Cohomology of affine Artin groups and applications

COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS FILIPPO CALLEGAR O, D A VIDE MOR ONI, AND MARIO SAL VETTI Abstra t. The result of this pap er is the determination of the oho- mology of Artin groups of t yp e A n , B n and ˜ A n with non-trivial lo al o eien ts. The main result (Theorem 1.1 ) is an expliit omputa- tion of the ohomology of the Artin group of t yp e B n with o eien ts o v er the mo dule Q [ q ± 1 , t ± 1 ] . Here the rst ( n − 1) standard genera- tors of the group at b y ( − q ) − m ultipliation, while the last one ats b y ( − t ) − m ultipliation. The pro of uses some te hnial results from previous pap ers plus omputations o v er a suitable sp etral sequene. The remaining ases follo w from an appliation of Shapiro's lemma, b y onsidering some w ell-kno wn inlusions: w e obtain the rational oho- mology of the Artin group of ane t yp e ˜ A n as w ell as the ohomology of the lassial braid group Br n with o eien ts in the n -dimensional represen tation presen ted in [TYM96 ℄. The top ologial oun terpart is the expliit onstrution of nite C W − omplexes endo w ed with a free ation of the Artin groups, whi h are kno wn to b e K ( π , 1) spaes in some ases (inluding nite t yp e groups). P artiularly simple form ulas for the Euler- harateristi of these orbit spaes are deriv ed. 1. Intr odution Reall that for ea h Co xeter group W one has a group extension G W , usually alled A rtin gr oup of t yp e W (see Setion 2). In this pap er w e giv e a detailed alulation of the ohomology of some Artin groups with non- trivial lo al o eien ts. Let R := Q [ q ± 1 , t ± 1 ] b e the ring of t w o-parameters Lauren t p olynomials. The main result (Theorem 1.1 ) is the omputation of the ohomology of the Artin group G B n (of t yp e B n ) with o eien ts in the mo dule R q ,t . The latter is the ring R with the mo dule struture dened as follo ws: the generators asso iated to the rst n − 1 no des of the Dynkin diagram of B n at b y ( − q ) − m ultipliation; the one asso iated to the last no de ats b y ( − t ) − m ultipliation. Let ϕ m ( q ) b e the m -th ylotomi p olynomial in the v ariable q . Dene the R -mo dules ( m > 1 , i ≥ 0 ) { m } i = R/ ( ϕ m ( q ) , q i t + 1) . Date : Ma y 2006. 2000 Mathematis Subje t Classi ation. 20J06; 20F36. Key wor ds and phr ases. Ane Artin groups, t wisted ohomology , group representa- tions. The third author is partially supp orted b y M.U.R.S.T. 40%. 1 2 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI and for m = 1 set: { 1 } i = R/ ( q i t + 1) . Notie that the mo dules { m } i are all non isomorphi as R -mo dules. { m } i and { m ′ } i ′ are isomorphi as Q [ q ± 1 ] -mo dules if and only if m = m ′ and are isomorphi as Q [ t ± 1 ] -mo dules if and only if φ ( m ) = φ ( m ′ ) ( φ is the Euler funtion) and m ( m,i ) = m ′ ( m,i ′ ) . Our main result is the follo wing Theorem 1.1. H i ( G B n , R q ,t ) =      L d | n, 0 ≤ k ≤ d − 2 { d } k ⊕ { 1 } n − 1 if i = n L d | n, 0 ≤ k ≤ d − 2 , d ≤ n j +1 { d } k if i = n − 2 j L d ∤ n, d ≤ n j +1 { d } n − 1 if i = n − 2 j − 1 .  