹ Y ; R) is always 0. In the case o a real blow-up, the connecting map is 0 i either R has characteristic 2 or each component o Y has odd codimension, and in general actors through multiplication by 2. Proo. Indeed, it suices to consider the case o C p or R p blown up at the origin, or which the claims are immediate. We now consider the implications o the blow-up long exact sequence or real De Concini Procesi models. Corollary 4.5. Suppose, that := \{} is a building set, and that / C ; let d = Y / be the exceptional divisor, and d = Y / its image. The homology groups o Y (R) and Y (R) are related as ollows. I A, then H k (Y )[A] = H k (Y )[A], and otherwise we have the long exact sequence H (Y /)[ A/] where the connecting map is induced by the composition φ H (Y )[A] H (Y )[A], H (Y ) H (Y d ) = H (N d ( ) d ) H 1 (d ), where H (X Y ):=H (X, X \ Y ) and the last map is induced rom the morphism H (R k {0}) = H 1 (S k 1 ) H 1 (P k 1 ). Proo. The key observation is that by Corollary 2.8, the map d = Y / d = Y / has a section, and thus the mapping cone long exact sequence breaks up into split short exact sequences 0 H +1 (d d ) H (d ) H (d ) 0.

20 Page 20 o 33 ERIC M. RAINS We may thus identiy the mapping cone homology with its image in H (d ): H k+1 (d d ) = H k (Y /)[ A]. The identiication o the map A / H +1 (d d ) H (Y ) ollows rom unctoriality and the commutative diagram d d φ φ Y Y. The remainder o the proo is straightorward. I is minimal in (which is the only case that need be considered when constructing De Concini Procesi models via repeated blow-ups), we ind that the projective bundle is trivial, and thus we can (noncanonically) compute the cohomology o the exceptional divisor via the Künneth ormula. We ind H (Y /)[ A/] = Tor Z ( H 1 (P ),H (Y / )[A/]) H (P ) H (Y / )[A/]. Since the torsion subgroup o H 1 (P ) has exponent 2, the Tor Z component consists entirely o 2-torsion; thus modulo that 2-torsion, we obtain the canonical isomorphism 2H +dim() 1 (Y /)[ A/] { H dim() 1 (P ) 2H (Y / )[A/], dim() = 0(2), = 0, dim() = 1(2). 5. A cell structure or Y We will construct the isomorphism o the main theorem via a chain map; this will require a careul choice o complex or the homology o Y. To each orest F in with root, we may associate a corresponding composition o operad maps: φ F : Y ( Fi )/ child F (F i) Y. i Indeed, i C is any sum o independent elements o F, we can take the composition φ C φ F C φ F/C where F C and F/C are the induced orests in C and /C. It then ollows easily rom the composition law that this composition is independent o C. Since each φ C is injective, it ollows that φ F is injective or all F, and it is thus reasonable to consider the image d F o φ F. In particular, i F consists only o (the components o) its root, then d F = Y, while i F consists only o the element in addition to its root, then d F = d. Now, suppose C has maximal element root() = k o dimension d. For each 0 i d k (the dimension o Y ), let X i be the union o d F or all orests F with root(f )=root() andd i nodes.

21 THE HOMOLOY OF REAL SUBSPACE ARRANEMENTS Page 21 o 33 Theorem 5.1. Suppose every chain in C can be extended to a complete lag (that is, with all codimensions 1). Then, or all n, H (X n+1,x n ) is ree, supported in degree n +1. Moreover, the induced map H n+1 (X n+1,x n ) H n (X n ) H n (X n,x n 1 ) provides these groups with a chain complex structure, with associated homology groups canonically isomorphic to H (Y ). In other words, the sequence X 0 X 1... X d k behaves homologically as the sequence o skeletons o a CW complex. (We conjecture that this is in act the sequence o skeletons o a regular cell complex, but the homological statement will suice or our purposes.) Proo o Theorem 5.1. It suices to prove the claim about H (X i+1,x i ), since then the derivation o cellular homology or CW complexes carries over mutatis mutandum. Now,itwas shownin[4] that the intersection o d F and d F is the submaniold d F F i F F is a orest, and empty otherwise. We thus have the canonical isomorphism H (X n+1,x n )= H (d F,d F X n ). F =d n 1 But then pulling this back through φ F reduces to the case F = k; thatis,h (Y, d ). Now, by the hypotheses, each contains a 1-dimensional space in the building set, or equivalently is contained in a hyperplane. In other words, Y / d is homeomorphic to the quotient o i P i by a nonempty hyperplane arrangement. But this in turn is homeomorphic to a wedge o spheres (one-point compactiications o intersections o open hal-spaces). Remark. Note that the hypothesis is necessary. Indeed, suppose F were a maximal orest with d i nodes or some i>0 (guaranteed to exist i the hypothesis ails). Then d F X i is disjoint rom X i 1, and thus its contribution to H (X i,x i 1 ) is simply H (d F ). But H 0 (d F )= Z 0. With this in mind, we will denote the above chain complex as C (Y ), and say that a building set satisying the hypotheses is cellular. For our purposes, the hypotheses are not particularly onerous; we can simply adjoin generic hyperplanes until they hold. This, o course, leads to the danger that constructions based on the cellular chain complex might depend on the choice o hyperplanes. To control this, we may use the chain map rom the ollowing trivial lemma. Note that the map goes in the reverse direction to the usual case o a morphism adjoining a subspace to a building set. Lemma 5.2. map Y Y Suppose = {H} with dim(h) =1 and cellular. Then the identity is cellular. It will also be helpul to have at our disposal various special cases o operad maps that respect the cell structure. The easiest is the case o a purely operadic weak morphism. Lemma 5.3. Let be a cellular building set, and C C. Then C and /C are cellular, as is the operad map φ C. The associated map on the cellular chain complex is injective.