The integer cohomology algebra of toric arrangements

The

integer

cohomology

algebra

of

toric

arrangements

Filippo Callegaroa,˚_{, Emanuele Delucchi}b a_{DipartimentodiMatematica,UniversityofPisa,Italy}

b_{Departementdemathématiques,UniversitédeFribourg,Switzerland}

a b s t r a c t Keywords: Toricarrangements Hyperplanearrangements Cellcomplexes Posets

Smallcategorieswithoutloops Combinatorialtopology Orlik–Solomonalgebra Salvetticomplex Arithmeticmatroids

We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement’s combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain:

– a “combinatorial” version of Brieskorn’s lemma in terms of Salvetti complexes of complexified arrangements, – a uniqueness result for realizations of arithmetic matroids

with at least one basis of multiplicity 1.

* Correspondingauthor.

E-mailaddresses:callegaro@dm.unipi.it(F. Callegaro),emanuele.delucchi@unifr.ch(E. Delucchi).

http://doc.rero.ch

Published in "Advances in Mathematics 313(): 746–802, 2017"

which should be cited to refer to this work.

1. Introduction

Thegoal ofthis paper isto givea presentationofthe cohomologyring withinteger coefficients ofthe complementof atoric arrangement – i.e., of afamilyof level sets of characters of the complex torus –and to investigate its dependencyfrom theposet of layersofthearrangement.

This line of research can be tracedback to Deligne’s seminal work oncomplements ofnormal crossingdivisorsinsmoothprojective varieties[13] andhasbeen extensively andsuccessfullycarriedoutinthecaseofarrangementsofhyperplanesincomplexspace, wheretheintegercohomologyringofthecomplementisawell-studiedobjectwithstrong combinatorialstructure.Inparticular,itcanbedefinedpurelyintermsoftheintersection poset of the arrangement, and in greater generality, for any matroid, giving rise to theclass ofso-called Orlik–Solomonalgebras.Wereferto Yuzvinsky’s survey[38] fora thoroughintroductionanda“tourd’horizon”oftherangeofdirectionsofstudyfocusing onOS-algebras.

Recently,the study of hyperplane arrangements hasbeen taken as astepping stone towards different kinds of generalizations. Among these let us mention the work of Dupont[16]developingalgebraicmodelsforcomplementsofdivisorswithhyperplane-like crossingsandofBibby[2]studyingtherationalcohomologyofcomplementsof arrange-mentsinabelianvarieties.Both applyindeedtothecaseof interesttous, thatoftoric arrangements.

Besidesbeinganaturalstepbeyondarrangementsofhyperplanesinthestudyof com-plementsofdivisors,ourmotivation forconsideringtoricarrangements stemsalsofrom recentworkofDeConcini,ProcesiandVergnewhichputstopologicalandcombinatorial propertiesoftoricarrangementsinamuchwidercontext(see[12]orthebook[11])and spurredaconsiderableamountofresearchaimedatestablishingasuitablecombinatorial framework.Thisresearchwastackledalongtwomain directions.

Onesuchdirection,fromalgebraiccombinatorics,ledMoci[25]tointroduceasuitable generalizationof theTuttepolynomials andthen, jointlywith d’Adderio[7],to the de-velopmentofarithmeticmatroids(foranup-todateaccountseeBrändénandMoci[4]). Theseobjects,aswellasotherslikematroidsoverrings[18],exhibitaninteresting struc-turetheoryandrecoverearlierenumerativeresultsbyEhrenborg,ReaddyandSlone[17]

andLawrence[20]but,asofyet,onlybearanenumerativerelationshipwithtopological or geometric invariants of toric arrangements – in particular,it is not knownwhether thesestructurescharacterizetheirintersectionpattern(oneattempttowardsclosingthis gaphasbeen madebyconsideringgroupactionsonsemimatroids[14]).

Theseconddirectionisthestudyofthecombinatorialinvariantsofthetopologyand geometryof toricarrangements:ourworkisacontributioninthisdirection,and there-forewenow briefly reviewearliercontributions.The Bettinumbersof thecomplement toatoricarrangementwereknownatleastsinceworkofLooijenga[23].De Conciniand Procesi[11] relatedtheseBetti numbersto thecombinatoricsoftheposetofconnected componentsofintersectionsinthecontextoftheircomputationofapresentationofthe cohomologyringoverC forunimodulararrangements(i.e.,thosearising fromkernelsof atotallyunimodularsetofcharacters), fromwhichtheyalsodeduceformalityforthese arrangements.A firstcombinatorialmodelforthehomotopytypeofcomplementsoftoric arrangementswasintroducedbyMociandSettepanella[27]for“centered”arrangements (i.e.,definedbykernelsofcharacters)whichinducearegularCW-decompositionofthe compact torus pS1_qd _{Ď pC}˚_qd_{, and was subsequently generalized to the case of}

“com-plexified”toricarrangements (S1_{-level sets ofcharacters) byd’Antonioand thesecond}

author[9]who,onthisbasis,alsogaveapresentationofthecomplement’sfundamental group.Inlaterwork[8],d’Antonioandthesecondauthoralsoprovedthatcomplements ofcomplexifiedtoric arrangementsareminimal spaces(i.e.,theyhavethehomotopy of aCW-complexwhere thei-dimensionalcellsarecounted bythei-th Bettinumber):in particular,the integercohomology groupsare torsion-free and are thus determined by theassociatedarithmeticmatroid.Thisraisesthequestionofwhether,asisthecasewith theOS-Algebra ofhyperplanearrangements,theintegercohomologyringis combinato-riallydetermined. Thework of Dupont [16] and Bibby [2]mentioned earlier, although moregeneralinscope,doesincludethecaseoftoricarrangementsbutfallsslightlyshort of our aim in that on the one hand it uses field coefficients1 _{and on the other hand}

computesonlythebigradedmoduleassociated toafiltrationofthecohomologyalgebra obtainedasthe abutmentofaspectral sequence.Moreover, DeshpandeandSutar[15], by anexplicit study of the Gysin sequence, gave asufficient criterion for the complex cohomology algebra of a toric arrangement to be generated in first degree and to be formal.

After afirst version of this paper De Concini and Gaiffi[10] constructed projective wonderful compactifications for complementsof toric arrangements, improvingon pre-vious(non-projective)modelsdescribedbyMoci [26].

Inthis paperwepairthe(bynow standard)spectral sequenceargumentwithavery explicitcombinatorialanalysis of thetoric Salvetti complexand can thuscompute the fullcohomologyalgebraovertheintegersofgeneralcomplexifiedtoricarrangements.The generalizationtonon-complexifiedcasereliesthenonatechnicalargument.Wegivetwo presentationsof the cohomology algebraand discuss its dependencyfrom theposet of connected componentsof intersections. Inthe caseof arrangements defined by kernels of characters there is also an associated arithmeticmatroid and in this case we prove that when the defining set of characters contains an unimodular basis the arithmetic

1 _{ArecentprivateconversationwithClémentDupontindicatedthatatleastpartsofhismethodscould} begeneralizedtointegercoefficients.

matroidsdetermine theintegercohomologyalgebra.Thepreciseresultswillbestatedin Section2,togetherwithabriefsurveyofthearchitectureoftheremainderofourwork. 2. Overviewandstatementof results

2.1. Main definitions

Let T “ pC˚qd be the complex torus and let Tc “ pS1qd be the compact subtorus

of T .

Atoric arrangement isafiniteset

A“ tY1,¨ ¨ ¨, Ynu

where,foreveryi “1,. . . ,n,

Yi:“χ´1i paiq

withχiPHompT,C˚qandai PC˚.ThearrangementA is calledcomplexified if ai PS1

foreveryi.

Alayer ofA isaconnectedcomponentofanon-emptyintersectionsofelementsofA. Therank ofalayer L is itscodimensionasacomplexsubmanifoldin T .Weorderlayers byreverseinclusion:L ďL1 ifL1 ĎL. LetC betheposetoflayers associatedto A and letCqbe thesubsetof C givenbythelayersL PC withrkpLq “q.