The pro of uses the sp etral sequene asso iated to a natural ltration of the algebrai omplex exhibited in [Sal94 ℄, plus some te hnial results from [DCPS01 ℄. W e apply Shapiro's lemma to a w ell kno wn inlusion of G ˜ A n − 1 in to G B n to deriv e the ohomology of G ˜ A n − 1 o v er the mo dule Q [ q ± 1 ] , the ation of ea h standard generator b eing ( − q ) − m ultipliation. By onsidering another natural inlusion of G B n in to the lassial braid group Br n +1 := G A n , w e also use Shapiro's lemma in order to iden tify the ohomology of G B n with o eien ts in R q ,t with that of Br n +1 with o e- ien ts in the irreduible ( n + 1) − dimensional represen tation of Br n +1 found in [TYM96℄, t wisted b y an ab elian represen tation. W e deriv e the trivial Q − ohomology of G ˜ A n − 1 as w ell as the ohomology of the braid group o v er the irreduible represen tation in [TYM96℄. Computation of the ohomology of Artin groups w as done b y sev eral p eople: for lassial br aid gr oups and trivial o eien ts it w as rst giv en b y F. Cohen [Coh76 ℄, and indep enden tly b y A. V a  n²te  n [ V a  78 ℄ (see also [Arn68 , Bri71 , BS72, F uk70 ℄). F or Artin groups of t yp e C n , D n see [Gor78℄, while for the exeptional ases see [Sal94 ℄, where the Z -mo dule struture w as giv en, while the ring struture w as omputed in [Lan00 ℄. The ase of non-trivial o eien ts o v er the mo dule of Lauren t p olynomials Q [ q ± 1 ] is in teresting b eause of its relation with the trivial Q -ohomology of the Mil- nor bre of the naturally asso iated bundle. F or the lassial braid groups see [F re88 , Mar96 , DCPS01 ℄, while for ases C n , D n see [DCPSS99 ℄. F or omputations o v er the in tegral Lauren t p olynomials Z [ q ± 1 ] , see [CS98 ℄ for the exeptional ases and reen tly [Cal06 ℄ for the ase of braid groups, and [DCSS97 ℄ for the top ohomologies in all ases. In the ase of Artin groups of non-nite t yp e, some omputations w ere giv en in [ SS97 ℄ and [CD95℄. The omputations of Theorem 1.1 ould b e partially extended to in te- gral o eien ts; ho w ev er, ma jor ompliations o ur b eause the Lauren t p olynomial ring Z [ q ± 1 ] is not a P .I.D.. In the last part w e also indiate (see [ CMS ℄) an expliit onstrution of nite CW-omplexes whi h are retrats of orbit sp a es asso iated to Artin groups. The onstrution w orks as in [ Sal94 ℄, with few v ariations neessary for innite t yp e Artin groups (see also [CD95 ℄ for a dieren t onstrution). COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 3 The Artin group iden ties with the fundamen tal group of the orbit spae, and the standard presen tation follo ws easily (see [Bri71 , D  un83, vdL83℄). The Euler  harateristi of the orbit spae redues to that of a simpliial omplex and in some ases one has a partiularly simple form ula. It is onjetured that su h orbit spaes are alw a ys K ( π , 1) spaes; for the ane groups, this is kno wn in ase ˜ A n , ˜ C n (see [Ok o79 , CP03 ℄) and reen tly also for ˜ B n ([CMS1℄) (see also [CD95℄ for a dieren t lass of Artin groups of innite t yp e). Notie also the geometrial meaning of the t w o-parameters ohomology of G B n : similar to the one-parameter ase, it is equiv alen t to the trivial ohomology of the homotop y-Milnor bre asso iated to the natural map of the orbit spae on to a t w o-dimensional torus. The main results of this pap er w ere announed (without pro of ) in [ CMS℄. 