THE HOMOTOPY TYPE OF THE COMPLEMENT OF A COORDINATE SUBSPACE ARRANGEMENT JELENAGRBIC´ ANDSTEPHENTHERIAULT Abstract. Thehomotopytypeofthecomplementofacomplexcoordinatesubspacearrangement 6 0 isstudiedbyfathoming outthe connection between its topological and combinatorial structures. 0 Afamilyofarrangementsforwhichthecomplementishomotopyequivalenttoawedgeofspheres 2 is described. One consequence is an application in commutative algebra: certain local rings are n a provedtobeGolod,thatis,allMasseyproductsintheirhomologyvanish. J 2 1 ] Contents T A 1. Introduction 2 . h 2. The main objects: their definitions and properties 6 t a 2.1. The Davis-Januszkiewicz space 7 m [ 2.2. The moment-angle complex 7 1 2.3. The cohomology of moment-angle complexes and shifted complexes 8 v 3. Preliminary homotopy decompositions 9 9 7 4. A review of homotopy actions 13 2 1 5. A special case of the Cube Lemma 14 0 6 6. Proper coordinate subspaces of the fat wedge 17 0 / 7. Homotopy fibres associated to regular sequences 20 h t 8. The existence of regular sequences 28 a m 9. The homotopy type of Z for shifted complexes 32 K : v 10. Topological extensions 36 i X 11. Algebra 40 r a References 42 2000 Mathematics Subject Classification. Primary13F55, 55P15,Secondary52C35. Key words and phrases. coordinate subspace arrangements, homotopy type, Golod rings, toric topology, cube lemma. 1 2 JELENAGRBIC´ ANDSTEPHENTHERIAULT 1. Introduction In this paper we study connections between the topology of the complements of certain complex arrangements,and algebraic and combinatorial objects associated to them. Let A={L ,...,L } 1 r be a complex subspace arrangement in Cn, that is, a finite set of complex linear subspaces in Cn. For suchanarrangementA, define its support |A| as|A|= r L ⊂Cn andits complement U(A) i=1 i S as U(A)=Cn\|A|. Arrangementsandtheir complements playa pivotalrole inmany constructionsofcombinatorics, algebraicandsymplectic geometry,etc.; they alsoariseas configurationspacesfordifferentclassical mechanical systems. Special problems connected with arrangements and their complements arise in different areas of mathematics and mathematical physics. The multidisciplinary nature of the subject results in ongoing theoretical improvements, a constant source of new applications and the penetrationofnew ideasandtechniques in eachof the componentresearchareas. Itis the interplay of methods from seemingly disparate areas that makes the theory of subspace arrangementsa vivid and appealing field of research. Inthe study ofarrangementsit is importantto get a detaileddescriptionof the topology oftheir complements,includingpropertiessuchashomologygroups,cohomologyrings,homotopytype,and so on. In this paper we are concerned with the homotopy type of the complement of a complex coordinate subspace arrangement. A complex coordinate subspace of Cn is given by L ={(z ,...,z )∈Cn| z =···=z =0} σ 1 n i1 ik where σ = {i ,...,i } is a subset of [n] = {1,...,n}, allowing us to define a complex coordinate 1 k subspace arrangement CAin Cn as afamily ofcoordinatesubspacesL for σ ⊂[n]. The main topo- σ logical space we study, naturally associated to the complex coordinate subspace arrangement CA, is the complement U(CA) in Cn. Our results are obtained by studying the topological and combi- natorial structures of U(CA) with the help of commutative and homologicalalgebra, combinatorics and homotopy theory. It has been known for some time that hyperplane arrangements have a torsion free cohomology ring. Recently it wasproved[S] thatafter suspending the complementofa hyperplanearrangement it becomes homotopy equivalent to a wedge of spheres. The case of complex coordinate subspace arrangementsis much more complicated. Already atthe cohomologylevel, there is a more intricate structure. The Buchstaber-Panov formula for H∗(U(CA)) [BP] detects torsion in special cases, implying that even stably U(CA) cannot always be homotopy equivalent to a wedge of spheres. Thatmakesthequestionofwhenthecomplementofacoordinatesubspacearrangementishomotopy THE HOMOTOPY TYPE OF THE COMPLEMENT OF AN ARRANGEMENTS 3 equivalent to a wedge of spheres more difficult and therefore more interesting. The main goal of this paper is to describe a family of coordinate subspace arrangementsfor which the complement is homotopy equivalent to a wedge of