ThecomplementofatoricarrangementA isthespace

MpAq:_“TzďA.

Remark 2.1.1. A toric arrangement is called essential if the layers of minimal dimen-sion havedimension 0 (equivalently, therank ofC_pA_qas aposetequalsthe dimension of T ). Notice thatfor any nonessential toric arrangement A there is an essential toric arrangementA1 withMpAq “ pC˚qrˆMpA1q, where r“rkpCpAqq,see[9,Remark 4].

Asinthecaseofanhyperplanearrangement,wedefinetherankofatoricarrangement rk_pA_q :_“rk_pC_pA_qq.

Toeverytoric arrangementA correspondsaperiodicaffinehyperplane arrangement

Aæ _{inthe universalcoverC}d _{of thecomplextorus. Thehyperplane arrangement A}æ _is

complexifiedexactlywhenA is.

Definition2.1.2.ForatoricarrangementA definethehyperplanearrangement

A0:“ tY1æ, . . . , Ynæu

where,fori “1,. . . ,n,Y_iæisthetranslateattheoriginofanyhyperplaneofAælifting Yi.

GivenalayerL PCpAq,definethen

ArLs:_{“ t}Y_jæPA0|LĎYju.

Remark 2.1.3. It is immediate to see that the intersection lattice of the hyperplane arrangementA_rLsisposet-isomorphictoC_ďL.

2.2. Background onhyperplanearrangements

Thefactthatthecohomologyringofanarrangement’scomplementiscombinatorial canbe madepreciseasfollows.

LetA beanarrangementof hyperplanesinCd_{.Themain combinatorialinvariant of}

A istheposet

LpAq:“ tXK|KĎAu

partiallyorderedbyreverseinclusion:X ěY ifX ĎY .NoticethatL containsaunique minimal element thatwe call ˆ0, correspondingto theintersection over the empty set. WhenA is central (i.e. XA ‰ H), this poset is ageometric lattice and thus defines a (simple)matroidassociatedto thearrangement.

Thej-thBetti numberof thecomplementMpAq :_“Cd

z YA canbe statedinterms ofL as

βjpMpAqq “ ÿ xPLj

μLpˆ0, xq

whereμL denotestheMöbiusfunctionofL andLj isthesetofelementsofL ofrankj.

Brieskorn [6]proved thatthecohomology ofMpAq istorsion-free, thus theadditive structure of H˚pMpAq;Z_q is determined by L. Moreover, we have the following fun-damental result expressing the cohomology of A in terms of the top cohomology of subarrangementsof theform

AX :“ tH PA|X ĎHu for XPLpAq. (1)

Lemma 2.2.1 (Brieskorn Lemma [30]). Let A be an arrangement of hyperplanes. For all k themap

à XPL,rkpXq“k

HkpMpAXq,Zq ÑHkpMpAq;Zq

inducedbytheinclusionsMpAq ãÑMpAXq isanisomorphismof groups.

Definition2.2.2.Given ahyperplanearrangement A andanintersectionX PL,wewill denote by bk _{: H}k

pMpAXq;Zq Ñ HkpMpAq;Zq the map given by inclusion into the

X-summandinthedecompositiongiveninBrieskorn’sLemma.

As faras thealgebrastructure isconcerned, Orlikand Solomondefinedan abstract algebraintermsofthematroidassociatedtoA,then proveditisomorphictothe coho-mologyalgebrausinginductiononrankviathedeletion–restriction recurrence,i.e.,the exactsequence

0ÑHkpMpA1q;Zq ÑHkpMpAq;Zq ÑHk´1pMpA2q;Zq Ñ0 (2) validforallką0,which,givenanyH0PA,connectsthecohomologyofthecomplement

ofthedeleted arrangementA1:“AztH0uandthecohomologyofthecomplementofthe restricted arrangementA2:_{“ t}HXH0|H PA1u.

TheabstractpresentationgivenbyOrlikandSolomonisthefollowing.

Definition2.2.3(Orlik–Solomonalgebraofahyperplanearrangement).Consideracentral arrangement of hyperplanes A “ tH1,. . . ,Hnu and let E˚ denote the graded exterior

algebragenerated byn elementse1,. . . ,en indegree1 overthe ringofintegers. Define

anidealJpAqas generatedbytheset:

 BeX ˇˇX Ď rns; codim č iPX Hiă |X| (

where,forX“ ti1,. . . ,iku P rns,wewrite BeX :“ei1¨ ¨ ¨eik and define

BeX “ k ÿ j“1

p´1_qj´1eXztiju.

TheOrlik–SolomonalgebraofA isthendefinedas thequotient OS˚_pA_q:_“E˚{JpAq.

Theorem 2.2.4 (Orlik and Solomon [29]). For every central arrangement of hyper-planesA,thereis anisomorphismof gradedalgebras

OS˚pAq »H˚pMpAq;Zq. 2.3. Results

Wenow brieflyformulateourmain results.Theremainder ofthepaper willbe then devotedtotheproofs.LetusconsideratoricarrangementA,writingC for theposetof layersofA.

2.3.1. The algebra A_pA_q

Definition2.3.1.LetL:C_{Ñ t}Z-algebras_ube thediagram definedby

LÞÑ LL:“H˚pL;Zq bH˚pMpArLsq;Zq

and

L1ďLÞÑ LL1_ďL“i˚_bb :LL1_{Ñ L}L,

where

i˚: H˚pL1;Zq ÑH˚pL;Zq

is the natural morphism induced by the inclusion L ãÑi L1 and b denotes the map of

Definition 2.2.2.

The algebra LL can be graded with the graduation induced by H˚pMpArLsq;Zq,

hencewehave

Lq

L“H˚pL;Zq bHqpMpArLsq;Zq.

Definition2.3.2.WedefinethealgebraA_pA_qas thedirectsum

à LPC

LrkpLq

L

with multiplication map defined as follows. Let L,L1 P C be two layers. Considertwo classesαP LLrkpLq andα1 P L

rkpL1_q

L1 . Wedefine theproduct

ppαq Ź pα1qqL2

:_“

#

LLďL2pαq Y LL1ďL2pα1q if LXL1 ďL2and rkpL2q “rkpLq `rkpL1q;

0 otherwise.

NoticethattheproductstructuredependsonthemapsLL1ďL fromDefinition 2.3.1,

henceonbothi˚andb.Whileitisknownthatthemapb iscombinatoriallydetermined (see[30,§5.4]),themapi˚ isnotcombinatorial.

TheoremA.Thereisanisomorphism ofalgebras H˚pMpAq;Zq »ApAq.

Example 2.3.3. As a first example we can consider the following arrangement in the 2-dimensionalcomplextorusT “ pC˚q2 withcoordinatesz1,z2.

A“ tH, H1u

whereH “ tpz1,z2q PT |z1 “1u, H1 “ tpz1,z2q PT |z1z22“1uand theirintersection

isgivenbythepointsp “ p1,1_qand q“ p1, ´1_q.

Fig. 1. The toric arrangement ofExample 2.3.3and its poset of layers.

Thearrangementshasposetoflayers describedinFig. 1.

Thelocal arrangementsare thefollowing:ArTsistheempty arrangementin dimen-sion 0,ArHsandArH1sareisomorphictoaone-pointarrangementinC,ArpsandArqs

areisomorphictoatwo-linesarrangement inC2_.

• TheringL0

T “H˚pT ;Zqisanalternatingalgebrageneratedindegree1 by x1,x2.

• ThefreeZ-module

L1

H “H˚pH;Zq bH1pMpArHsq;Zq »H˚pH;Zq bH1pC˚;Zq

is generated in degree 1 by i˚px2q and in degree 2 by i˚px2q bw1, where i is the

inclusionH ãÑi T .Moreover,wehavei˚px1q “0.

