Arc Operads and Arc Algebras

Transcription

1 ISSN (on line) (prined) 5 Geomery & Topology Volume 7 (003) Pulihed: 8 Augu 003 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Ar Operd nd Ar Alger Rlph M Kufmnn Muriel Liverne RC Penner RMK: Oklhom Se Univeriy, Sillwer, USA nd Mx Plnk Iniu für Mhemik, Bonn, Germny ML: Univerié Pri 3, Frne RCP: Univeriy of Souhern Cliforni, Lo Angele, USA Emil: rlphk@mpim-onn.mpg.de, liverne@mh.univ-pri3.fr rpenner@mh.u.edu Ar Severl opologil nd homologil operd ed on fmilie of projeively weighed r in ounded urfe re inrodued nd udied. The pe underlying he i operd re idenified wih open ue of ominoril ompifiion due o Penner of pe loely reled o Riemnn moduli pe. Alger over hee operd re hown o e Blin Vilkoviky lger, where he enire BV ruure i relized impliilly. Furhermore, our i operd onin he i operd up o homoopy. New operd ruure on he irle re lified nd omined wih he i operd o produe geomerilly nurl exenion of he lgeri ruure of BV lger, whih re lo ompued. AMS Clifiion numer Primry: 3G5 Seondry: 8D50, 7BXX, 83E30 Keyword Moduli of Surfe, Operd, Blin Vilkoviky lger Propoed: Shigeyuki Mori Reeived: 6 Deemer 00 Seonded: Rlph Cohen, Vughn Jone Aeped: 8 June 003 Geomery & Topology Puliion

2 5 Rlph M Kufmnn, Muriel Liverne nd R C Penner Inroduion Wih he rie of ring heory nd i ompnion opologil nd onforml field heorie, here h een very fruiful inerion eween phyil ide nd opology. One of i mnifeion i he renewed inere nd reerh iviy in operd nd moduli pe, whih h unovered mny new invrin, ruure nd mny unexpeed reul. The preen reie dd o hi generl undernding. The ring poin i Riemnn moduli pe, whih provide he i operd underlying onforml field heory. I h een exenively udied nd give rie o Blin Vilkoviky ruure nd grviy lger. Uully one i lo inereed in ompifiion or pril ompifiion of hi pe whih led o oher ype of heorie, for inne, he Deligne Mumford ompifiion orrepond o ohomologil field heorie re preen in Gromov Wien heory nd qunum ohomology. There re oher ype of ompifiion however h re eqully inereing like he new ompifiion of Loev nd Mnin [0] nd he ominoril ompifiion of Penner []. The ler ompifiion, whih we will employ in hi pper, i for inne well uied o hndle mrix heory nd onruive field heory. The ominoril nure of he onruion h he gre enefi h ll lulion n e mde explii nd ofen hve very nie geomeri deripion. The ominoril ruure of hi pe i deried y omplexe of emedded weighed r on urfe, upon whih we hll ommen furher elow. Thee omplexe hemelve re of exreme inere hey provide very generl deripion of urfe wih exr ruure whih n e een ing vi oordim. The urfe devoid of r give he uul oordim defining opologil field heorie y Aiyh nd Segl. Tking he r omplexe he i oje, we onru everl operd nd uoperd ou of hem, y peilizing he generl ruure o more reried one, herey onrolling he informion rried y he oordim. The mo inereing one of hee i he yli operd ARC h orrepond o urfe wih weighed r uh h ll oundry omponen hve r emning from hem. We will ll lger over he orreponding homology operd Ar lger. The impoed ondiion orrepond o open ue of he ominoril ompifiion whih onin he operor h yield he Blin Vilkoviky ruure. Moreover, hi ruure i very nurl nd i opologilly explii in len nd euiful wy. Thi operd in genu zero wih no punure n lo e emedded ino n operd in rirry genu

3 Ar Operd nd Ar Alger 53 wih n rirry numer of punure wih he help of genu generor nd punure operor, whih ulimely hould provide good onrol for ilizion. A numer of operd of urren inere [, 6, 6, ] re governed y ompoiion whih, in effe, depend upon omining fmilie of r in urfe. Our operd ompoiion on ARC depend upon n explii mehod of omining fmilie of weighed r in urfe, h i, eh omponen r of he fmily i igned poiive rel numer; we nex riefly derie hi ompoiion. Suppoe h F,F re urfe wih diinguihed oundry omponen F, F. Suppoe furher h eh urfe ome equipped wih properly emedded fmily of r, nd le i,...,i p i denoe he r in F i whih re iniden on i, for i =,, illured in pr I of figure. Idenify wih o produe urfe F. We wih o furhermore omine he r fmilie in F,F o produe orreponding r fmily in F, nd here i evidenly no well-defined wy o hieve hi wihou mking furher hoie or impoing furher ondiion on he r fmilie (uh p = p ). Figure : Gluing nd from weighed r: I, r fmilie in wo urfe, II, omining nd. Our ddiionl d required for gluing i given y n ignmen of one rel numer, weigh, o eh r in eh of he r fmilie. The weigh w i j on i j i inerpreed geomerilly he heigh of rengulr nd Ri j = [0,] [ wj i/,wi j /] whoe ore [0,] {0} i idenified wih i j, for i =, nd j =,...,p i. We hll ume h p j= w j = p j= w j for impliiy, o h he ol heigh of ll he nd iniden on gree wih h of ; in ligh of hi umpion, he nd in F n e enily hed long o he nd in F long o produe olleion of nd in he urfe F illured in pr II of figure ; noie h he horizonl edge of he

4 54 Rlph M Kufmnn, Muriel Liverne nd R C Penner rengle {Rj }p deompoe he rengle {Rj }p ino u-rengle nd onverely. The reuling fmily of u-nd, in urn, deermine weighed fmily of r in F, one r for eh u-nd wih weigh given y he widh of he u-nd; hu, he weighed r fmily in F o produed depend upon he weigh in non-rivil u ominorilly explii wy. Thi derie he i of our gluing operion on fmilie of weighed r, whih i derived from he heory of rin rk nd pril meured foliion [4]. In f, he implifying umpion h he ol heigh gree i ovied y onidering no weighed fmilie of r, u rher weighed fmilie of r modulo he nurl overll homohei ion of R >0 o-lled projeively weighed r fmilie; given projeively weighed r fmilie, we my de-projeivize in order o rrnge h he implifying umpion i in fore (uming h p 0 p ), perform he onruion ju deried, nd finlly re-projeivize. We hll prove in eion h hi onruion indue well-defined operd ompoiion on uile le of projeively weighed r. The reuling operd ruure provide ueful frmework i evidened y he f h here i n emedding of he u operd of Voronov [6] ino hi operd uoperd, we diovered. Thi how how o view he Ch Sullivn [] ring opology, whih w he inpirion for our nlyi nd o whih eion owe oviou inelleul griude, inide he ominoril model. In priulr, we reover in hi wy urfe deripion of he i whih in urn n e viewed reduion of he urfe ruure in very preie wy. In f, here i reduion of ny urfe wih r o onfigurion in he plne, whih i no neerily u nd my hve muh more omplex ruure. For he urfe whoe reduion i u in he ene of [6], one rein he nurl ion on loop pe y forgeing ome of he inernl opologil ruure of he urfe u keeping n eenil pr of he informion rried y he r. For more generl onfigurion, imilr u more omplied ruure i expeed. One virue of viewing he i urfe ined of ingulr level e i h in hi wy he rnhing ehvior i niely depied while he ingulriie of one dimenionl Feynmn grph do no pper Wien h poined ou mny ime. Furhermore, here re nurl uoperd governing he Gerenher nd he BV ruure, where he ler uoperd i genered y he former nd he -ry operion of ARC. On he level of homology, hee operion ju dd one l, h of he BV operor. For i, he iuion i muh more omplied nd i given y i roed produ [7]. I i onjeured [] h he full r omplex of urfe wih oundry i pheril nd hu will no rry muh operdi informion on he homologil