spheres. Thebasicconnectionsbetweenthetopology,combinatoricsandcommutativealgebraofcoordinate subspace arrangements are established as follows. Let K be a simplicial complex on the vertex set [n]. We shall consider only complexes that are finite, abstract simplicial complexes represented by their collection of faces. Every simplicial complex K on the vertex set [n] defines a complex arrangement of coordinate subspaces in Cn via the correspondence K ∋σ 7→span{e :i6∈σ} i where {e }n is the standard basis for Cn. Equivalently, for each simplicial complex K on the set i i=1 [n], we associate the complex coordinate subspace arrangement CA(K)={L |σ 6∈K} σ and its complement (1) U(K)=Cn\ L . σ σ[6∈K On the other hand, to K and a commutative ring R with unit there is an associated algebraic object, the Stanley-Reisner ring R[K], also known in the literature as the face ring of K. Denote by R[v ,...,v ] the graded polynomial algebra on n variables where deg(v )=2 for each i over R. 1 n i The Stanley-Reisner ring of a simplicial complex K on the vertex set [n] is the quotient ring R[K]=R[v ,...,v ]/I 1 n K whereI is thehomogeneousidealgeneratedbyallsquarefreemonomialsvσ =v ···v suchthat K i1 is σ ={v ,...v }6∈K. i1 is Coming back to topology and following the Buchstaber-Panov approach [BP] to toric topology, thereareanothertwotopologicalspacesassociatedtoasimplicialcomplexK anditsStanley-Reisner ring R[K]. The first space arises as a topological realisation of the Stanley-Reisner ring. It is the Davis-Januszkiewicz space DJ(K), whose cohomology ring is isomorphic to the Stanley-Reisner ring R[K]. The Davis-Januszkiewicz space maps by an inclusion into the classifying space of the n-dimensionaltorus. Thehomotopyfibreofthisinclusioncanbeidentifiedwithanothertorusspace, the moment-angle complex Z , which has as a deformation retract the complement U(K) of the K complex coordinate subspace arrangement[BP, 8.0]. Different models of DJ(K) and Z as well as K their additional properties will be addressed later on in Section 2. These homotopic identifications show that the problemof determining the homotopy type of the complementof complex coordinate 4 JELENAGRBIC´ ANDSTEPHENTHERIAULT subspacearrangementsisequivalenttodeterminingthehomotopytypeofthemoment-anglecomplex Z . To do this we need to closely examine the homotopy fibration sequence K Z −→DJ(K)−in→cl BTn. K The main technique employed for understanding of this filtration is Mather’s Cube Lemma [M], which relates homotopy pullbacks and homotopy pushouts in a cubical diagram. This is applied iteratively as K is built up one face at a time, in a prescribed order. An analysis of the component homotopy fibration and cofibration sequences produces our main result, Theorem 1.1(see below). To find a suitable simplicial complex K whose U(K) will be homotopy equivalent to a wedge of spheres, we first look at its cohomology ring. As U(K) is homotopy equivalent to Z , this is the K same as lookingat the cohomologyring of Z . The integralcohomologyof Z has been calculated K K in [BP, 7.6 and 7.7]. If Z is to be homotopy equivalent to a wedge of spheres then we need to K consider simplicial complexes K for which all Massey products in H∗(Z ) vanish. That will not K imply that Z is itself homotopic to a wedge of spheres but at least on the cohomological level K there will be no obstructions to that claim. Combinatorists, from their point of view, have studied simplicialcomplexesandassociatedtothemcertainToralgebrasthatcorrespondtothecohomology of Z as in our case. They have determined severalclasses of complexes for which it can be shown K that all Massey products in associated Tor algebras vanish. One such class is of shifted complexes. A simplicial complex K is shifted if there is an ordering onthe vertexsetsuchthatwheneverσ isasimplex ofK andv′ <v,then(σ−v)∪v′ isasimplexof K. Gasharov,Peeva and Welker [GPW] showed that when K is a shifted complex, then all Massey products in H∗(Z ) are trivial. In this case we obtain much stronger result by determining the K homotopy type of Z . K Theorem 1.1. Let K be a shifted complex. Then U(K) is homotopy equivalent to a wedge of spheres. Previously,the only known cases of simplicial complexes K for which the complement U(K) has the homotopytype ofawedgeofspheresoccurredwhenK wasa disjointunionofn