Analogously,thefreeZ-moduleL1

H1»H˚pH1;Zq bH1pC˚;Zqisgeneratedindegree

1 by i1 ˚px2q and in degree 2 by i1 ˚px2q bw2, where i1 is the inclusion H1 i 1

ãÑ T .

Moreover, i1 ˚px1q “2i1 ˚px2q.

• Both L1

p and Lp1 are isomorphic to H2ppC˚q2;Zq, and are generated respectively

by wp₁wp₂ and ω “ wq₁wq₂ where w_ip (resp. w_iq) is the image in L_p˚ (resp. L_q˚) of the class wi with respect to the map H1pMpArHsq;Zq Ñ H1pMpArpsq;Zq (resp.

H1

pMpArH1sq;Z_{q Ñ}H1

pMpArqsq;Z_q)inducedbytheBrieskorndecomposition. HencethecohomologyringH˚pMpAq;Zqisanalternatingalgebrageneratedbyx1,x2, w1,w2 indegree1 andω indegree2,withrelations

x1w1“0, px1´2x2qw2“0, and ωxi“ωwi“0 for all i.

InparticularrkH1

pMpAq;Z_{q “}4,rkH2

pMpAq;Z_{q “}5.

2.3.2. The algebra BpAq

Definition 2.3.4. Let α be an element in the direct sum À_LPCLL. We say that α is

coherent ifforeveryintegerq and foreveryL PCąqwehavethat

ÿ L1_PpCďLqq

LL1ďLpαqL1q “αqL

whereαq_L (resp.αq_L1)is thecomponentofαL (resp.αL1)inL_Lq (resp.L_Lq1).

CoherentelementsinÀ_LPCL_Lq generate asubgroup, infacttheyformasubalgebra (seeProposition 5.2.7)thatwecallBpAq(see Definition 5.2.8).

TheoremB(see Proposition 5.2.10).The algebrasA_pA_qandB_pA_qareisomorphic.

2.3.3. Combinatorial aspects

Afterobtainingagrasponthecohomologyalgebraitisnatural,especiallyin compar-isonwith thecase ofhyperplane arrangements, to askthe questionof whether(andin whatsense)itiscombinatoriallydetermined.Themostnaturalcombinatorialstructure to consider in this context is of course the poset of layers C, both because this is the directcounterpart of the intersection poset of a hyperplane arrangement and because we already know it determines the Betti numbers and hence (by torsion-freeness) the cohomology groups. As an additional element of similarity, we provein §7.1 that just as in the caseof hyperplane arrangements the cohomology groups can be obtainedas theWhitneyhomologyofC.Whenthearrangementiscentered(i.e.,definedbykernels of characters), another associated structure is the arithmetic matroid of the defining characters [4]. Whilefor hyperplanearrangements thetwo counterparts – (semi)lattice offlatsand(semi)matroid – areequivalent combinatorialstructures,inoursituationit isstill truethatinthecenteredcase C determinesan arithmeticmatroid,butitis not knownat present howto construct C fromthe associated abstract arithmeticmatroid. Thusthequestionisthefollowing.

Question2.3.5.Istheisomorphismtypeoftheintegercohomologyringofthecomplement ofacomplexified toric arrangementdetermined bytheposetof layers?

Thestrongestaffirmativeresultwecanproveatthemomentisthatforcenteredtoric arrangements which possessa unimodular basisthe poset C does determine the coho-mologyalgebra.Indeed,inthiscasethearithmeticmatroiddeterminesthearrangement itself:ourTheorem 7.2.1showsthatifanarithmeticmatroidwithaunimodularbasisis representable,then therepresentationis uniqueuptosignreversalof thevectors.

WecannotatthismomentsolveQuestion 2.3.5inthegeneral(non-centered,without unimodular bases) case, and will close our work with an example that we hope will illustratesomeofthedelicacyofthesituation,namely:eveniftwocohomologyringsare isomorphic,thereneedsnotbea“natural”isomorphism.

Remark2.3.6.Inthefollowingsectionswewillconsideronlycomplexifiedtoric arrange-ments.Theextensionof ourresultstogeneral,non-complexified toricarrangementwill begiveninSection6.

2.4. Structure ofthepaper

GivenacomplexifiedtoricarrangementA (definedin§2.1),ourcombinatorialmodel forthe homotopytypeof thecomplementofA is thetoric SalvetticomplexSal_pA_q,in theformulationgivenin[8],inparticularasthenerveofanacycliccategoryobtainedas homotopycolimitofadiagramofposets.InSection3wereviewsomebasicfactsabout thecombinatoricsandtopologyofacycliccategories andestablishsomefactsaboutthe combinatorialtopologyofSalvetticomplexes ofcomplexifiedhyperplane arrangements. Inparticular,

(a) we identify maps between poset of cells of Salvetti complexes which induce the Brieskornisomorphisms(Proposition 3.3.3,whichwecalla“combinatorialBrieskorn Lemma”forcomplexifiedarrangements).

Thenextstepiscarried outinSection4,where

(b) for every connected component L of an intersection of elements of A we define a subcomplex

SLãÑSalpAq (3)

withthehomotopytypeoftheproductL ˆMpArLsq,whereArLsisthearrangement ofhyperplanesinCd_{definedbyA inthetangentspaceto pC}˚_qd_{atanygenericpoint}

inL andMpArLsq :“Cdz ŤArLs.

(c) Moreover, using (a) we can identify,and study at the level of cell complexes, the maps that are induced in cohomology by the inclusions(3) and between H˚pSLq

andH˚pSL1_qforL ĎL1.

Section5.1isdevotedtotheinspectionofthespectralsequence pEp,q

r forSalpAqcoming

fromtheformulationofthetoricSalvetticomplexasahomotopycolimit(seeSegal[33]) (whichisindeedequivalenttotheLerayspectralsequenceoftheinclusionofMpAqinto thetorus) and the(trivial) spectral sequencesLErp,q forSL comingfromprojectionon

thetorusfactor.These spectralsequencesalldegenerateat thesecond page.

(d) Themapofspectralsequencesinducedbytheinclusions(3)leadsustoconsiderthe followingcommutingdiagram(ofgroups).

H˚pSal_pAqq ÝÝÝÝÑ H˚pšLSLq § § 𠧧đ p E2p,q ÝÝÝÝÑ À L LE p,q 2 “ÀLH˚pLq bH˚pMpArLsqq

http://doc.rero.ch

AftersomepreparationinSection5.2,thegistofourproofisreachedinSection5.3, where we use the (explicit) bottom map (of groups) to prove injectivity and to characterize, via(c), the image of the top map (of rings).We do this by present-ing the image as an algebra ApAq obtained by defining “the natural product” on

À

LH˚pLq bHcodimLpMpArLsqq (Definition 2.3.2) as well as an algebra BpAq of

“coherentelements”ofÀ_LH˚pLq bH˚pMpArLsqq(Definition 2.3.4).

InSection6weextendourresultsto general(non-complexified)toric arrangements, using a deletion–restriction type argument which allows us to reduceto the complex-ified case. We then close with Section 7 where we investigate the dependency of the cohomology ringstructure fromthe posetC of connected componentsof intersections, tryingto identifysimilaritiesanddifferences withthecaseofhyperplane arrangements, wherethiscohomologystructureiscompletelydeterminedbytheposetofintersections. We will show that the cohomology groupsare, as in thehyperplane case, obtainedas Whitneyhomologyoftheintersectionposet(§7.1),andweprovethatC determines the cohomology ringof every toric arrangement which is definedas the set of kernels of a familyofcharacters whichcontainsat least oneunimodularbasis(Theorem 7.2.1). We conclude by givingtwo examples (§7.3) which illustrate the subtle relationship of the combinatorics of the poset of layers with the ring structure of the cohomology. First, wepresenttwo(centered)arrangementsin _pC˚_q2_{withisomorphicposetsoflayerswhich}

doindeedhaveisomorphiccohomology rings,butno‘natural’ isomorphismexists (i.e., noisomorphismwhichfixestheimageoftheinjectionsincohomologyobtainedfrom in-cludingthecomplementsintothefullcomplextorus).Last,wegiveanotherarrangement (alsoinrank2) which showsthat a“natural”conditionfor thecohomology ringto be generatedindegree1 isnotsufficient.