2. Preliminar y resul ts In this setion w e briey x the notation and reall some preliminary results. 2.1. Co xeter groups and Artin groups. A Coxeter gr aph is a nite undi- reted graph, whose edges are lab elled with in tegers ≥ 3 or with the sym b ol ∞ . Let S b e the v ertex set of a Co xeter graph. F or ev ery pair of v erties s, t ∈ S ( s 6 = t ) joined b y an edge, dene m ( s, t ) to b e the lab el of the edge joining them. If s, t are not joined b y an edge, set b y on v en tion m ( s, t ) = 2 . Let also m ( s, s ) = 1 (see [Bou68 , Hum90℄). T w o groups are asso iated to a Co xeter graph: the Coxeter gr oup W dened b y W = h s ∈ S | ( st ) m ( s,t ) = 1 ∀ s, t ∈ S su h that m ( s, t ) 6 = ∞i and the A rtin gr oup G dened b y (see [BS72 ℄): G = h s ∈ S | stst . . . | {z } m ( s,t ) − terms = tsts . . . | {z } m ( s,t ) − terms ∀ s, t ∈ S su h that m ( s, t ) 6 = ∞i . Lo osely sp eaking, G is the group obtained b y dropping the relations s 2 = 1 ( s ∈ S ) in the presen tation for W . In this pap er, w e are primarily in terested in Artin groups asso iated to Co xeter graph of t yp e A n , B n and ˜ A n − 1 (see Figure 1). 2.2. Inlusions of Artin groups. Let Br n +1 := G A n b e the braid group on n + 1 strands and Br n +1 n +1 < Br n +1 b e the subgroup of braids xing the ( n + 1) -st strand. The group Br n +1 n +1 is alled the ann ular braid group. Let K n +1 = { p 1 , . . . , p n +1 } b e a set of n + 1 distint p oin ts in C . The lassial braid group Br n +1 = G A n an b e realized as the fundamen tal group of the spae of unordered ongurations of n + 1 p oin ts in C with basep oin t K n +1 (see the left part of Figure 2), with K 6 = { 1 , . . . , 6 } ). W e an no w think to the subgroup Br n +1 n +1 < Br n +1 as the fundamen tal group of the spae of unordered ongurations of n p oin ts in C ∗ : in fat if w e tak e p n +1 = 0 and p i ∈ S 1 ⊂ C for i ∈ 1 , . . . , n , sine for a braid β ∈ Br n +1 n +1 the orbit of the ( n + 1) -st p oin t an b e though t onstan t, up to homotop y , w e an think to β as a braid with n strands in the an ulus (see the righ t part of Figure 2 ). 4 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI σ n ˜ σ 2 ˜ σ 3 ˜ σ 1 ˜ A n − 1 ǫ 1 ǫ 2 4 B n ǫ n − 1 σ 1 σ 2 σ n − 1 A n ˜ σ n − 1 ˜ σ n ¯ ǫ n Figure 1. Co xeter graph of t yp e A n , B n ( n ≥ 2 ) and ˜ A n − 1 ( n ≥ 3 ). Lab els equal to 3 , as usual, are not sho wn. More- o v er, to x notation, ev ery v ertex is lab elled with the orre- sp onding generator in the Artin group. 