5 Ar Operd nd Ar Alger 55 level. The pheriiy of he full ompifiion n gin e ompred o he Deligne Mumford iuion where he ompifiion in he genu zero e leve no odd ohomology nd in Kozul dul wy i omplemenry o he grviy ruure of he open moduli pe. In generl he ide of oining uoperd y impoing erin ondiion n e een prllel o he philoophy of Gonhrov Mnin [5]. The Spheriiy Conjeure [] would provide he i for lulion of he homology of he operd under oniderion here, k h will e underken in [9]. One more virue of hving onree urfe i h we n lo hndle ddiionl ruure on he oje of he orreponding oordim egory, viz. he oundrie. We inorpore hee ide in he form of dire nd emi dire produ of our operd wih operd ed on irle. Thee give geomerilly nurl exenion of he lgeri ruure of BV lger. Thi view i lo inheren in [7] where he i operd re deompoed ino i roed produ. The operd uil on irle whih we onider hve he geomeri mening of mrking ddiionl poin in he oundry of urfe. Their lgeri onruion, however, i no linked o heir priulr preenion u rher relie on he lgeri ruure of S monoid. They re herefore lo of independen lgeri inere. The pper i orgnized follow: In eion we review he lien feure of he ominoril ompifiion of moduli pe nd he Spheriiy Conjeure. Uing hi kground, we define he operdi produ on urfe wih weighed r h underlie ll of our onruion. In eion we unover he Gerenher nd Blin Vilkoviky ruure of our operd on he level of hin wih explii hin homoopie. Thee hin homoopie mnifely how he ymmery of hee equion. The BV operor i given y he unique, up o homoopy, yle of r fmilie on he ylinder. Seion 3 i devoed o he inluion of he i operd ino he r operd oh in i originl verion [, 6] well in i pinele verion [7]. Thi explii mp, lled frming, lo how h he pinele i govern he Gerenher ruure while he Voronov i yield BV; hi f n lo e red off from he explii lulion of eion for he repeive uoperd of ARC, whih re lo defined in hi eion. The differene eween he wo u operd re operion orreponding o Fenhel Nielen ype deformion of he ylinder, i.e., he -ry operion of ARC. A generl onruion whih forge he opologil ruure of he underlying urfe nd rein olleion of prmeerized loop in he plne wih inidene/ngeny ondiion died y he r fmilie i onined in eion 4. Uing hi pril forgeful mp we define he ion of uoperd of ARC on loop pe of mnifold. For he priulr uoperd

6 56 Rlph M Kufmnn, Muriel Liverne nd R C Penner orreponding o i, hi operion i invere o he frming of eion 3. In eion 5 we define everl dire nd emi dire produ of our operd wih yli nd non yli operd uil on irle. One of hee produ yield for inne dgbv lger. The operd uil on irle whih we uilize re provided y our nlyi of hi ype of operd in he Appendix. The pproh i lifiion of ll operion h re liner nd lol in he oordine of he omponen o e glued. The reul re no oningen on he priulr hoie of S, u only on erin lgeri properie like eing monoid nd re hu of more generl nure. Aknowledgmen The fir uhor wihe o hnk he IHÉS in Bure ur Yvee nd he Mx Plnk Iniu für Mhemik in Bonn, where ome of he work w rried ou, for heir hopiliy. The eond uhor would like o hnk USC for i hopiliy. The finnil uppor of he NSF under grn DMS# i lo knowledged y he fir uhor i he uppor of NATO y he eond uhor. The operd of weighed r fmilie in urfe I i he purpoe of hi eion o define our i opologil operd. The ide of our operd ompoiion on projeively weighed r fmilie w lredy deried in he Inroduion. The ehnil deil of hi re omewh involved, however, nd we nex riefly urvey he meril in hi eion. To egin, we define weighed r fmilie in urfe ogeher wih everl differen geomeri model of he ommon underlying ominoril ruure. The olleion of projeively weighed r fmilie in fixed urfe i found o dmi he nurl ruure of impliil omplex, whih deend o CW deompoiion of he quoien of hi impliil omplex y he ion of he pure mpping l group of he urfe. Thi CW omplex hd rien in Penner erlier work, nd we diu i relionhip o Riemnn moduli pe in n exended remrk. The diuion in he Inroduion of nd whoe heigh re given y weigh on n r fmily i formlized y Thuron heory of pril meured foliion, nd we nex riefly rell he lien deil of hi heory. The pe underlying our opologil operd re hen inrodued; in effe, we onider r fmilie o h here i le one r in he fmily whih i iniden on ny given oundry omponen. In hi eing, we define olleion of r meure pe, one uh pe for eh oundry omponen of he urfe, oied o eh pproprie projeively weighed r fmily.

7 Ar Operd nd Ar Alger 57 We nex define he operd ompoiion of hee projeively weighed r fmilie whih w kehed in he Inroduion nd prove h hi ompoiion i well-defined. Our i operd re hen defined, nd everl exenion of he onruion re finlly diued.. Weighed r fmilie.. Definiion Le F = F g,r e fixed oriened opologil urfe of genu g 0 wih 0 punure nd r oundry omponen, where 6g 7 + 4r + 0, o he oundry F of F i neerily non-empy y hypohei. Fix n enumerion,,..., r of he oundry omponen of F one nd for ll. In eh oundry omponen i of F, hooe one nd for ll loed r W i i, lled window, for eh i =,,...,r. The pure mpping l group PMC = PMC(F) i he group of ioopy le of ll orienion-preerving homeomorphim of F whih fix eh i W i poinwie (nd fix eh W i ewie), for eh i =,,...,r. Define n eenil r in F o e n emedded ph in F whoe endpoin lie mong he window, where we demnd h i no ioopi rel endpoin o ph lying in F. Two r re id o e prllel if here i n ioopy eween hem whih fixe eh i W i poinwie (nd fixe eh W i ewie) for i =,,...,r. An r fmily in F i he ioopy l of n unordered olleion of dijoinly emedded eenil r in F, no wo of whih re prllel. Thu, here i well-defined ion of P M C on r fmilie... Severl model for r We hll ee h olleion of r fmilie in fixed urfe led o nurl impliil omplex, n open ue of whih will form he opologil pe underlying our i operd. Before urning o hi diuion, hough, le u riefly nlyze he role of he window in he definiion ove nd derie everl geomeri model of he ommon underlying ominori of r fmilie in ounded urfe in order o pu hi role ino perpeive. For he fir uh model, le u hooe diinguihed poin d i i, for i =,,...,r, nd onider he pe of ll omplee finie-re meri on