3. Preparations

3.1. Categoriesanddiagrams

Givenacategory C,wewill denoteby |C| thegeometricrealization ofthenerve ofC (inparticular,thisis apolyhedral complexinthesense of[19]).A functorF :C1ÑC2

induces a cellularmap _|F| between the geometric realizations. Wewill for brevity say thattwo categoriesare‘homotopyequivalent’ meaningthattheirnervesare.

A kind of categories of special interest for us are face categories of polyhedral com-plexes.We refer e.g.to [8, Section 3] for aprecise definitionand hereonly recall that theface category FpKq of apolyhedral complex K has the cellsof K as objects, and onemorphism P Ñ Q for every attachment of the polyhedral cell P to a face of the polyhedralcellQ.

Itisastandardfactthat,ifK isapolyhedralcomplex, |FpKq|canbeembeddedinto

K asitsbarycentricsubdivision(see[35] forathoroughinvestigationofthissituation).

Facecategoriesofpolyhedralcomplexesareexamplesofcategorieswheretheidentity morphismsaretheonlyinvertiblemorphisms,aswell astheonlyendomorphisms.Such categories are called scwols (for “small categories without loops”) in the terminology of[5]or“acycliccategories”,e.g.,in[19].

Itisstandardtoconsiderapartiallyorderedset pP, ďqasascwolP withsetofobjects Ob_pP_{q “}P and | Mor_Ppp,qq| ď1 for allp,qPP , with | Mor_Ppp,qq| “1 if and onlyif

p ďq:inthis casewe willsimplyspeakof“themorphismp ďq”.

Adiagram overacategoryI (whichinourcasewill alwaysbeascwol)isafunctor

D:IÑX

where,inthis paper,X canbe thecategoryTop of topologicalspaces,Scwol ofscwols, orthecategoriesofAbeliangroupsorZ-algebras.A morphismbetweendiagramsD1,D2

overthesameindexcategory I isafamilyα“ pαi :D1piq Ñ D2piqqiPObI ofmorphisms

of X thatcommutewithdiagram maps –thatis, suchthat,for everymorphismi Ñj

ofI, αj˝ D1pi Ñjq “ D2pi Ñjq ˝αi.

3.1.1. X “Top

There is an extensive literature on diagrams of spaces, in particular studying their homotopy colimits.Wecontent ourselveswith listing somefactswe’llhaveusefor and referto[37]or [19]foranintroductiontothesubjectandproofs.

Lemma 3.1.1. Let α be a morphism between twodiagrams D1,D2 over the same index categoryI.Ifeveryαiisahomotopyequivalence,thenα inducesahomotopyequivalence

of homotopycolimits

hocolimD1ÑhocolimD2.

Thatthere isacanonicalprojection

π : hocolimD Ñ |I|.

The Leray spectral sequence of this projection then can be used to compute the (co)homologyofthehomotopycolimit.Itisequivalent tothespectralsequence studied bySegal[33]and hassecond page

E2p,q“Hpp|I|,Hqpπ´1;Zqq ñH˚phocolimD;Zq.

3.1.2. X “Scwol

The topological spaces we will be studying will come with anatural combinatorial stratificationandcanthereforebe writtenasnervesofacycliccategories.Recallfrom[36, Definition 1.1]theGrothendieckconstructionşDassociatedtoadiagramD:I_ÑScwol. This is the category with object set consisting of all pairs _pi,xq with i P Ob_pI_q and

x P Ob_pDpiqq, and with morphisms pi1,x2q Ñ pi2,x2q corresponding to pairs pf,μq, f PMor_Ipi1,i2qandμ PMorDpi2qpfpi1q,i2q,composedintheobviousfashion.

Lemma3.1.2(Theorem 1.2of[36]).GivenadiagramD:I ÑScwol,wehaveanatural homotopyequivalence

hocolim_{|D| » |}şD|.

Remark3.1.3.Inthiscasethecanonicalprojectionofthehomotopycolimitonthenerve oftheindex categorybecomesthemap ofpolyhedralcomplexesinducedbytheevident functor

ş

D ÑI; pi, xq ÞÑi.

3.2. Arrangementsof hyperplanes

LetA bealocally finitearrangementof hyperplanesinCd_.Wewillwrite

MpAq:“CdzďA

foritscomplement.

Recall the definitionsof Section2.2 and, given X P LpAq, the arrangement AX :“ tH PA |X ĎHu(Equation(1)).If A iscomplexified,theassociated realarrangement

AR induces a polyhedral cellularization of Rd with poset of faces FpAq, ordered by inclusion, whose maximal elements (the maximal cells) are called chambers of A. We writeT_pA_qfortheset ofallchambersofA.

NoticethateveryG PFpAqiscontainedinaunique(relativelyopen)faceofAX,that

we denote by GX. One readilychecks thatthis defines aposet map FpAq ÑFpAXq,

sinceF1ěF2 implies pF1qX ě pF2qX.

3.2.1. Sign vectorsandoperationson faces

A standard way of dealing with such polyhedral subdivisions is by choosing a real defining form H for every H P A and thus defining H` :“ tx P Rd | Hpxq ą 0u,

H´ :“ tx PRd_|

Hpxq ă0u, H0 :“H.Eachface F isthenidentified byitssign vector

γF : A Ñ t`1, ´1,0u with γFpHq :“σ if and only ifF ĎHσ. If we consider the set t`1, ´1,0_u partiallyordered accordingto `1 ą 0, ´1 ą 0 and `,1, ´1 incomparable weseethat,withournotation,foranyF,G PFpAqwehave

FďG if and only if, for all HPA, γFpHq ďγFpHq.

Also,foreveryX PL we havethatγFX istherestrictionofγF toAX.Givenchambers

C1,C2PTpAq,recalltheset

SpC1, C2q “ tH PA|γC1pHq “ ´γC2pHq ‰0u

ofhyperplanesseparating C1 fromC2.

For F PFpAqwe letAF :“A_{|F |}, where |F| denotes theaffine spanof F . This will

thenmeanthatH PAF ifandonlyifγFpHq “0.

Definition 3.2.1. Given F,G P FpAq, we define GF P FpAq to be the face uniquely

determinedby pGFq|F |“G|F | andGF ěF .

Inparticular,there isaninclusion

iF :FpAFq ÑFpAq

whereintermsofsignvectorswehave

γGFpHq:“ # γFpHq if H RAF γGpHq if H PAF and γiFpGqpHq:“ # γFpHq if HRAF γGpHq if HPAF.

Thefollowingaresomepropertiesthatshow thattheaboveobjectsarewell-defined, andwhichwe listasalemmaforlater reference.Theirproofisastraightforwardcheck ofsignvectors.

Lemma3.2.2.

(1) piFpGqq|F |“G, hence iF mapsbijectivelyontoFpAqěF.

(2) IfG1ěG2PFpAFq,then iFpG1q ěiFpG2q.

(3) _pGF1qF2 “GF_1F2 forallG,F1,F2PFpAq.

(4) FG “F if F ěG.

Definition 3.2.3. Let A be a complexified real central arrangement, X Ă A and σ P t˘1,0u.Define

Δσ_pA; X_q:_{“ t}F PFpAq |γFpHq “σ if HPXu.

Fordisjoint(possiblyempty)subsetsN,Z,P ĎA definethen Δ_pA; N, Z, P_q:_“Δ´1_pA; N_{q X}Δ0_pA; Z_{q X}Δ`1_pA; P_q.

Wewillhaveoccasionaluseforthefollowingresult.

Lemma 3.2.4(See Proposition 4.3.6 of [3]).Let A be a nonempty, centralcomplexified arrangement.ThesubposetΔ_pA;N,Z,Pq,ifnot empty,iscontractible.