4 3 4 3 3 4 2 5 1 2 1 5 1 2 5 6 Figure 2. A braid in Br 6 6 represen ted as an ann ular braid on 5 strands. It is w ell kno wn that the ann ular braid group is isomorphi to the Artin group G B n of t yp e B n . F or a pro of of the follo wing Theorem see [Cri99 ℄ or [Lam94 ℄. Theorem 2.1. L et σ 1 , . . . , σ n and ǫ 1 , . . . , ǫ n − 1 , ¯ ǫ n b e r esp e tively the stan- dar d gener ators for G A n and G B n . Then, the map G B n → Br n +1 n +1 < Br n +1 ǫ i 7→ σ i for 1 ≤ i ≤ n − 1 ¯ ǫ n 7→ σ 2 n is an isomorphism.  Using the suggestion giv en b y the iden tiation with the ann ular braid group, a new in teresting presen tation for G B n an b e w ork ed out. Let τ = ¯ ǫ n ǫ n − 1 · · · ǫ 2 ǫ 1 . It is easy to v erify that: τ − 1 ǫ i τ = ǫ i +1 for 1 ≤ i < n − 1 COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 5 Figure 3. As an ann ular braid the elemen t τ is obtained turning the b ottom ann ulus b y a rotation of 2 π /n . i.e. onjugation b y τ shifts forw ard the rst n − 2 standard generators. By analogy , let ǫ n = τ − 1 ǫ n − 1 τ . W e ha v e the follo wing Theorem 2.2 ([KP02 ℄) . The gr oup G B n has pr esentation hG |Ri wher e G = { τ , ǫ 1 , ǫ 2 , . . . , ǫ n } R = { ǫ i ǫ j = ǫ j ǫ i for i 6 = j − 1 , j + 1 }∪ { ǫ i ǫ i +1 ǫ i = ǫ i +1 ǫ i ǫ i +1 }∪ { τ − 1 ǫ i τ = ǫ i +1 } wher e ar e al l indexes should b e  onsider e d mo dulo n .  Letting ˜ σ 1 , ˜ σ 2 , . . . , ˜ σ n b e the standard generator of the Artin group of t yp e ˜ A n − 1 , w e ha v e the follo wing straigh tforw ard orollary: Corollary 2.3 ([KP02 ℄) . The map G ˜ A n − 1 ∋ ˜ σ i 7→ ǫ i ∈ G B n gives an isomorphism b etwe en the gr oup G ˜ A n − 1 and the sub gr oup of G B n ge- ner ate d by ǫ 1 , . . . , ǫ n . Mor e over, we have a semidir e t pr o dut de  omp osition G B n ∼ = G ˜ A n − 1 ⋊ h τ i .  W e ha v e th us a urious inlusion of the Artin group of innite t yp e ˜ A n − 1 in to the Artin group of nite t yp e B n . R emark 2.4 . The pro of of Theorem 2.2 presen ted in the original pap er is algebrai and based on Tietze mo v es; a somewhat more oinise pro of an ho w ev er b e obtained b y standard top ologial onstrutions. Indeed, one an exhibit an expliit innite yli o v ering K ( G ˜ A n − 1 , 1) → K ( G B n , 1) (see [All02℄). 6 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI 2.3. ( q , t ) -w eigh ted P oinaré series for B n . F or future use in ohomol- ogy omputations, w e are in terested in a ( q , t ) -analog of the usual P oinaré series for B n , that is an analog of the P oinaré series with o eien ts in the ring R = Q [ q ± 1 , t ± 1 ] of Lauren t p olynomials. This result and similar ones are studied in [Rei93℄, to whi h w e refer for details. W e also use lassial results from [Bou68 , Hum90 ℄ without further referene. Consider the Co xeter group W of t yp e B n with its standard generating reetions s 1 , s 2 , . . . , s n . F or w ∈ W , let n ( w ) b e the n um b er of times s n app ears in a redued expression for w . By standard fats, n ( w ) is w ell- dened. W e dene the ( q , t ) -w eigh ted P oinaré series for the Co xeter group of t yp e B n as the sum W ( q , t ) = X w ∈ W q ℓ ( w ) − n ( w ) t n ( w ) , where ℓ is the length funtion. W e reall some notation. W e dene the q -analog of the n