8 58 Rlph M Kufmnn, Muriel Liverne nd R C Penner u v w u v w u v w u v w I II III IV Figure : I, r running o poin on he oundry, II, r running o poin infiniy, III, r in window, IV, nd in window F of onn Gu urvure (o-lled hyperoli meri ) o h eh i = i {d i } i olly geodei (o-lled qui hyperoli meri ) on F. To explin hi, onider hyperoli meri wih geodei oundry on one-punured nnulu A nd he imple geodei r in i ympoi in oh direion o he punure; he indued meri on omponen of A give model for he qui hyperoli ruure on F = F {d i } r ner i. The fir geomeri model for n r fmily α in F i e of dijoinly emedded geodei in F, eh omponen of whih i ympoi in oh direion o ome diinguihed poin d i ; ee pr II of figure. In he homoopy l of eh i, here i unique geodei i F. Exiing from F { i }r ny omponen whih onin poin of, we oin hyperoli ruure on he urfe F F wih geodei oundry (where in he peil e of n nnulu, F ollpe o irle). Tking α F, we find olleion of geodei r onneing oundry omponen (where in he peil e of he nnulu, we find wo poin in he irle). Thi i our eond geomeri model for r fmilie. We my furhermore hooe diinguihed poin p i i nd regulr neighorhood U i of p i in i, for i =,,...,r. Provided p i / α, we my ke U i uffiienly mll h U i α =, o he r V i = i U i form nurl window onining α i. There i hen n mien ioopy of F whih hrink eh window V i down o mll r W i i, under whih α i rnpored o fmily of (non-geodei) r wih endpoin in he window W i. In e p i doe lie in α, hen le u imply move p i mll moun in he direion of he nurl orienion ( oundry omponen of F ) long i nd perform he me onruion; ee pr III of figure. Thi led o our finl geomeri model of r fmilie, nmely, he model

9 Ar Operd nd Ar Alger 59 defined in.. for he purpoe of hi pper, of r in ounded urfe wih endpoin in window. Thi hird model i in he piri of rin rk nd meured foliion (f. [4]) we hll ee nd i mo onvenien for deriing he operdi ruure...3 The r omplex Le u induively uild impliil omplex Ar = Ar (F) follow. For eh ingleon r fmily in F, here i diin verex of Ar (F). Hving hu induively onrued he (k )-keleon of Ar for k, le u dd k-implex σ(α ) o Ar for eh r fmily α in F of rdinliy k+. The impliil ruure on σ(α ) ielf i he nurl one, where fe orrepond o u-r fmilie of α, nd we my herefore idenify he proper fe of σ(α ) wih implie in he (k )-keleon of Ar. Adjoining k-implie in hi mnner for eh uh r fmily α of rdinliy k + define he k-keleon of Ar. Thi omplee he induive definiion of he impliil omplex Ar. Ar i hu impliil omplex, upon whih PMC = PMC(F) oninuouly, nd we define he quoien opologil pe o e Ar = Ar(F) = Ar (F)/PMC(F). If α = { 0,,..., k } Ar, hen he r i ome in nonil liner ordering. Nmely, he orienion of F indue n orienion on eh window, nd rvering he window in he order W,W,...,W r in hee orienion, one fir enouner n endpoin of he r 0,,..., k in ome order, whih prerie he limed liner ordering. Thu, ell in Ar nno hve finie ioropy in PMC, nd he impliil deompoiion of Ar deend o CW deompoiion of Ar ielf...4 Spheriiy Conjeure [] Fix ny urfe F = F g,r wih 6g 7 + 4r + 0. Then Ar(F g,r) i pieewie-linerly homeomorphi o phere of dimenion 6g 7 + 4r +. Reen work eem o provide proof of hi onjeure le in he e g = 0 of plnr urfe will e ken up elewhere. I i worh emphizing h he urren pper i independen of he Spheriiy Conjeure, u hi give ueful perpeive on he relionhip eween Riemnn moduli pe nd he operd udied here...5 Remrk I i he purpoe of hi remrk o explin he geomery underlying Ar(F) whih w unovered in [, 3]. Rell he diinguihed

10 50 Rlph M Kufmnn, Muriel Liverne nd R C Penner poin p i i hoen in our eond geomeri model for r fmilie. The moduli pe M = M(F) of he urfe F wih oundry i he olleion of ll omplee finie-re meri of onn Gu urvure - wih geodei oundry, ogeher wih diinguihed poin p i in eh oundry omponen, modulo puh forwrd y diffeomorphim. There i nurl ion of R + on M y imulneouly ling eh of he hyperoli lengh of he geodei oundry omponen, nd we le M/R + denoe he quoien. The min reul of [3] i h M/R + i proper homoopy equivlen o he omplemen of odimenion-wo uomplex Ar (F) of Ar(F), where Ar (F) orrepond o r fmilie α o h ome omponen of F α i oher hn polygon or one-punured polygon. I i remrkle o ugge h y dding o pe homoopy equivlen o M/R + uile impliil omplex we oin phere. There i muh known [] ou he geomery nd ominori of Ar(F) nd Ar (F). Furhermore, he Spheriiy Conjeure hould e ueful in luling he homologil operd of he r operd...6 Noion We hll lwy dop he noion h if α i he PMC-ori of n r fmily in F, hen α Ar denoe ome hoen r fmily repreening α. Furhermore, i will omeime e onvenien o peify he omponen 0,,..., k, for k 0, of n r fmily α repreening α Ar, nd we hll wrie imply α = { 0,,..., k } in hi e. Sine he liner ordering on omponen of α Ar i invrin under he ion of PMC, i deend o well-defined liner ordering on he omponen underlying he P M C-ori α. Of oure, k-dimenionl ell in he CW deompoiion of Ar i deermined y he PMC-ori of ome r fmily α = { 0,,..., k } Ar, nd (gin relying on he PMC-invrin liner ordering diued previouly) poin in he inerior of hi ell i deermined y he projeive l of orreponding (k + )-uple (w 0,w,...,w k ) of poiive rel numer. A uul, we hll le [w 0 : w : : w k ] denoe ffine oordine on he projeive le of orreponding non-negive rel (k + )-uple, nd if w i = 0, for i in proper ue I {0,,,k}, hen he poin of Ar orreponding o (α,[w 0 : w : : w k ]) i idenified wih he poin in he ell orreponding o { j α : j / I} wih projeive uple goen y deleing ll zero enrie of [w 0 : w : : w k ]. To remline he noion, if α = { 0,,..., k } i in Ar nd if (w 0,w,..., w k ) R k+ + i n ignmen of numer, lled weigh, one weigh o eh

11 Ar Operd nd Ar Alger 5 omponen of α, hen we hll omeime uppre he weigh y leing (α ) denoe he r fmily α wih orreponding weigh w(α ) = (w 0,w,...,w k ). In he me mnner, he projeivized weigh on he omponen of α will omeime e uppreed, nd we le [α ] denoe he r fmily α wih projeive weigh w[α ] = [w 0 : w :... : w k ]. Thee me noion will e ued for PMC-le of r fmilie well, o, for inne, (α) denoe he PMCori α wih weigh w(α) R k+ +, nd poin in Ar my e denoed imply [α] Ar(F), where he projeive weigh i given y w[α].. Pril meured foliion Anoher poin of view on elemen of Ar, whih i ueful in he uequen onruion, i derived from he heory of rin rk (f. [4]) follow. If α = { 0,,..., k } Ar i given weigh (w 0,w,...,w k ) R k+ +, hen we my regrd w i rnvere meure on i, for eh i = 0,,...,k o deermine meured rin rk wih op nd orreponding pril meured foliion, onidered in [4]. For he onveniene of he reder, we nex riefly rell he lien nd elemenry feure of hi onruion. Chooe for he purpoe of hi diuion ny omplee Riemnnin meri ρ of finie re on F, uppoe h eh i i mooh for ρ, nd onider for eh i he nd B i in F oniing of ll poin wihin ρ-dine w i of i. If i i neery, le he meri ρ o λρ, for λ >, o gurnee h hee nd re emedded, pirwie dijoin in F, nd hve heir endpoin lying mong he window. The nd B i ou i ome equipped wih foliion y he r prllel o i whih re fixed ρ-dine o i, nd hi foliion ome equipped wih rnvere meure inheried from ρ; hu, B i i regrded rengle of widh w i nd ome irrelevn lengh, for i = 0,,...,k. The folied nd rnverely meured nd B i, for i = 0,,..., k omine o give pril meured foliion of F, h i, foliion of loed ue of F upporing n invrin rnvere meure (f. [4]). The ioopy l in F rel F of hi pril meured foliion i independen of he hoie of meri ρ. Coninuing o uppre he hoie of meri ρ, for eh i =,,...,r, onider i ( k j=0 B j), whih i empy if α doe no mee i nd i oherwie olleion of loed inervl in W i wih dijoin inerior. Collpe o poin eh omponen omplemenry o he inerior of hee inervl in W i o oin n inervl, whih we hll denoe i (α ). Eh uh inervl i (α ) inheri n oluely oninuou meure µ i from he rnvere meure on he nd.