Proof. IfN “P “ H, theposetsunderconsiderationcontain theuniqueminimal ele-ment _XA thusare contractible.Otherwise, ourΔ_pA;N,Z,Pqcorresponds to theposet ofcellsintheinterioroftheshellablePL-balldenotedby Δ´_NYZXΔ`_PYZ in[3, Propo-sition 4.3.6.],andweconcludewiththegeneralfactthattheordercomplexoftheposet

P ofcellsof theinterior ofaPL-ballB is contractible. Thislast factcan be provedas follows:theordercomplexofP isthemaximalsubcomplexofallcellsofΔ_pF_pBqqthat are disjoint from the (full) subcomplex Δ_pF_pBqzPq (which triangulates the boundary of B).HenceΔpPqisaretractoftheinteriorof B and,assuch,contractible. l

3.3. Acombinatorial Brieskornlemma

ThedataofFpAqcanbeusedtoconstructaregularCW-complexduetoSalvetti[32]

whichembedsinMpAqasadeformationretract.ThiscomplexiscalledSalvetticomplex

ofA anddenotedSalpAq.Itsface category(in fact,a poset)SpAq :“FpSalpAqqcanbe describedasfollows:

SpAq “ trF, Cs PFpAq ˆTpAq |F ďC inFpAqu rF, Cs ě rF1, C1sif F ďF1, CF1_“C1.

Definition 3.3.1. From now in this section we will assume that the arrangement A is

central,i.e.,that _XA _{‰ H}. Then,lettingP :“ XA,thecomplexSal_pA_qcan be decom-posedasaunionof(combinatoriallyisomorphic)closedpolyhedralcellsofdimensiond,

correspondingtothepairs rP,CswithCPTpAq.Wedefinesubposets

SC:“SďrP,Cs for CPTpAq

correspondingtothefaces oftheclosureofthemaximalcells _rP,Cs.

Ournextgoalwillbeto offeracombinatorialversionofLemma 2.2.1,i.e.,toexpress Brieskorn’smap asinducedbyposetmaps betweenSalvetticomplexes.

Definition3.3.2.GivenX PLpAqdefine

bX :SpAq ÑSpAXq,rG, Cs ÞÑ rGX, CXs.

Moreover, for everyF PFpAq we have thefollowing naturalinclusion of posets, well-definedbyLemma 3.2.2.(2).

jF :SpAFq ÑSpAq, rG, Cs ÞÑ riFpGq, iFpCqs.

ThefollowingisthenacombinatorialversionofBrieskorn’sLemma.

Proposition 3.3.3 (Combinatorial Brieskorn Lemma). Let A be a central complexified arrangement of hyperplanes and for every X P LpAq choose an FpXq P FpAq with

|FpXq| “X.

The maps of posets jFpXq,bX induce an injective map b˚X : H˚pSalpAXqq Ñ

H˚pSalpAqq and a surjective map j_F˚_pXq : H˚pSalpAqq Ñ H˚pSalpAFqq, such that

j_F˚ ˝ˆbX“idH˚_pSalpAXqq. Thenforallk theinclusion

jkp“ \rkpXq“kjFpXqq: ğ

rkpXq“k

SpAFpXqq ÑSpAq

inducestheBrieskornisomorphism

pbkq´1: HkpSpAqq Ñ à rkpXq“k HkpSpAFpXqqq “ à rkpXq“k HkpSpAXqq.

Inparticular,themapinducedincohomologyby jFpXq doesnotdepend onthechoiceof

FpXqamong themaximalcells ofitsaffine span.

Proof. For thefirst partofthe claim, notice(e.g., by acheck ofsign vectors)thatthe compositionbX˝jFpXq istheidentity onSpAFpXqq.Forthesecondpartweprovethat,

infact,themapbX ishomotopic totheinclusionMpAq ĎMpAXq.

Firstofall,noticethattheradialmapρ: z ÞÑz{|z|definesahomotopybetweenthe inclusion MpAq ĎMpAXqand the inclusion SzA Ď SzAX, where S denotes theunit

sphereinCd »R2d.

Wefollow[30,Chapter 5]andconsiderthearrangementsA andAX asframedbythe

arrangement

H“A1YA2:“ tHˆRd|H PARu Y tRdˆH |H PARu

in R2d_{. This defines, as usual, a cellularization of R}2d _{with poset of faces F} pHq »

FpAq ˆFpAq (productof posets,see e.g.[34, Section 3.2])and, after barycentric sub-division, a triangulationTH of the sphere S realizing the order complex of FpHqztˆ0u. TheintersectionSXŤA (resp.SXŤAX)isafullsubcomplexNA(resp.NAX)ofTH, thus NAX Ď NA as a full subcomplex. Let MHpAq be the biggest subcomplex of TH whichisdisjoint toN_A,andsimilarly forM_HpAXq.Then,MHpAq ĎM_HpAXqisafull

subcomplex.

Itisastandardfact(seee.g.[28, Lemma 70.1])thatTHz ŤAX deformationretracts

onto MHpAXq (say, by a retraction fX) and MHpAXqzA deformation retracts onto

MHpAq (say,byf ).We thenhavethatthe inclusionMHpAq ĎMHpAXqis homotopic

totheoriginalinclusionMpAq ĎMpAXq.

NownoticethatthesimplicialcomplexesMHpAXq,MHpAqareinfactrealizationsof

theordercomplexesoftheposets

MHpAXq “ tpF, Gq PFpAq ˆFpAq |AFXAGXAX “ Hu

MHpAq “ tpF, Gq PFpAq ˆFpAq |AFXAG“ Hu

and the inclusion of complexes is induced by the inclusion of posets ι : M_HpAq Ñ

MHpAXq.Wesummarizebysayingthatthefollowingdiagram commutes upto

homo-topy MpAq ÝÝÝÝÝÑρ|MpAq THzNHpAq ÝÝÝÝÑfX MHpAXqzNHpAq f ÝÝÝÝÑ MHpAq Ď § § đ Ď § § đ Ď § § đ |ι| § § đ MpAXq ρ ÝÝÝÝÑ THzNHpAXq fX ÝÝÝÝÑ MHpAXq MHpAXq

Inorder tostudythemap |ι|further,itisenoughtoargueatthelevelofposets. In[30, Chapter 5]itis provedthatthemap

φ :MHpAq ÑSpAqop, pF, Gq ÞÑ rF, GFs

isahomotopyequivalence. Wedefine amap

ψ :M_HpAXq ÑSpAXqop, pF, Gq ÞÑ rFX,pGFqXs

sothat,bydefinition,thefollowing diagramcommutes:

MHpAq ÝÝÝÝÑφ SpAqop ι § § đ bX § § đ MHpAXq ψ ÝÝÝÝÑ SpAXq

Nowit is enoughto provethatψ isahomotopy equivalence, and wewill then have provedthatthegeometric map |bX|inducedbybX ishomotopicto |ι|, whichinturn is

homotopictotheinclusionMpAq ĎMpAXq.

To prove that ψ is a homotopy equivalence, consider some rF,Cs P SpAXqop and pF1,Gq PMHpAXqsuchthatψpF1,Gq ě rF,Cs inSpAXqop.Thisis thecaseexactlyif

F_X1 ěF inFpAqand pGF1qX “CpF1_qX.

Intermsofsignvectors,this willbeverifiedexactlyif: • γF1_pH_{q ě}γFpHqforallHPAX;

• γGpHq “γCpHqforallH PAXXAF1;

• AF1XAXXAG“ H,

wherethelastconditionjustensuresthatindeed _pF1,Gq PMHpAXq.