vertices. When n = 2 or n = 3, these are classical results of low dimensional topology, while the general case was proved by the authors [GT]. The result in Theorem 1.1 is much more general. Notice for example thatany fullk-dimensionalskeletonofthe standardsimplicialcomplex onn vertices∆n is ashifted complex. Let F be the family of simplicial complexes K for which the moment-angle complex Z has the t K property that ΣtZ is homotopy equivalent to a wedge of spheres. Our next theorem describes the K influence that combinatorial operations on simplicial complexes have with respect to F . t THE HOMOTOPY TYPE OF THE COMPLEMENT OF AN ARRANGEMENTS 5 Theorem 1.2. Let K ∈ F and K ∈ F for some non-negative integers t and s. The effect on 1 t 2 s family membership of the simplicial complex K resulting from the following operations on K and 1 K is: 2 (1) the disjoint union of simplicial complexes: if K =K K , then K ∈F where m=max{t,s}; 1 2 m ` (2) gluing along a common face: ifK =K K ,thenK ∈F whereσ isacommonfaceofK andK andm=max{t,s}; 1 σ 2 m 1 2 S (3) the join of simplicial complexes: if K =K ∗K , then K ∈F where m=max{t,s}+1. 1 2 m As a corollary we specify the operations on simplicial complexes for which F is closed. 0 Corollary 1.3. Let K and K be simplicial complexes in F . Then F is closed for the following 1 2 0 0 operations on simplicial complexes: (1) the disjoint union of simplicial complexes, K =K K ∈F ; 1 2 0 ` (2) gluing along a common face, K =K K ∈F , where σ is a common face of K and K . 1 σ 2 0 1 2 S The information we have obtained on complex subspace arrangements has an application in commutative algebra. Let R be a local ring. One of the fundamental aims of commutative algebra is to describe the homologyring of R,that is Tor (k,k), where k is a groundfield. The firststep in R understanding Tor (k,k) is to obtain information about its Poincar´eseriesP(R), more specifically, R whether P(R) is a rational function. A far reaching contribution to this problem was made by Golod. A local ring R is Golod if all Massey products in Tor (R,k) vanish. Golod [G] k[v1,...,vn] proved that if a local ring is Golod, then its Poincar´e series represents a rational function and it is determined by P(Tor (R,k)). Although being Golod is an important property, not many k[v1,...,vn] Golod rings are known. Using our results on the homotopy type of the complement of a coordinate subspace arrangement, we are able to use homotopy theory to gain some insight into these difficult homological-algebraicquestions. The main results are as follows. Theorem 1.4. For a simplicial complex K, t(1+t)n P(k[K])≤ . t−P(H∗(U(K))) Equality is obtained when k[K] is Golod. Theorem 1.5. If K ∈F , then k[K] is a Golod ring. 0 Combining Theorems 1.4 and 1.5, we obtain the following result. 6 JELENAGRBIC´ ANDSTEPHENTHERIAULT Corollary 1.6. For a simplicial complex K ∈F , 0 t(1+t)n P(k[K])= . t−P(H∗(U(K))) To close, let us remark that all the techniques used in this paper can be also applied to real and quaternioniccoordinatesubspacearrangementsbychangingthe groundringfromcomplexnumbers to real, quaternion numbers respectively. In those cases Theorem 1.1 describes the homotopy type of the complement of real, quaternionic coordinate subspace arrangements. For real arrangements instead of torus spaces and CP∞, we look at spaces with an action of Z/2 (also considered as S0) and RP∞, respectively; while in the case of quaternionic arrangements we deal with S3 spaces and HP∞. The disposition of the paper is as follows. Section 2 catalogues the main objects of study and states various properties they satisfy. Sections 3 through 9 build up to and deal with the primary focus of the paper, Theorem 1.1. Sections 3 through 6 establish the preliminary homotopy theory. Includedareidentificationsofthehomotopytypesofvariouspushouts,areviewofhomotopyactions, the general statement of Mather’s Cube Lemma and a finer analysis of a special case involving homotopy actions, and several properties of the fat wedge. Section 7 considers a particular pattern of successive inclusions of one coordinate subspace into another which we term a regular sequence. Such a sequence need not always exist, but when it does we show there is a measure of control over the homotopy types of the successive homotopy fibres obtained from including the