12 5 Rlph M Kufmnn, Muriel Liverne nd R C Penner.. Definiion Given wo repreenive (α ) nd (α ) of he me weighed PMC-ori (α), he repeive meure pe ( i (α ),µi ) nd ( i (α ),µi ) re nonilly idenified, whih llow u o onider he meure pe ( i (α),µ i ) of weighed PMC-ori (α) ielf, whih i lled he i h end of (α), for eh i =,,...,r..3 The pe underlying he opologil operd.3. Definiion An r fmily α in F i id o e exhuive if for eh oundry omponen i, for i =,,...,r, here i le one omponen r in α wih i endpoin in he window W i. Likewie, PMC-ori α of r fmilie i id o e exhuive if ome (h i, ny) repreening r fmily α i o. Define he opologil pe Ar g (n) = {[α] Ar(F g,n+ ) : α i exhuive}, Ãr g (n) = Ar g (n) (S ) n+, whih re he pe h omprie our vriou fmilie of opologil operd. In ligh of Remrk..5, Ar g(n) i idenified wih n open upe of Ar(F) properly onining pe homoopy equivlen o M(F g,n+ )/S, nd Ar(F) i pheril in le he plnr e. Enumering he oundry omponen of F 0,,..., n one nd for ll (where 0 will ply peil role in he uequen diuion) nd leing S p denoe he p h ymmeri group, here i nurl S n+ -ion on he leling of oundry omponen whih reri o nurl S n -ion on he oundry omponen leled {,,...,n}. Thu, S n nd S n+ on Ar g(n), nd exending y he digonl ion of S n+ on (S ) n+, he ymmeri group S n nd S n+ likewie on Ãr g (n), where S n y definiion rivilly on he fir oordine in (S ) n+. Coninuing wih he definiion, if [α] ( 0,,..., n ) Ãr g(n), le u hooe orreponding deprojeivizion (α), fix ome oundry omponen i, for i = 0,,...,n, nd le m i = µ i ( i (α)) denoe he ol meure. There i hen unique orienion-preerving mpping (α) i : i (α) S () whih mp he (l of he) fir poin of i (α) in he orienion of he window W i o 0 S, where he meure on he domin i π m i µ i nd on he rnge

13 Ar Operd nd Ar Alger 53 i he Hr meure on S. Of oure, eh (α) i i injeive on he inerior of i (α) while ( (α) i ) ({0}) = ( i (α) ). Furhermore, if (α ) nd (α ) re wo differen deprojeivizion of ommon projeive l, hen (α ) i ( (α ) i ) exend oninuouly o he ideniy on S..3. Remrk We do no wih o peilize o one or noher priulr e hi ge, u if F ome equipped wih n priori oluely oninuou meure (for inne, if F ome equipped wih nonil oordinizion, or if F or F ome equipped wih fixed Riemnnin meri), hen we n idenify i (α) wih i F, for eh i = 0,,...,n, in he oviou mnner (where, if he priori ol meure of i i M i, hen we ler he Riemnnin meri ρ employed in he definiion of he nd o h he µ i ol meure m i = µ i ( i (α)) gree wih M i for eh i = 0,,...,n nd finlly ollpe he endpoin of i (α) o ingle poin). Thu, he ide i h he i h end of (α) give model or oordinizion for i, u of oure, lering he underlying (α) in urn ler he oordinizion (α) i of i. The rivil produ Ãr g(n) = Ar g(n) (S ) n+ led in hi wy o fmily of oordinizion of F whih re wied y Ar g (n)..3.3 Pioril repreenion of r fmilie A explined efore, here re everl wy in whih o imgine weighed r fmilie ner he oundry. They re illured in figure. I i lo onvenien, o view r ner oundry omponen oleed ino ingle wide nd y ollping o poin eh inervl in he window omplemenry o he nd; hi inervl model i illured in figure 3, pr I. I i lo omeime onvenien o furher ke he imge under he mp o produe he irle model i depied in figure 3, pr II. w u v w u v I II Figure 3: I, nd ending on n inervl, II, nd ending on irle

14 54 Rlph M Kufmnn, Muriel Liverne nd R C Penner.4 Gluing r fmilie.4. Gluing weighed r fmilie Given wo weighed r fmilie (α ) in Fg,m+ nd (β ) in Fh,n+ o h µ i ( i (α )) = µ 0 ( 0 (β )), for ome i m, we hll nex mke hoie o define weighed r fmily in F + g+h,m+n follow. Fir of ll, le 0,,..., m denoe he oundry omponen of Fg,m+, le 0,,..., n denoe he oundry omponen of Fh,n+, nd fix ome index i m. Eh oundry omponen inheri n orienion in he ndrd mnner from he orienion of he urfe, nd we my hooe ny orienion-preerving homeomorphim ξ : i S nd η : 0 S eh of whih mp he iniil poin of he repeive window o he e-poin 0 S. Gluing ogeher i nd 0 y idenifying x S wih y S if ξ(x) = η(y) produe pe X homeomorphi o F + g+h,m+n, where he wo urve i nd 0 re hu idenified o ingle epring urve in X. There i no nurl hoie of homeomorphim of X wih F + g+h,m+n, u here re nonil inluion j : Fg,m+ X nd k : F h,n+ X. We enumere he oundry omponen of X in he order 0,,..., i,,,... n, i+, i+,... m nd re-index leing j, for j = 0,,...,m + n, denoe he oundry omponen of X in hi order. Likewie, fir enumere he punure of Fg,m+ in order nd hen hoe of F h,n+ o deermine n enumerion of hoe of X, if ny. Le u hooe n orienion-preerving homeomorphim H : X F + g+h,m+n whih preerve he leling of he oundry omponen well hoe of he punure, if ny. In order o define he required weighed r fmily, onider he pril meured foliion G in Fg,m+ nd H in F h,n+ orreponding repeively o (α ) nd (β ). By our umpion h µ i ( i (α )) = µ 0 ( 0 (β )), we my produe orreponding pril meured foliion F in X y idenifying he poin x i (α ) nd y 0 (β ) if (α) i (x) = (β) 0 (y). The reuling pril meured foliion F my hve imple loed urve leve whih we mu imply dird o produe ye noher pril meured foliion F in X. The leve of F hu run eween oundry omponen of X nd herefore, in he previou eion, deompoe ino olleion of nd B i of ome widh w i, for i =,,... I. Chooe lef of F in eh uh nd B i nd oie o i he weigh w i given y he widh of B i o deermine weighed r fmily