Going back, wesee that,under theisomorphism pFpAqq2»FpHqthese pF1,Gq are exactlythefaces pF ofH with

γFppHˆR

d

q “γFpHqfor H PAXzAF, γFppR d

ˆHq “γCpHqfor H PAX,

andthusψ´1pSpAXqěrF,CsqisasubposetofFpHqconsistingofallfaceswithprescribed

sign on a certain set of hyperplanes. This set is of the formΔ_pH;N,Z,Pq, nontrivial becauseX‰ˆ0 andnonemptybecauseitcontains _pF,Cq,andtheirordercomplexisthus contractiblebyLemma 3.2.4. l

Wenextproveapropositionwhichexpresses,inthelanguageofposets,thefactthat givenany faceF ofacentralarrangement,theunionofthecells rP,CswithC running

throughallchambers adjacent toF is asubcomplexofSalpAq homotopyequivalent to

MpAFq.

Definition3.3.4.LetA beacentral,complexifiedarrangementofhyperplanes,andwrite

P fortheminimalelementofF_pA_q.Wedefine asubposetofS_pA_qas

SF

pAq:“ ď CěF

SC

(whereweviewSC asasubposetofSpAqasinDefinition 3.3.1),considertherestriction

j0F :SpAFq ÑSFpAq

ofthemapjF of Definition 3.3.2anddefine

ξF :SFpAq ÑSpA|F |q, rG, Cs ÞÑ rG_{|F |}, C_{|F |}s.

Proposition3.3.5. Theposetmaps jF

0,ξF are homotopyinverse toeach other.

TheproofofthispropositionwillrestonsometechnicalfactsaboutthemapsξF that

weproveasseparatelemmas forlaterreference.

Lemma3.3.6. LetFPFpAq,C,G PFpA|F |qwith C achamber.Then

ξF´1prG, Csq “ trK, iFpCqKs PSC|K|F |“Gu.

Proof. Wefirstprovetheright-to-leftinclusion.ForK suchthatK|F |“G wehave

ξFprK, iFpCqKsq “ rK|F |,piFpCqKq|F |s “ rK_{|F |}, iFpCqK|F |s “ rK|F |, CGs “ rG, Cs

where in thesecond equalitywe used Lemma 3.2.2.(3) and inthe last equality simply thefactthatbydefinitionCěG.

Now to the left-to-right inclusion. ConsiderK PFpAq and R PTpAq with K ďR

andsuchthatξFprK,Rsq “ rG,Cs.ImmediatelybydefinitionwehaveKF “G,and we

areleftwithprovingthatR“iFpCqK. Forthat,wecheck thedefinitions.

– IfγKpHq ‰0,γiFpCqKpHq “γKpHqand,sinceRěK,γKpHq “γRpHq. – IfγKpHq “0, γiFpCqKpHq “ # γFpHq if γFpHq ‰0, γiFpCqpHq “γRpHq else, because C“R|F |.

It remains to see that γFpHq “ γRpHq when γKpHq “ 0 and γFpHq ‰ 0.

In-deed, since rK,Rs P SFpAq, it must be R “ pC1qK (hence γRpHq “ γC1_pH_qwhen γKpHq “0)forsomeC1 ěF (thusγC1pHq “γFpHqwheneverγFpHq ‰0). l

Corollary3.3.7.Forevery SPSpA|F |q,theposetξ_F´1pSqiscontractible.

Proof. TheexpressiongiveninLemma 3.3.6showsthatξ_F´1prG,Csqisposet-isomorphic totheorderdualofthesubposetofFpAqconsistingofallKPFpAqwithK|F |“G: in-deed,giventwosuchK1,K2withK1ěK2,then piFpCqK1qK2 “iFpCqK_1K2 “iFpCqK1,

hence rK1,iFpCqK1s ď rK2,iFpCqK2s,andthereverseimplicationistrivial.

Now,theKPFpAqwithK_{|F |}“G areexactlythoseinthesubposet

ΔpA, AFXγ_G´1p´1q,AFXγ_G´1p0q,AFXγ´1_G p`1qq,

whichisnonemptythuscontractiblebyLemma 3.2.4. l

Lemma 3.3.8. Let A be a central, complexified arrangement of hyperplanes. For every F PFpAqandevery rG,Cs PSpAFq,theposetξ´1_F pSpAFq_ďrG,Csqis contractible.

Proof. Consider an element rG,Cs PSpAFq (thus G ě F and C ě G in FpAq) and

considerthepreimageof

SpAFqďrG,Cs“ trG1, C1s|G1ěG, CG1 “C1u

withrespectto ξF.

ByLemma 3.3.6,thispreimageisthesubposetofSF _{consistingofelements} ď

G1ěG C1“C_G1

trK, CK1 s |K|F |“G1u “ trK, RKs |K|F |ěG, R“iFpCG1_{q “}iFpCK_{|F |}qu

which is isomorphic, as inthe proof of Corollary 3.3.7 to the subposet of F_pA_q given by

P :“ tKPFpAq |K_{|F |}ěGu “Δ_pA; AFXG´;H;AF XG`q,

whichisnonempty(itcontainse.g.iFpGq),hencecontractible byLemma 3.2.4. l

Proofof Proposition 3.3.5. ThecompositionξF ˝j0F equalsobviously theidentity. We

provethatjF

0 ˝ξF ishomotopictotheidentity onSF.Tothisend consider

α : ΔpSFq Ñ2|SF|, αpσq:_{“ |}ξ_F´1pSpAFqďmax ξFpσqq|.

Clearly,thecarriermapα carriestheidentity.Moreover, aneasycheck showsthat

pξF ˝j0F ˝ξFqpσq “ξFpσq

andthus

pjF

0 ˝ξFqpσq Ďξ´1pSpAFqďmax ξFpσqq, henceα carriesboth theidentityand jF

0 ˝ξF.Weconcludeby theCarrierLemma [24, Proposition II.9.2]andLemma 3.3.8. l

4. Combinatorialtopologyoftoricarrangements

4.1. The toric Salvetticategory

LetA beacomplexifiedtoricarrangement.OnewaytoobtainananalogueofSalvetti’s complexistonoticethatthecanonicalembeddingofSalpAæqintoMpAæqisequivariant withrespecttotheactionoftherank-d integerlatticeonCd_{asthegroupofdeck}

trans-formationsoftheuniversalcoverofT .Thisleadsustolookforaconvenientdescription of thequotient of the Salvetticomplex, as was firstdone in[27] inthe casewhere the resultingcomplexisagainsimplicial.Ingeneral,oneseesthatthisactionrestrictstoan actiononFpAæq, andthefacecategory FpAqisisomorphicto thequotient

FpAq –FpAæq{Zd, with covering functor Q :FpAæq ÑFpAq.

Thisactioninduces anactiononSpAæqvia

grF, Cs:_{“ r}gF, gCs (4)

for everyF,C PFpAæqwith C ěF , andhence to acellularaction onSalpAæq.In [9], takingadvantageofthegeneralityofacycliccategories,a descriptionofthequotient cat-egoryS_pAæ_q{Zd_{isgiven,togetherwiththeproofthatindeed |S}

pAæq{Zd| » |SpAæq|{Zd»

MpAq.

Fig. 2. Apictorialrepresentationofthenotationsintroducedhere.The“abstract”arrangementsArF s serve

asamodel forthe localizationsAæ

F æ,andthe mapim isinducedat theabstractlevelby theinclusion

definedbya(ny)liftofthemorphismm toFpAæ_q.

Infact,much ofthe notationof theprevious sectionhas been introducedin[9,8]in order todescribe the relationshipsinvolvedinthese covering morphisms. Forinstance, ifF PFpAq,thearrangement ArFs isan‘abstract’copyofeveryAæ_Fæ withQpFæq “F

for which we can choose linear forms according to those defining Aæ. Then, there are canonicalisomorphisms

FpAæqěFæ –FpAæ_Fæq “FpArFsq, SalpAæ_Fæq –SalpArFsq, (5)

givenbytheidenticalmappingofsignvectors.Ifm: F ÑG isamorphismofF_pA_q,then

ArGs “ArFsG0 whereG0istheintersectionofallhyperplanesofArFsthatcorrespond

tohypertoricontainingG.Moreover,foranychoiceofFæPQ´1pFqthereisexactlyone face Gæ suchthatQpFæ ďGæq “m: wecall Fm thecorrespondingface of ArFs under

theisomorphismofEquation(5)(noticethat |Fm| “G0).Following[9]wedefine

im:FpArGsq ãÑFpArFsq

tobethemapcorrespondingtotheinclusionFpAæqěGæ ĎFpAæq_ěFæ.Foranillustration

ofthesedefinitionsseeFig. 2.