um b er m b y the p olynomial [ m ] q := 1 + q + · · · q m − 1 = q m − 1 q − 1 . Notie that [ m ] = Q i | m, i 6 =1 ϕ m ( q ) , where w e denote with ϕ m ( q ) the m -th ylotomi p olynomial in the v ariable q . Moreo v er w e dene the q -fatorial analog [ m ] q ! as the pro dut m Y i =1 [ i ] q and the q -analog of the binomial  m i  as the p olynomial  m i  q := [ m ] q ! [ i ] q ![ m − i ] q ! . W e an also dene the ( q , t ) -analog of an ev en n um b er [2 m ] q ,t := [ m ] q (1 + tq m − 1 ) and of the double fatorial [2 m ] q ,t !! := m Y i =1 [2 i ] q ,t = [ m ] q ! m − 1 Y i =0 (1 + tq i ) . Finally , w e dene the p olynomial  m i  ′ q ,t := [2 m ] q ,t !! [2 i ] q ,t !![ m − i ] q ! =  m i  q m − 1 Y j = i (1 + tq j ) . Prop osition 2.5 ([Rei93 ℄) . W ( q , t ) = [2 n ] q ,t !! . Pr o of. Consider the parab oli subgroup W I asso iated to the subset of re- etions I = { s 1 , . . . , s n − 1 } . Notie that W I is isomorphi to the symmetri group on n letters A n − 1 and that it has index 2 n in B n . Let W I b e the set of COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 7 minimal oset represen tativ es for W /W I . Then, b y m ultipliativ e prop erties on redued expressions: W ( q , t ) = X w ∈ W q ℓ ( w ) − n ( w ) t n ( w ) =  X w ′ ∈ W I q ℓ ( w ′ ) − n ( w ′ ) t n ( w ′ )  ·  X w ′′ ∈ W I q ℓ ( w ′′ ) − n ( w ′′ ) t n ( w ′′ )  . (2.1) Clearly , for elemen ts w ′′ ∈ W I , w e ha v e n ( w ′′ ) = 0 ; so the seond fator in (2.1 ) redues to the w ell-kno wn P oinaré series for A n − 1 : X w ′′ ∈ W I q ℓ ( w ′′ ) − n ( w ′′ ) t n ( w ′′ ) = [ n ] q ! . T o deal with the rst fator, instead, w e expliitly en umerate the elemen ts of W I . Let p i = s i s i +1 · · · s n for 1 ≤ i ≤ n . Then, it an b e easily v eried that W I = { p i r p i r − 1 · · · p i 2 p i 1 | i 1 < i 2 < · · · < i r − 1 < i r } . Notie that n ( p i r p i r − 1 · · · p i 2 p i 1 ) = r and ℓ ( p i r p i r − 1 · · · p i 2 p i 1 ) = P r j =1 ℓ ( p i j ) = P r j =1 ( n + 1 − i j ) . Th us, X w ′ ∈ W I q ℓ ( w ′ ) − n ( w ′ ) t n ( w ′ ) = n − 1 Y i =0 (1 + tq i ) . Finally , W ( q , t ) =  n − 1 Y i =0 (1 + tq i )  [ n ] q ! = [2 n ] q ,t !! .  3. The ohomology of G B n 3.1. Pro of of the Main Theorem. In this Setion w e pro v e Theorem 1.1 en uniated in the in tro dution. W e use the notations giv en in the In tro du- tion. T o p erform our omputation w e will use the omplex diso v ered in [Sal94 ℄, [DCS96 ℄ (notie: an equiv alen t omplex w as diso v ered b y dieren t metho ds in [Squ94 ℄), and the sp etral sequene indued b y a natural ltration. The omplex that omputes the ohomology of G B n o v er R q ,t is giv en as follo ws (see [Sal94 ℄): C ∗ n = M Γ ⊂ I n R. Γ where I n denote the set { 1 , . . . , n } and the graduation is giv en b y | Γ | . The set I n orresp onds to the set of no des of the Dynkin diagram of B n and in partiular the last elemen t, n , orresp onds to the last no de. It is useful to onsider also the omplex C ∗ n for the ohomology of G A n on the lo al system R q ,t . In this ase the ation asso iated to a standard generator