15 Ar Operd nd Ar Alger 55 (δ ) in X whih i evidenly exhuive. Le (γ ) = H(δ ) denoe he imge in under H of hi weighed r fmily. F + g+h,m+n.4. Lemm The PMC(F + g+h,m+n )-ori of (γ ) i well-defined (α ) vrie over PMC(F g,m+ )-ori of weighed r fmilie in F g,m+ nd (β ) vrie over PMC(F h,n+ )-ori of weighed r fmilie in F h,n+. Proof Suppoe we re given weighed r fmilie (α ) = φ(α ), for φ PMC(Fg,m+ ), nd (β ) = ψ(β ), for ψ PMC(F h,n+ ), well pir H l : X l F + g+h,m+n of homeomorphim ove ogeher wih he pir j,j : Fg,m+ X l nd k,k : Fh,n+ X l of indued inluion, for l =,. Le F l, F l denoe he pril meured foliion nd le (δ l ) nd (γ l ) denoe he orreponding weighed r fmilie in X l nd F + g+h,m+n, repeively, onrued ove from (α l ) nd (β l ), for l =,. Le l = j l ( 0 ) = k l ( i ) X l, nd remove uulr neighorhood U l of l in X l o oin he uurfe X l = X l U l, for l =,. Ioope j l,k l off of U l in he nurl wy o produe inluion j l : F g,m+ X l nd k l : F h,n+ X l wih dijoin imge, for l =,. φ indue homeomorphim Φ : X X uppored on j (F g,m+ ) o h j φ = Φ j, nd ψ indue homeomorphim Ψ : X X uppored on k (F h,n+ ) o h k ψ = Ψ k. Beue of heir dijoin uppor, Φ nd Ψ omine o give homeomorphim G : X X o h j φ = G j nd k ψ = G k. We my exend G y ny uile homeomorphim U U o produe homeomorphim G : X X. By onruion nd fer uile ioopy, G mp F X o F X, nd here i power τ of Dehn wi long uppored on he inerior of U o h K = τ G lo mp F U o F U. K hu mp F o F nd hene (δ ) o (δ ). I follow h he homeomorphim H K H : F + g+h,m+n F + g+h,m+n mp (γ ) o (γ ), o (γ ) nd (γ ) re indeed in he me PMC(F + g+h,m+n )-ori..4.3 Remrk I i worh emphizing gin h, owing o he dependene upon weigh, he r in γ re no imply deermined ju from he r in α nd β ; he r in γ depend upon he weigh. I i lo worh poining ou h he ompoiion ju deried i no well-defined on projeively weighed r fmilie u only on pure mpping l ori of uh. In f, y mking hoie of ndrd model for urfe well ndrd inluion of

16 56 Rlph M Kufmnn, Muriel Liverne nd R C Penner hee ndrd model, one n lif he ompoiion o he level of projeively weighed r fmilie..4.4 Remrk We imply dird imple loed urve omponen whih my rie in our onruion, nd J L Lody h propoed inluding hem here in nlogy o [6]. In f, hey nurlly give rie o he onjugy l of n elemen of he rel group ring of he fundmenl group of F..4.5 Definiion Given [α] Ar g (m) nd [β] Ar h (n) nd n index i m, le u hooe repeive deprojeivizion (α ) nd (β ) nd wrie he weigh w(α ) = (u 0,u,...,u m ), w(β ) = (v 0,v,...,v n ). Define ρ 0 = v i, ρ i = { β: 0 } { α: i } u i, where in eh um he weigh re ken wih mulipliiy, e.g., if h oh endpoin 0, hen here re wo orreponding erm in ρ 0. Sine oh r fmilie re exhuive, ρ i 0 ρ 0, nd we my re-le ρ 0 w(α ) = (ρ 0 u 0,ρ 0 u,...,ρ 0 u m ), ρ i w(β ) = (ρ i v 0,ρ i v,...,ρ i v n ), o h he 0 h enry of ρ i w(β ) gree wih he i h enry of ρ 0 w(α ). Thu, we my pply he ompoiion of.4. o he re-led r fmilie o produe orreponding weighed r fmily (γ ) in F + g+h,m+n, whoe projeive l i denoed [γ] Ar + g+h (m + n ). We le in order o define he ompoiion [α] i [β] = [γ], i : Ar g(m) Ar h (n) Ar+ g+h (m + n ), for ny i =,,...,m. A in Remrk.4.3, hi ompoiion i no impliil mp u ju opologil one.

17 Ar Operd nd Ar Alger A pioril repreenion of he gluing A grphil repreenion of he gluing n e found in figure 4, where we preen he gluing in hree of he differen model. More exmple of gluing in he inervl nd irle model n e found in figure 5 nd hroughou he ler eion. p q p q p p q p q = q q p q I II III Figure 4: The gluing: I, in he inervl piure, II, in he window wih nd piure nd III, in he r running o mrked poin verion..5 The i opologil operd.5. Definiion For eh n 0, le ARC p (n) = Ar 0 0 (n) (where he p nd for omp plnr), nd furhermore, define he dire limi ARC(n) of Ar g (n) g, under he nurl inluion..5. Theorem The ompoiion i of Definiion.4.5 imue he olleion of pe ARC p (n) wih he ruure of opologil operd under he nurl S n ion on lel on he oundry omponen. The operd h uni ARC p () given y he l of n r in he ylinder meeing oh oundry omponen, nd he operd i yli for he nurl S n+ ion. Proof The fir emen follow from he ndrd operdi mnipulion. The eond emen i immedie from he definiion of ompoiion. For he hird emen, noie h he ompoiion re he wo urfe ymmerilly, nd he xiom for yliiy gin follow from he ndrd operdi mnipulion. In preiely he me wy, we hve he following heorem.

18 58 Rlph M Kufmnn, Muriel Liverne nd R C Penner.5.3 Theorem The ompoiion i of Definiion.4.5 indue ompoiion i on ARC(n) whih imue hi olleion of pe wih he nurl ruure of yli opologil operd wih uni..5.4 The deprojeivized pe DARC For he following i i onvenien o inrodue deprojeivized r fmilie. Thi moun o dding for R >0 for he overll le. Le DARC(n) = ARC(n) R >0 e he pe of weighed r fmilie; i i ler h DARC(n) i homoopy equivlen o ARC(n). A he definiion.4.5 of gluing w oined y lifing o weighed r fmilie nd hen projeing k, we n promoe he ompoiion o he level of he pe DARC(n). Thi endow he pe DARC(n) wih ruure of yli operd well. Moreover, y onruion he wo operdi ruure re ompile. Thi ype of ompoiion n e ompred o he ompoiion of loop, where uh reling i lo inheren. In our e, however, he ling i performed on oh ide whih render he operd yli. In hi onex, he ol weigh given oundry omponen given y he um of he individul weigh w of iniden r mke ene, nd hu he mp n e nurlly viewed mp o irle of rdiu w..6 Noion We denoe he operd on he olleion of pe ARC(n) y ARC nd he operd on he olleion of pe DARC(n) y DARC. By n Ar lger, we men n lger over he homology operd of ARC. Likewie, ARC p nd DARC p re ompried of he pe ARC p (n) nd DARC p (n) repeively, nd n Ar p lger i n lger over he homology of ARC p.6. Remrk In f, he reriion h he r fmilie under oniderion mu e exhuive n e relxed in everl wy wih he idenil definiion reined for he ompoiion. For inne, if he projeively weighed r fmily [α] Ar g(m) fil o mee he i h oundry omponen nd [β] Ar h (n), hen we n e [α] i [β] o e [α] regrded n r fmily in Fg,m+ F + g+h,m+n, where he oundry omponen hve een re-leled. In oh formulion, we mu require h 0 [β] in order h [α] i [β] i non-empy, u hi ymmeri remen deroy yliiy. Anoher poiiliy i o inlude he empy r fmily in he operd ny r fmily wih ll weigh zero o preerve yliiy. The ompoiion hen imue