Remark4.1.1.Intermsofsignvectors,themapim isdeterminedas follows:

γimpKqpHq:“ #

γFmpHq if HRArGs

γKpHq else.

In particular, if X is a flat of both A_rGs and of A_rFs, then impKqX “ KX for all

KPFpArGsq.

Theorder relationFæďGæ alsodefinesaninclusion

jrFæďGæs: Sal_pAæ_{Gæq ãÑ}Sal_pAæ_Fæq,

i.e., themapinduced oncomplexesby theinclusionjG of Definition 3.3.2withrespect

tothe“ambient”arrangementAæ_F.

Given a morphism m: F Ñ G ofFpAq, there is acorresponding inclusioninduced by im(seealso [8,Definition 5.9]),

jm:SpArGsq ãÑSpArFsq,rK, Cs ÞÑ rimpKq, impGqs.

Wecannow re-statethefollowingdefinitionfrom[8].

Definition4.1.2.LetA beacomplexifiedtoricarrangement,recallthenotationsof Def-inition 2.1.2 and, forF P FpAq, write ArFs :“Ar|F|s, where |F| denotes thesmallest dimensionallayerthatcontainsF .Defineadiagram

D: F_pA_qop

ÑScwol

F ÞÑSpArFsq

m : F ÑGÞÑjm:SpArGsq ãÑSpArFsq.

In[8]itis provedthat _| colimD|is homotopyequivalent toMpAq.For ourpurposes weneedtoproveanotherconnectionbetweenD andthehomotopytypeofMpAq. Theorem4.1.3. hocolim_{|D| »}MpAq.

Proof. Wewillprovethathocolim|D|ishomotopyequivalenttothequotientofSalpAæq

bytheinducedZd_-action.

Write Zd _{for the one-object category (we write ˚ for this object) representing the}

group.WethenwritethequotientSal_pAæ_qas thecolimitofthediagram

S :ZdÑTop ˚ ÞÑSalpAæq

gÞÑg : SalpAæq ÑSalpAæq

where theactionofsomegPZd onSal_pAæ_qisdescribedinEquation(4).

NownoticethatbyconstructionSalpAæqiscoveredbythesubcomplexesSalpAæFqand

isthusthecolimitof

E :F_pAæ_qop_ÑTop, F _ÞÑSal_pAæ_Fq, F ďGÞÑjrF ďGs: Sal_pAæ_{Gq Ñ}Sal_pAæ_Fq,

where jrF ďGs denotes the map inducedon complexes bythe inclusionjG of Defini-tion 3.3.2withrespecttothe“ambient”arrangementAæ_F.Considernowthepush-forward

ˆ

E of E along the functor Q : F_pAæ_{q Ñ} F_pA_q(and thus with colimE “colim ˆE). We havethefollowing explicitform.

ˆ E :F_pA_qop_ÑTop F ÞÑ ğ Fæ_PQ´1_{pF q} SalpAæFæq m : F ÑGÞÑ \pFæ_ďGæ_qPQ´1_pmqj_rFæ_ďGæ_s:_pS, Gæ_{q ÞÑ p}j_rFæ_ďGæ_spS_q, Fæ_q.

Noticethat,for gPZd and anymorphism FæďGæ ofthe(poset-) category FpAæq, we haveg˝jrFæ ď Gæs “ jrgpFæ ďGæqs, where we write g forthe automorphism of

FpAæq definedbyg.Withthis, thefollowingdiagram describingtheaction ofZd _onEˆ iswell-defined.

ˆ

G :ZdˆFpAqopÑTop p˚, Fq ÞÑE pˆFq pg, m : F ÑGq ÞÑg˝E pˆmq.

Claim1.colim_FpAqG “ Sˆ .

Proof. Wechecktheequalitypointwise.Onobject(s)wehave

pcolim_FpAqG qp˚q “ˆ colim ˆE “colimE “Sal_pAæ_q.

The morphism pcolim_FpAqG qpˆ gq : pcolim_FpAqG qp˚q Ñ pˆ colim_FpAqG qp˚qˆ is induced by thenaturaltransformationG pˆg,id_´q : ˆG p˚, ´q ñG p˚ˆ , ´qwhichactsoverobjectsas

ˆ

G pg, idFæq: ˆG p˚, Fq ÑG p˚ˆ , Fq,

pS, Fæq ÞÑ pg˝idFqpS, Fæq “ pgpSq, gpFæqq.

(6)

NowonechecksexplicitlythatthisinducesthemapS ÞÑgpSqoncolim ˆE “Sal_pAæ_q, asrequired. l

Claim2.colim_ZdG – |D|ˆ .

Proof. Again we can verify the isomorphism pointwise using the fact that preimages under the functor Q are exactly orbits of the action of Zd _{on F}

pAæq. For every F P

ObpFpAqq,withEquation(6)we have

colim_ZdG p´ˆ , Fq “E pˆFq{Zd–SalpArFsq – |DpFq|,

where the last two congruence symbols denote isomorphismof complexes and homeo-morphism(bybarycentricsubdivision),and onmorphisms m PMorpFpAqq:

colim_ZdG p´ˆ , mq “E pˆmq{Zd– |jm|. l

WiththesetwoclaimstheTheorem willfollow,because thenwehave Sal_pAæ_q{G“colim_ZdS “colim ˆG

“colim_FpAqcolim_ZdG “ˆ colim|D| »hocolim|D|,

where the last homotopy equivalence is given by the Projection Lemma [37, Proposi-tion 3.1]. l

Corollary4.1.4. MpAq » | şD|.

Proof. ImmediatewiththeTheoremaboveandLemma 3.1.2. l

This prompt us to deviate from the conventions of [8] and to define the Salvetti complexofacomplexified toricarrangementasfollows.

Definition4.1.5.ForeverycomplexifiedtoricarrangementA let

SpAq:_“şD; Sal_pA_q:_{“ |}S_pA_q|.

Remark4.1.6.Wehaveimmediately SalpAq »MpAq. Moreover,with Remark 3.1.3we haveacellularmap

π : SalpAq Ñ |FpAq| »Tc

inducedbythecanonicalprojectionfromhocolim_|D|.

4.2. Inclusions

Thegoalofthissection willbe toassociateto everylayerL asubcomplexofSal_pA_q homotopy equivalent to the product of L times the complement of the (hyperplane) arrangement A_rLs. Wewill do this ina way thatis compatible with theprojection to the compact torus and so that the maps induced in cohomology by the inclusions of these subcomplexes satisfy aBrieskorn-type compatibilitycondition which will be the steppingstonetowardsapresentationof thecohomology algebra.

Definition 4.2.1.Given alayer L PC we write Lc forthe intersectionL XTc of L with

thecompacttorus.

Definition4.2.2.LetAL _{denotethecomplexifiedtoricarrangement definedinthetorus}

L by allhypertorinotcontainingL,thatis thearrangementof allhypertoriappearing asaconnectedcomponentofanintersectionL XK forKPA, L ĘK.

Noticethatthecellularization _|F_pA_q|ofTcrestrictstoacellularization |FpALq|ofLc

(i.e.,thereisacellularhomeomorphismh: TcÑ |FpAq|with |FpALq| “ |FpAq| XhpLq).

Theorem 4.2.3. For every layer L P C and every chamber B0 P TpA0q adjacent to L0:“ XArLs PLpA0q, letF0:“B0XL0. Then, there isasubcomplex SF0 of thetoric Salvetticomplex Sal_pA_qsatisfying

(1) under thecanonicalprojection SalpAq ÑTc,SF0 mapstothelayerLc;

(2) there isahomotopyequivalenceΘF0 :SF0 Ñ |FpALq| ˆSalpArLsq;

(3) the firstcomponent of ΘF0 istheprojection from(1).