coordinate subspaces into the full coordinate space X ×···×X . Section 8 gives conditions guaranteeing the 1 n existence of regular sequences, which are based on the properties of a shifted complex. Section 9 puts together all the material in Sections 3 through 8 to prove Theorem 1.1. At this point, the class of simplicial complexes for which Z is homotopy equivalent to a wedge of spheres includes K the shiftedcomplexes. Section10showsthatthereareothersimplicialcomplexesK whichhaveZ K homotopyequivalent(orstablyhomotopyequivalent)toawedgeofspheresbyprovingTheorem1.2 and Corollary 1.3. Finally, Section 11 turns to commutative algebra considering Golods rings and their properties, and proves Theorems 1.4 and 1.5. Acknowledgements. The authors would like to thank Professors Victor Buchstaber and Taras Panov for their stimulating work, as well as for their helpful suggestions and kind encouragement. The first author would also like to thank Professor Volkmar Welker for explaining to her the con- nection between combinatorics and arrangements and for making it possible for her to visit the University of Marburg for a week. 2. The main objects: their definitions and properties As mentioned in the introduction the main objective of this paper is the study of arrangements and their complements from topological point of view. To pass from the combinatorial concept of arrangements to a topological one, we use different topological models associated to simplicial THE HOMOTOPY TYPE OF THE COMPLEMENT OF AN ARRANGEMENTS 7 complexes K and their algebraic counterparts,the Stanley-Reisner rings Z[K] (or the face rings) of K. Thepurposeofthissectionistopresentthemainobjectswhichwearegoingtouseandtosetthe notation. We rely heavily on constructions in toric topology introduced and studied by Buchstaber and Panov [BP]. 2.1. TheDavis-Januszkiewicz space. ThetopologicalrealisationoftheStanley-ReisnerringZis calledtheDavis-JanuszkiewiczspaceDJ(K). ThefirstmodelofDJ(K)isaBorel-typeconstruction due to Davis and Januszkiewicz [DJ]. For our purposes we use another model of DJ(K) given by Buchstaber-Panov [BP]. In what follows, we identify the classifying space of the circle S1 with the infinite-dimensional projective space CP∞, and therefore the classifying space BTn of the n-torus with the n-fold product of CP∞. For an arbitrary subset σ ⊂[n], define the σ-power of BT as BTσ ={(x ,...,x )∈BTn| x =∗ if i∈/ σ}. 1 n i Definition 2.1. LetK beasimplicialcomplexontheindexset[n]. TheDavis-Januszkiewiczspace is given as the cellular subcomplex DJ(K)= BTσ ⊂BTn. σ[∈K Buchstaber and Panov justified the name of this topological model by proving the following. Proposition2.2(Buchstaber-Panov[BP]). ThecohomologyofDJ(K)isisomorphictotheStanley- Reisner ring Z[K]. Moreover, the inclusion of cellular complexes i: DJ(K) −→ BTn induces the quotient epimorphism i∗: Z[v ,...,v ]−→Z[K] 1 n in cohomology. Recently, Notbohm-Ray [NR] showed that the Davis-Januszkiewicz spaces are uniquely deter- mined, up to homotopy equivalence, by their cohomology ring. This implies that all models of Davis-Januszkiewicz spaces are mutually homotopy equivalent. 2.2. The moment-angle complex. Realise the torus Tn as a subspace of Cn Tn = (z ,...,z )∈Cn| |z |=1, for i=1,...,n 1 n i (cid:8) (cid:9) contained in the unit polydisc (D2)n = (z ,...,z )∈Cn| |z |≤1, for i=1,...,n . 1 n i (cid:8) (cid:9) For an arbitrary subset σ ⊂[n], define B = (z ,...,z )∈(D2)n| |z |=1 i∈/ σ . σ 1 n i (cid:8) (cid:9) 8 JELENAGRBIC´ ANDSTEPHENTHERIAULT Definition 2.3. Let K be a simplicial complex on the index set [n]. Define the moment-angle complex Z by K Z = B ⊂(D2)n. K σ σ[∈K Observethat since each B is invariantunder the actionof Tn, the moment-anglecomplex Z is σ k a Tn-space. Buchstaber and Panov showed that the moment-angle complex is another topological model of the Stanley-Reisner ring Z[K] by proving that the Tn-equivariant cohomology H∗ (Z ) Tn K is isomorphic to Z[K]. The following description of the moment-angle complex Z together with its relation to the K complement of an arrangement plays the pivotal role in our approach to determine the homotopy type of the complement of a complex coordinate subspace arrangement. Proposition 2.4 (Buchstaber-Panov [BP]). The moment-angle complex Z is the homotopy fibre K of the embedding i: DJ(K)−→BTn. Recall