19 Ar Operd nd Ar Alger 59 he deprojeivized Ar(Fg,n+ ) hemelve wih he ruure of opologil operd, u he orreponding homology operd, in ligh of he Spheriiy Conjeure..4 would e rivil i.e., he rivil one-dimenionl S n module for eh n..6. Turning on punure nd genu When llowing genu or he numer of punure o e differen from zero, here re wo operor for he opologil operd h genere r fmilie of ll gener repeively wih ny numer of punure from r fmilie of genu zero wih no punure. Thee operor re depied in figure 5. I II Figure 5: I, he genu generor, II, he punure operor For he orreponding liner homology operd, we expe imilriie wih [0], in whih he punure ply he role of he eond e of poin. Thi men h he liner operd will on enor power of wo liner pe, one for he oundry omponen nd one for he punure. In hi exenion of he operdi frmework, here i no gluing on he punure nd he repeive liner pe hould e regrded prmeerizing deformion whih perur eh operion eprely..6.3 Oher Prop nd Operd There re reled u-operd nd u-prop of Ar(Fg,n+ ) oher hn ARC p nd ARC whih re of inere. In he generl e, one my peify ymmeri (n + )-y-(n + ) mrix A (n) well n (n + )-veor R (n) of zeroe nd one over Z/Z nd onider he upe of Ar(Fg,n+ ) where r re llowed o run eween oundry omponen i nd j if nd only if A (n) ij 0 nd re required o mee oundry omponen k if nd only if R (n) k 0. For inne, he e of inere in hi pper orrepond o A (n) he mrix nd R (n) he veor whoe enrie re ll one. In Remrk.6., we lo menioned he exmple wih A (n) he mrix oniing enirely of enrie one nd R (n) he ndrd fir uni i veor.

20 530 Rlph M Kufmnn, Muriel Liverne nd R C Penner For l of exmple, onider priion of {0,,... n} = I (n) O (n), ino inpu nd oupu, where A (n) ij = if nd only if {i,j} I (n) nd {i,j} O (n) re eh ingleon, nd R (n) i he veor whoe enrie re ll one. The orreponding prop i preumly reled o he ring prop of []. There re mny oher inereing poiiliie. For inne, he i operd of [6] nd he pinele i of [7] will e diued nd udied in eion nd 3, nd oher reled u-like exmple re udied in [7]. A furher vriion i o rify he pe y inidene mrie wih non negive ineger enrie orreponding o he numer of r eh oundry omponen. The Gerenher nd BV ruure of ARC In hi eion we will how h here i ruure of Blin-Vilkoviky lger on Ar lger nd Ar p lger. More preiely, ny lger over he ingulr hin omplex operd C (ARC p ) or C (ARC) i BV lger up o erin hin homoopie. For onveniene, C (n) will denoe C (ARC p (n)) or C (ARC(n)), ine for our lulion, we only require ARC p. An elemen in C will e lo lled n r fmily. We hll relize he underlying urfe F0,n+ 0 in he plne wih he oundry 0 eing he ouide irle. The onvenion we ue for he drwing i h ll irle re oriened ounerlokwie. Furhermore if here i only one mrked poin on he irle, i i he eginning of he window. If here re wo mrked poin on he oundry, he window i he mller r eween he wo. Vi gluing, ny r fmily in C (n) give rie o n n ry operion on r fmilie in C. Here one h o e reful wih he prmeerizion. To e ompleely explii we will lwy inlude hem if we ue priulr fmily n operion. A menioned in he Inroduion, mny of he lulion of hi eion re inpired y [].. A reminder on ome lgeri ruure In hi eion, we would like o rell ome i definiion of lger nd heir relion whih we will employ in he following. The proof in hi ueion re omied, ine hey re righforwrd ompuion. They n e found in [3, 4].

21 Ar Operd nd Ar Alger 53.. Definiion (Gerenher) A pre Lie lger i Z/Z grded veor pe V ogeher wih iliner operion h ifie (x y) z x (y z) = ( ) y z ((x z) y x (z y)) Here x denoe he Z/Z degree of x... Definiion An odd Lie lger i Z/Z grded veor pe V ogeher wih iliner operion {, } whih ifie () {, } = ( ) ( +)( +) {, } () {, {, }} = {{, }, } + ( ) ( +)( +) {, {, }}..3 Remrk Pre Lie lger of he ove ype re omeime lo lled righ ymmeri lger. There i lo he noion of lef ymmeri lger, whih ifie (x y) z x (y z) = ±(y x) z y (x z) Given lef ymmeri lger i oppoie lger i righ ymmeri nd vie ver. Here he mulipliion for he oppoie lger A opp i opp =. Hene i i mer of e, whih lger ype one regrd. To mh wih ring opology, one h o ue lef ymmeri lger, while o mh wih he Hohhild ohin, one will hve o ue righ ymmeri lger...4 Definiion A Gerenher lger or n odd Poion lger i Z/Z grded, grded ommuive oiive lger (A, ) endowed wih n odd Lie lger ruure {, } whih ifie he ompiiliy equion {, } = {, } + ( ) ( +) {, }..5 Propoiion (Gerenher) For ny pre-lie lger V, he rke {, } defined y {, } := ( ) ( +)( +) () endow V wih ruure of odd Lie lger...6 Remrk The me hold rue for righ ymmeri lger.

22 53 Rlph M Kufmnn, Muriel Liverne nd R C Penner..7 Definiion A Blin Vilkoviky (BV) lger i n oiive uper ommuive lger A ogeher wih n operor of degree h ifie = 0 () = () + ( ) () + ( ) () () ( ) () ( ) + () Here uper ommuive men uul Z/Z grded ommuive, i.e. = ( )...8 Propoiion (Gezler) For ny BV lger (A, ) define {, } := ( ) () ( ) () () (3) Then (A, {, }) i Gerenher lger...9 Definiion We ll riple (A, {, }, ) GBV lger if (A, ) i BV lger nd {, } : A A A ifie he equion (3). By he Propoiion ove (A, {, }) i Gerenher lger. The purpoe of hi definiion i h in ome e in he e of he Ar operd i hppen h rke well he BV operor pper nurlly. In our e he rke ome nurlly from pre Lie ruure whih one n lo view produ, while he BV operor pper from yle nurlly prmeerized y S = Ar(F 0 0, = ARC p(). We ue he nme GBV lger o indie h hee ruure lhough hving independen origin re indeed ompile. A we will how elow, he independen origin n e inerpreed ying h he Gerenher ruure i governed y one uoperd nd he BV ruure y noher igger uoperd whih onin he previou one. Moreover he rke of he BV ruure oinide wih he rke of he Gerenher ruure lredy preen in he mller uoperd.. Ar fmilie nd heir indued operion The poin in ARC p () re prmeerized y he irle, whih i idenified wih [0, ], where 0 i idenified o. To derie prmeerized fmily of weighed r, we hll peify weigh h depend upon he prmeer [0,]. Thu, y king [0,] figure 6 derie yle δ C () h pn H (ARC p ()).