Wekeepthenotationsofthetheorem’sclaim(F0,B0,L0)splittheproofinmultiple

stepsforeasierunderstandingandlaterreference.

Definition4.2.4.RecallDefinition 2.1.2:F0isafaceofA0 andwecan define BpF0q:“ tCPTpA0q |CěF0inFpA0qu,

theset of all chambers of A0 thatare adjacent to F0. Moreover, forevery F P FpALq

define

μF :TpA0q ÑTpArFsq; μFpCq ĚC.

Lemma4.2.5.

(a) For everymorphismm: F ÑG inFpAq andeveryCPTpA0q, SpμFpCq, impμGpCqqq XArGs “ H.

(b) ForallF PFpALqwe haveF0PFpArFsq,and

μFpBpF0qq “ tCPTpArFsq |CěF0 inFpArFsqu.

Proof. For part (a) notice that C Ď μFpCq Ď μGpCq, thus for H P ArGs clearly

γμGpCqpHq “ γμFpCqpHq, and moreover with Remark 4.1.1 we have γimpμGpCqqpHq “

γμGpCqpHq whenever H P ArGs. For part (b) notice first that μF maps chambers to chambers,thusitis enoughtocheck thatforCPBpF0qwe haveμFpCq ěF0. Butthe

definitionofμF isthatγμFpCqpHq “γCpHqforallH PArFs,thusF0ďC inA0implies

F0ďμFpCqinArFs. l

Definition4.2.6.Wenowdefinethefollowing diagram:

DF0 :FpA L qopÑScwol F ÞÑ ď BPBpF0q SpArFsqμFpBq

andthemapsaredefinedasrestrictionsofthecorrespondingmaps ofthediagramD.

Lemma4.2.7. Thediagram iswell-defined,and DF0pFq “S

F0_pArFsq.

Proof. Thediagramis well-definedbecause,byLemma 4.2.5.(a)and[8,Remark 5.13],for everym: F ÑG inFpALqandeveryCPTpA0qtheinclusionjm:SpArGsq ÑSpArFsq

restrictstoaninclusionSμFpCqÑSμGpCq(compare[8,Lemma 5.12]).Thesecondclaim followsfromLemma 4.2.5.(b). l

Definition4.2.8.Define

SF0 :“ |

ş

DF0|.

Remark4.2.9.SinceşDF0 isasubcategoryof

ş

D,SF0 isasubcomplexofSalpAq.

Notation 4.2.10. We will form now on use a ‘column’ notation for the Grothendieck construction. For a diagram D : I _Ñ Scwol we will write “x_i‰ for the object of şD associatedto i PObI andx PObDpiq,and

“x1 i1 ‰ ” μf ı ÝÝÝÑ“xi22 ‰

forthemorphismcorrespondingto f PMor_Ipi1,i2qandμ PMorDpi2qpDpfqpx1q,x2q.

Lemma 4.2.11. The canonical projection π : SalpAq Ñ |FpAq| restricts to πL : SF0 Ñ

|FpALq|.

Proof. Thisisacheck ofthedefinitions,e.g.withRemark 4.2.9. l

Definition4.2.12.LetKL be theconstantdiagram

KL :FpALqopÑScwol

F ÞÑSpArLsq

mÞÑid.

Definition4.2.13.GivenF PFpALqlet

ξrFsF0 :S

F0_pA_{rFsq Ñ}S_pA_rFs

|F0|“Lq “SpArLsq

denote the map described in Definition 3.3.4 referred to the ‘ambient’ arrange-mentArFs.

Lemma4.2.14.The mapsξrFsF0 of Definition 4.2.13induceanatural transformation DF0 ñ KL

andthusafunctor

ΞF0 :

ş

DF0 Ñ

ş

KL

which,moreover, induceshomotopyequivalenceof nerves.

Proof. Inordertocheck thatthediagram

DF0pFq ξrF s_F0 SpArLsq DF0pGq jm ξrGs_F0 SpArLsq “

commutesitisenoughtoseethat,foreveryKPFpArGsq,impKq|F0|“K|F0|,asisproved

inRemark 4.1.1. Thus, themaps ξrFsF0 induceawell-defined naturaltransformation,

andtherebytherequiredfunctorΞF0,actingonobjects

“_K F ‰

andmorphisms“_mě‰(notice thateveryDF0pFqisindeedaposet),as

ΞF0 “_K F ‰ “ ” ξrF s_F0pKq F ı , ΞF0 ”_ě m ı “ ”_ě m ı .

Since each map ξrFsF0 is a homotopy equivalence (by Lemma 3.3.8 via Quillen’s

Theorem A [31]), the homotopy theorem [37, Proposition 2.3] ensures that the natu-raltransformation induceshomotopyequivalencebetweenhomotopy colimits,thusalso betweentheGrothendieckconstructions,as claimed. l

Weconsidernervesassimplicialsets,andthusdenotecellsinthegeometricrealization ofacategory bythecorrespondingchainofmorphisms.

Lemma4.2.15.Acell ofSF0 has theform

σ“ ” D0 G0 ı ” ě m1 ı ÝÝÝÝÑ ” D1 G1 ı ¨ ¨ ¨ ” Dk Gk ı

where Gi P ObpFpAlqq, mi : Gi´1 Ñ Gi P MorpFpALqq, Di P SpArGisq and Di´1 ě

jmipDiq.

Thenthefunctionmappingσ to σÞÑ pG0 m1 ÝÑG1¨ ¨ ¨ mk ÝÑGk,ξ{F0D0ě. . .ě {ξF0Dkq

http://doc.rero.ch

(whereforDPSpArGisq wewrite pD :“jmi˝¨¨¨˝m1pDiq PSpArG0sq) inducesahomotopy equivalenceΘF0 :SF0 Ñ |FpALq| ˆ |SpArLsq|.

Proof. Notice that there is an evident equivalence of categories şKL – FpALqopˆ

SpArLsq–thusweseethat

ΘF0 “ΣL˝ |ΞF0|,

thecompositionofthehomotopyequivalenceinducedbyΞF0andthecanonical

(“reverse-subdivision”,seee.g.[19,4.2.2])homeomorphismΣL:|FpALqopˆSpArLsq| Ñ |FpALq| ˆ |SpArLsq|. l

Definition 4.2.16.Wenow fix foreverylayer L a face F0“F0pLq ofA0 with |F0| “L

andachamberB0ofA0 adjacenttoF0,anddefine

DL:“ DF0, SL :“SF0, ΘL:“ΘF0, ΞL:“ΞF0.

WecallϕL theinclusionmapSLãÑSalpAqfromTheorem 4.2.3.

Ournextgoalisthefollowingtheorem,whichwilljustifytheideaofcoherentelement giveninDefinition 2.3.4.

Theorem4.2.17. Fix aninteger q andlet L bealayer with rk_pLq ąq. Considertheset

pCďLqq of all thelayers L1 such that L ĎL1 andq “rkpL1q. The followingdiagram of

groupsiscommutative. H˚pSalpAq;Zq ϕ˚_L À L1PpCďLqq ϕ˚ L1 À L1_PpCďLqqH q_pMpArL1_{sq;Zq bH}˚_pL1_;Zq ř L1PpCďLqq LpL1ďLq Hq pMpArLsq;Z_{q b}H˚pL;Zq

Lemma4.2.18.The map

CL:Fp|FpALq| ˆ |SpArLsq|q Ñ | ş

DL|, xÞÑΘ´1L pxq

isacontractiblecarrier map(inthesense of [24,Chapter II]).

Proof. Letusconsider acellx asintheclaim, say

x“ |pFk mk´1

ÝÑ Fk´1¨ ¨ ¨ÝÑm1 F1q| ˆ |pSlě ¨ ¨ ¨ ěS1q|.