from (1) that U(K) denotes the complement of the complex coordinate subspace arrange- ment associated to a simplicial complex K. Theorem 2.5 (Buchstaber-Panov [BP]). There is an equivariant deformation retraction U(K)−→Z . K 2.3. The cohomology of moment-angle complexes and shifted complexes. Theorem 2.5 insuresthatthe homotopytype ofthe complementU(K)ofacomplexcoordinatesubspacearrange- ment can be obtained by finding the homotopy type of the moment-angle complex Z . K InourstudyofthehomotopytypeofU(K)wespecialisedbyaskingforwhichsimplicialcomplexes K the complement U(K) is homotopy equivalent to a wedge of spheres. To begin we look at the cohomology ring of Z finding those simplicial complexes for which there is no cohomological K obstruction for Z to be homotopic to a wedge of spheres. Buchstaber and Panov [BP] described K the cohomology algebra of Z by proving that there is an isomorphism K H∗(ZK;k)∼=Tork[v1,...,vn](k[K],k). as graded algebras. Definition2.6. TheStanley-Reisnerringk[K]isGolodifallMasseyproductsinTor (k[K],k) k[v1,...,vn] vanish. This definition provides a class of rings k[K] for which Z might be homotopic to a wedge of K spheres. Although being Golod is an interesting property of a ring, there are not many examples of Golod rings. The one that is going to be of use for us comes from combinatorics. THE HOMOTOPY TYPE OF THE COMPLEMENT OF AN ARRANGEMENTS 9 Definition 2.7. A simplicial complex K is shifted if there is an ordering on its set of vertices such that whenever σ ∈K and v′ <v, then (σ−v)∪v′ ∈K. Noticethatanyfulli-thskeleton∆i(n−1)ofthestandardsimplicialcomplex∆n−1 (alsodenoted by ∆(n)) on n vertices is shifted. Proposition 2.8 (Gasharov,Peeva and Welker [GPW]). If K is shifted, then its face ring k[K] is Golod. 3. Preliminary homotopy decompositions The purpose of this section is to identify the homotopy type of several pushouts. We begin by stating Mather’s Cube Lemma [M], which relates homotopy pullbacks and homotopy pushouts in a cubical diagram. Lemma 3.1. Suppose there is a homotopy commutative diagram // E F @@@@@@@(cid:31)(cid:31) AAAAAAAA // G H (cid:15)(cid:15) (cid:15)(cid:15) // A B @@@@@@@(cid:31)(cid:31) (cid:15)(cid:15) AAAAAAAA (cid:15)(cid:15) // C D. Suppose the bottom face A−B−C−D is a homotopy pushout and the sides E−G−A−C and E−F −A−B are homotopy pullbacks. (a) If the top face E−F−G−H is also a homotopy pushout then the sides G−H− C−D and F −H −B−D are homotopy pullbacks. (b) If the sides G−H−C−D and F−H−B−D are also homotopy pullbacks then the top face E−F −G−H is a homotopy pushout. (cid:3) Wenextsetsomenotation. LetX andX bespaces,andlet1≤j ≤2. Letπ :X ×X −→X 1 2 j 1 2 j bethe projectionontothejth factorandleti :X −→X ×X be theinclusionintothejth factor. j j 1 2 Let q :X ∨X −→X be the pinch map onto the jth wedge summand. As well, unless otherwise j 1 2 j specified, we adopt the Milnor-Moore notation of denoting the identity map on a space X by X. 10 JELENAGRBIC´ ANDSTEPHENTHERIAULT Lemma 3.2. Let A, B and C be spaces. Define Q as the homotopy pushout ∗×B // A×B C×B π1 (cid:15)(cid:15) (cid:15)(cid:15) // A Q. Then Q≃(A∗B)∨(C⋊B). Proof. Consider the diagram of iterated homotopy pushouts π2 // i2 // A×B B C×B π1 ∗ s (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) ∗ // t // A A∗B Q. Here, it is well known that the left square is a homotopy pushout, and the right homotopy pushout defines Q. Note thati ◦π ≃∗×B. The outerrectangleinaniteratedhomotopypushoutdiagram 2 2 isitselfahomotopypushout,soQ≃Q. Therightpushoutthenshowsthatthehomotopycofibreof C×B −→QisΣB∨(A∗B). Thusthasalefthomotopyinverse. Further,s◦i ≃∗sopinchingout 2 B in the right pushout gives a homotopy cofibration C ⋊B −→ Q −r→A∗B with r◦t homotopic to the identity map. (cid:3) Lemma 3.3. Let A, B, C and D be spaces. Define Q as the homotopy pushout ∗×B // A×B C×B A×∗ (cid:15)(cid:15) (cid:15)(cid:15) // A×D Q. Then Q≃(A∗B)∨(C⋊B)∨(A⋉D). Proof. Let Q be the homotopy pushout of the maps A×D −→Q and A×D −π→1 A. Then there 1 is a diagram of iterated homotopy pushouts ∗×B // A×B C×B A×∗ (cid:15)(cid:15) (cid:15)(cid:15) // A×D Q π1 (cid:15)(cid:15) (cid:15)(cid:15) // A Q1. Observe that the outer rectangle is also a homotopy pushout, so by Lemma 3.2 we have Q ≃ 1 (A∗B)∨(C ⋊B). Further, the outer rectangle shows that the map A −→ Q is null homotopic. 1