23 Ar Operd nd Ar Alger 533 = I II Figure 6: I, he ideniy nd II, he r fmily δ yielding he BV operor A ed ove, here i n operion oied o he fmily δ. For inne, if F i ny r fmily F : k ARC p, δf i he fmily prmeerized y I k ARC p wih he mp given y he piure y inering F ino he poiion. By definiion, = δ C (). In C () we hve he i fmilie depied in figure 7 whih in urn yield operion on C. The do produ The r δ (,) Figure 7: The inry operion To fix he ign, we fix he prmeerizion we will ue for he glued fmilie follow: y he fmilie F,F re prmeerized y F : k ARC p nd F : k ARC p nd I = [0,]. Then F F i he fmily prmeerized y k k ARC p defined y figure 7 (i.e., he r fmily F inered in oundry nd he r fmily F inered in oundry ). Inerhnging lel nd nd uing hin homoopy in figure 8 yield he ommuiviy of up o hin homoopy d(f F ) = ( ) F F F F F F (4)

24 534 Rlph M Kufmnn, Muriel Liverne nd R C Penner Noie h he produ i oiive up o hin homoopy. Likewie F F i defined o e he operion given y he eond fmily of figure 7 wih I = [0,] prmeerized over k I k ARC p. By inerhnging he lel, we n produe yle {F,F } hown in figure 8 where now he whole fmily i prmeerized y k I k ARC p. {F,F } := F F ( ) ( F +)( F +) F F. * {,} ( +)( +) ( ) * Figure 8: The definiion of he Gerenher rke.. Definiion We hve defined he following elemen in C : δ nd = δ in C (); in C 0 (), whih i ommuive nd oiive up o oundry. nd {, } in C () wih d( ) = τ nd {, } = τ. Noe h δ, nd {, } re yle, where i no... Remrk We would like o poin ou h he ymol in he ndrd uper noion of odd Lie rke { }, whih i igned o hve n inrini degree of, orrepond geomerilly in our iuion o he one dimenionl inervl I.

25 Ar Operd nd Ar Alger The BV operor The operion orreponding o he r fmily δ i eily een o qure o zero in homology. I i herefore differenil nd nurl ndide for derivion or higher order differenil operor. I i eily heked h i i no derivion, u i i BV operor, we hll demonre. I i onvenien o inrodue he fmily of operion on r fmilie δ whih re defined y figure 9, where he fmilie re prmeerized over I k k n. δ() δ (,) δ (,,) n δ (,,, ) n Figure 9: The definiion of he n ry operion δ We noie he following relion whih re he rion d êre for hi definiion: δ() δ(,) + ( ) δ(,) δ() δ(,,) + ( ) ( + ) δ(,,) (5) δ( n ) +( ) ( + ) δ(,,) n ( ) σ(i,) δ( i (),..., i (n)) (6) i=0

26 536 Rlph M Kufmnn, Muriel Liverne nd R C Penner where i he yli permuion (,...,n) nd σ( i,) i he ndrd uperign of he permuion. The homoopy here i ju reprmeerizion of he vrile I. There i furher relion immedie from he definiion whih how h he only new operion i δ(,) δ(,,..., n ) δ(, 3 n ) (7) where we ue homoopy o le ll weigh of he nd no hiing he oundry o he vlue..3. Lemm δ(,,) ( ) ( +) δ(,) + δ(,) δ() (8) Proof The proof i onined in figure 0. Le : k ARC p, : k ARC p nd : k ARC p, e r fmilie hen he wo prmeer fmily filling he qure i prmeerized over I I k k k. Thi fmily give u he deired hin homoopy..3. Propoiion The operor ifie he relion of BV operor up o hin homoopy. 0 () () + ( ) () + ( ) () () ( ) () ( ) + () (9) Thu, ny Ar lger nd ny Ar p lger i BV lger. Proof The proof follow lgerilly from Lemm.3. nd equion (5). We n lo mke he hin homoopy explii. Thi h he dvnge of illuring he ymmeri nure of hi relion in C direly. Given r fmilie : k ARC p, : k ARC p nd : k ARC p, we define he wo prmeer fmily defined y he figure where he fmilie in he rengle re he depied wo prmeer fmilie prmeerized over I I k k k nd he ringle i no filled, u rher i oundry i he operion δ().

27 Ar Operd nd Ar Alger 537 ( +) ( ) δ () = δ() ( ) ( ) ( )( ) δ (,) ( +) ( ) δ(,) δ (,,) Figure 0: The i hin homoopy reponile for BV From he digrm we ge he hin homoopy oniing of hree, nd repeively welve, erm. δ() δ(,,) + ( ) ( + ) δ(,,) + ( ) ( + ) δ(,,) ( ) ( +) δ(,) + δ(,) δ() + ( ) δ(,) +( ) δ(,) ( ) δ() + ( ) ( + ) δ(,) +( ) ( + )+ δ(,) ( ) + δ() δ() + ( ) δ() + ( ) + δ() δ() ( ) δ() ( ) + δ() (0)

28 538 Rlph M Kufmnn, Muriel Liverne nd R C Penner ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) Figure : The homoopy BV equion.4 Gerenher Sruure We hve lredy defined he operion whoe odd ommuor i given y he BV operor..4. Theorem The Gerenher rke indued y i given y he operion {,} = ( ) ( +)( +) In oher word ARC i GBV Alger up o homoopy.

29 Ar Operd nd Ar Alger 539 ( +)( +) ( ) * δ ( ) δ () ( ) δ * Figure : The odd ommuor relizion of he rke Proof We onider he r fmily depied in figure, where he wo prmeer r fmilie re prmeerized vi k T k ARC p, where T i he ringle T := {(,) [0,] [0,] : + } wih indued orienion, nd where he middle loop i ( ) F δ(f F ) uily re prmeerized. Inering F in he oundry, nd F in oundry, nd ping o homology, we n red off he relion: ( ) F δ(f F ) = or wih = δ: ( ) ( F +)( F +) F F + ( ) F δ(f ) F F F + F (δf ) {F,F } = ( ) F (F F ) ( ) F (F ) F F ( F ).4. Remrk Algerilly, he Joi ideniy nd he derivion propery of he rke follow from he BV relion. For he le lgerilly inlined we n gin mke everyhing opologilly explii. Thi lo h he virue of howing how differen weigh onriue o opologilly diin gluing.

30 540 Rlph M Kufmnn, Muriel Liverne nd R C Penner Thi remen lo how h we n reri ourelve o he e of liner (Chinee) ree of eion 3 or o i wihou pine (f. eion 5 nd [7]) nd herefore hve Gerenher ruure on hi level..4.3 The oior I i inruive o do he lulion in he r fmily piure wih he operdi noion. For he gluing we oin he elemen in C () preened in figure 3 o whih we pply he homoopy of hnging he weigh on he oundry 3 from o while keeping everyhing ele fixed. We ll hi normlizion. o = ~ 3 3 Figure 3: The fir iered gluing of Unrveling he definiion for he normlized verion yield figure 4, where in he differen e he gluing of he nd i hown in figure 5. The gluing in r fmilie i impler nd yield he gluing depied in figure 6 o whih we pply normlizing homoopy y hnging he weigh on he nd emning from oundry from he pir (,( )) o (,( )) uing poinwie he homoopy ( +, + ( )) for [0,]: Comining figure 4 nd 6 while keeping in mind he prmeerizion we n red off he pre Lie relion: F (F F 3 ) (F F ) F 3 ( ) ( F +)( F +) (F (F F 3 ) (F F ) F 3 ) () whih how h he oior i ymmeri in he fir wo vrile nd hu following Gerenher [3] we oin:.4.4 Corollry The rke {, } ifie he odd Joi ideniy.

31 Ar Operd nd Ar Alger I =0 II < IV <<+ V 3 =+ 3 VII = 3 III = VI ( ) (+) 3 >+ 3 3 ( ) 3 3 Figure 4: The glued fmily fer normlizion.4.5 The Gerenher ruure The derivion propery of he rke follow from he ompiiliy equion whih re proved y he relion repreened y he wo digrm 7 nd 8. The fir e i ju lulion in he r fmily piure. In he eond e he r fmily piure lo mke i very ey o wrie down he fmily induing he hin homoopy expliily. We fix r fmilie,, prmeerized over k,k,k repeively. Fir noie h he fmily prmeerized y k I k k, whih i depied

32 54 Rlph M Kufmnn, Muriel Liverne nd R C Penner I =0 II < III = IV <<+ V =+ VI >+ VII = ( ) ( ) ( ) ( ) ( ) Figure 5: The differen e of gluing he nd ( ) o = ~ 3 3 Figure 6: The oher ierion of in figure 7, illure h () ( ) + ( ) ( +) ( ) Seond, he peil wo prmeer fmily hown in figure 8 where he wo (*) ( +) ( ) (*) ~ ~ ~ = *() Figure 7: The fir ompiiliy equion prmeer r fmily relizing he hin homoopy i prmeerized over k k T k wih T := {(,) [0,] [0,] : + } give he hin

33 Ar Operd nd Ar Alger 543 homoopy () ( ) + ( ) ( +) ( ) In oh e, we ued normlizing homoopie efore. ~ (*) ( +) ~ ( ) (*) ~ ()* Figure 8: The eond ompiiliy equion Thee wo equion imply h nd we oin: {, } {,} + ( ) ( +) {,}, ().4.6 Propoiion The rke {, } i Gerenher rke up o hin homoopy on C (ARC p ) or C (ARC) for he produ. Summing up, we oin:.5 Theorem There i BV ruure on C (ARC p ) or C (ARC) up o explii hin oundrie. The indued Gerenher rke i lo given y uh explii oundrie. Thi rke i ompile wih produ (oiive nd ommuive up o explii oundrie) given y poin in C 0 (ARC p ()).

34 544 Rlph M Kufmnn, Muriel Liverne nd R C Penner.6 Corollry All Ar p nd Ar lger re Blin-Vilkoviky lger. The Gerenher ruure indued y he BV operor oinide wih he rke indued y he pre Lie produ. Hene ll Ar p nd Ar lger re GBV lger. 3 Ci uoperd of ARC In he l eion (eion ), we exhiied GBV ruure up o hin homoopy on C (ARC). Inpeing he r fmilie relizing he relevn homoopie, we oerve h he Gerenher ruure i lredy preen in uoperd whih we hll diu here. Thi uoperd, lled liner ree, orrepond o he pinele i of [7]. Furhermore for he BV operor, we only need o dd he one more operion, o h he BV ruure i relized on he uoperd genered y pinele i nd ARC p (). Lly he uoperd genered y pinele i i onined in he uoperd genered y i nd he wo Gerenher ruure gree, i.e. he one oming from he BV operor nd he previouly defined rke. We will furhermore how h hi operd i he imge of n emedding (up o homoopy) of Voronov i [6] ino ARC. By he reul of [, 6], lger over he i operd hve BV ruure. The mp of operd we onru will hu indue he ruure of BV lger for lger over he homology of ARC. Thi indued ruure i indeed he me h defined in eion n e een from he emedding nd our explii relizion of ll relevn operion. 3. Suoperd 3.. Definiion The ree uoperd i defined for r fmilie in urfe wih g = = 0 in he noion of.6.3 y he llowed inidene mrix A (n), whoe non zero enrie re 0i = = i0, for i =,...,n, nd required inidene relion R (n), whoe enrie re ll equl o one. Thi i uoperd of ARC p, nd repreenion of i olleion of leled ree n e found in [7]. Dropping he requiremen h g = = 0, we oin uoperd of ARC lled he rooed grph or Chinee ree uoperd.

35 Ar Operd nd Ar Alger Remrk We hve lredy oerved h here i liner nd y forgeing he ring poin yli order on he e of r iniden on eh oundry omponen. In he (Chinee) ree uoperd ll nd mu hi he 0 h omponen, whih indue liner nd yli order on ll of he nd. Furhermore he yli order i repeed for ree, in he ene h he nd meeing he i h omponen form yli uhin in he yli order of ll nd, for Chinee ree, hi i n exr ondiion. And gin ll Chinee ree whih ify hi ondiion form uoperd whih we ll he yli Chinee ree. The liner order i, however, no even repeed for ree, n eily e een in ARC p () Lineriy Condiion We y h n elemen of he (yli Chinee) ree uoperd ifie he Lineriy Condiion if he liner order mh, i.e., he nd hiing eh oundry omponen in heir liner order re uhin of ll he nd in heir liner order derived from he 0 h oundry. I i ey o hek h hi ondiion i le under ompoiion. We ll he uoperd of elemen ifying he Lineriy Condiion of he (yli Chinee) ree operd he (yli Chinee) liner ree operd Propoiion The uoperd genered y (yli Chinee) liner ree nd ARC p () inide ARC oinide wih (yli Chinee) ree. Proof Given (Chinee) ree we n mke i liner y gluing on wi from ARC() he vriou oundry omponen hee wi hve he effe of moving he mrked poin of he oundry round he oundry. Sine he yli order i lredy repeed, uh wi my e pplied o rrnge h he liner order gree. Sine (Chinee) liner ree nd ARC() lie inide he (Chinee) ree operd he revere inluion i oviou. 3. Ci There re everl peie of i, whih re defined in [7], o whih we refer he reder for deil. By i, we men Voronov i defined in [6], i.e., onneed, plnr ree-like onfigurion of prmeerized loop, ogeher wih mrked poin on he onfigurion. Thi poin, lled glol zero, define n ouide irle or perimeer y king i o e he ring poin nd hen going round ll loop in ounerlokwie fhion y jumping ono he nex loop (in he indued yli order) he inereion poin. The pinele

36 546 Rlph M Kufmnn, Muriel Liverne nd R C Penner vriey of i i oined y pouling h he lol zero defined y he prmeerizion of he loop oinide wih he fir inereion poin of he perimeer wih loop (omeime lled loe ) of he u. The gluing i of wo i C nd C i done y fir ling in uh wy h he lengh of i h loe of C mhe he lengh of he ouide irle of C nd hen inering he u ino he i h loe y uing he prmeerizion gluing d. 3.. Sling of u Ci nd pinele i oh ome wih univerl ling operion of R >0 whih imulneouly le ll rdii y he me for λ R >0. Thi ion i free ion nd he gluing deend o he quoien y hi ion. 3.. Lef, Righ nd Symmeri Ci For he operdi gluing one h hree i poiiliie o le in order o mke he ize of he ouer loop of he u h i o e inered mh he ize of he loe ino whih he inerion hould e mde. () Sle down he u whih i o e inered. Thi i he originl verion we ll i he righ ling verion. () Sle up he u ino whih will e inered. We ll i he lef ling verion. (3) Sle oh i. The one whih i o e inered y he ize of he loe ino whih i will e inered nd he u ino whih he inerion i going o e king ple y he ize of he ouer loop of he u whih will e inered. We ll hi i he ymmeri ling verion. All of hee verion re of oure homoopy equivlen nd in he quoien operd of i y overll ling, he projeive i i/r >0 hey ll deend o he me glueing Frming of pinele u We will give mp of pinele i ino ARC lled frming. Fir noie h pinele u n e deompoed y he iniil poin nd he inereion poin ino equene of r following he nurl orienion given