is alw a ys the ( − q ) -m ultipliation and so the omplex C ∗ n and its ohomology are free as Q [ t ± ] -mo dules. The omplex C ∗ n is isomorphi to C ∗ n as a R -mo dule. In b oth omplexes the ob oundary map is (3.1) δ ( q , t )(Γ) = X j ∈ I n \ Γ ( − 1) σ ( j, Γ) W Γ ∪{ j } ( q , t ) W Γ ( q , t ) (Γ ∪ { j } ) 8 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI where σ ( j, Γ) is the n um b er of elemen ts of Γ that are less than j . In the ase A n W Γ ( q , t ) is the P oinaré p olynomial of the parab oli subgroup W Γ ⊂ A n generated b y the elemen ts in the set Γ , with w eigh t − q for ea h standard generator, while in the ase B n W Γ ( q , t ) is the P oinaré p olynomial of the parab oli subgroup W Γ ⊂ B n generated b y the elemen ts in the set Γ , with w eigh t − q for the rst n − 1 generators and − t for the last generator. Using Prop osition 2.5 w e an giv e an expliit omputation of the o ef- ien ts W Γ ∪{ j } ( q , t ) W Γ ( q , t ) . F or an y Γ ⊂ I n , let Γ b e the subgraph of the Dynkin diagram B n whi h is spanned b y Γ . Reall that if Γ is a onneted omp o- nen t of the Dynkin diagram of B n without the last elemen t, then W Γ ( q , t ) = [ m + 1] q ! , where m = | Γ | . If Γ is onneted and on tains the last elemen t of B n , then W Γ ( q , t ) = [2 m ] q ,t !! , where m = | Γ | . If Γ is the union of sev eral onneted omp onen ts of the Dynkin diagram, Γ = Γ 1 ∪ · · · ∪ Γ k , then W Γ ( q , t ) is the pro dut k Y i =1 W Γ i ( q , t ) of the fators orresp onding to the dieren t omp onen ts. If j / ∈ Γ w e an write Γ( j ) for the onneted omp onen t of Γ ∪ { j } on- taining j . Supp ose that m = | Γ( j ) | and i is the n um b er of elemen ts in Γ( j ) greater than j . Then, if n ∈ Γ ( j ) w e ha v e W Γ ∪{ j } ( q , t ) W Γ ( q , t ) =  m i  ′ q ,t and W Γ ∪{ j } ( q , t ) W Γ ( q , t ) =  m + 1 i + 1  q otherwise. It is on v enien t to represen t generators Γ ⊂ I n b y their  harateristi funtions I n → { 0 , 1 } so, simply b y strings of 0 s and 1 s of length n . W e dene a dereasing ltration F on the omplex ( C ∗ n , δ ) : F s C n is the sub omplex generated b y the strings of t yp e A 1 s (ending with a string of s 1 s) and w e ha v e the inlusions C n = F 0 C n ⊃ F 1 C n ⊃ · · · ⊃ F n C n = R. 1 n ⊃ F n +1 C n = 0 . W e ha v e the follo wing isomorphism of omplexes: (3.2) ( F s C n /F s +1 C n ) ≃ C n − s − 1 [ s ] where C n − s − 1 is the omplex for G A n − s − 1 and the notation [ s ] means that the degree is shifted b y s . Let E ∗ b e the sp etral sequene asso iated to the ltration F . The equalit y 3.2 tells us ho w the E 1 term of the sp etral sequene lo oks lik e. In fat for 0 ≤ s ≤ n − 2 w e ha v e (3.3) E s,r 1 = H r ( G A n − s − 1 , R q ,t ) = H r ( G A n − s − 1 , Q [ q ± 1 ] q )[ t ± 1 ] COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 9 sine the t -ation is trivial. F or s = n − 1 and s = n the only non trivial elemen ts in the sp etral sequene are (3.4) E n − 1 , 0 1 = E n, 0 1 = R. In order to pro v e Theorem 1.1 w e need to state the follo wing lemmas. Lemma 3.1. L et I ( n, k ) b e the ide al gener ate d by the p olynomials  n n − d  ′ q ,t for d | n and d ≤ k If k | n the map α n,k : R / ( ϕ k ( q )) → R /I ( n, k − 1) indu e d by the multipli ation by  n n − k  ′ q ,t is wel l dene d and is inje tive. Remark. The fat that this map is w ell dened will follo w automatially from the general theory of sp etral sequenes, as it is lear from the pro of of Theorem 1.1. Ho w ev er, b elo w w e pro v e it b y other means. Pr o of. Let d, k b e p ositiv e in tegers su h that d | n and k | n . W e an observ e that ϕ d ( q ) |  n k  q =  n n − k  q if and only if d ∤ k . Moreo v er ea h fator ϕ d app ears in  n k  q at most with exp onen t 1 . Let J ( n, k ) b e the ideal generated b y the p olynomials  n n − d  q for d | n and d ≤ k . It is easy to see that w e ha v e the follo wing inlusion: n − 1 Y i = n − k (1 + tq i ) J ( n, k ) ⊂ I ( n, k ) . Moreo v er J ( n, k ) is a prinipal ideal and is generated b y the pro dut p n,k ( q ) = Y d | n,k · · · > d h = 1 b e the list of all the divisors of n in dereasing order. If w e set P i := ϕ d i ( q ) and Q i := d i Y j = d i +1 +1 (1 + tq n − j ) w e an rewrite our ideal as e I ( n ) =  n n − d h  ,  n n − d h − 1  Q h − 1 ,  n n − d h − 2  Q h − 2 Q h − 1 , . . . . . . ,  n n − d 2  Q 2 · · · Q h − 1 , Q 1 · · · Q h − 1  (3.5) W e laim that w e an redue to the follo wing set of generators: e I ( n ) = ( P 1 · · · P h − 1 , P 1 · · · P h − 2 Q h − 1 , P 1 · · · P h − 3 Q h − 2 Q h − 1 . . . . . . , P 1 Q 2 · · · Q h − 1 , Q 1 · · · Q h − 1 ) (3.6) The rst generator is the same in b oth equations and the j -th generator in Equation 3.6 divides the orresp onding generator in Equation 3.5 . No w sup- p ose that a fator ϕ m ( q ) divides  n n − d j  but do es not divide P 1 · · · P j − 1 . W e ma y distinguish t w o ases: i) Supp ose that m ∤ n . Then w e an get rid of the fator ϕ m ( q ) in  n n − d j  with an opp ortune om bination with the p olynomial P 1 · · · P h − 1 ii) Supp ose m | n . Then m = d l for some l > j and w e an get rid of ϕ m ( q ) using a suitable om bination with the p olynomial P 1 · · · P l − 1 Q l · · · Q h − 1 W e ma y no w pro eed indutiv ely . Supp osing w e ha v e already redued the rst j − 1 terms, w e an redue the j -th term of the ideal in Equation 3.5 to the orresp onding term in Equation 3.6 . COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 11 No w w e observ e that if J, I 1 , I 2 are ideals and I 1 + I 2 = (1) , then ( J, I 1 I 2 ) = ( J, I 1 )( J, I 2 ) . Sine the p olynomials P i are all oprime, w e an apply this fat to the ideal e I ( n ) h − 2 times. A t the i -th step w e set I 1 = ( P i ) , I 2 = ( P i +1 · · · P h − 1 , P i +1 · · · P h − 2 Q h − 1 , . . . , Q i +1 · · · Q h − 1 ) , J = ( Q i · · · Q h − 1 ) . So w e an fator e I ( n ) as ( P 1 , Q 1 · · · Q h − 1 )( P 2 · · · P h − 1 , P 2 · · · P h − 2 Q h − 1 , Q 2 · · · Q h − 1 ) = · · · · · · = ( P 1 , Q 1 · · · Q h − 1 )( P 2 , Q 2 · · · Q h − 1 ) · · · ( P h − 1 , Q h − 1 ) . Finally w e an split ( P s , Q s · · · Q h − 1 ) as the pro dut ( P s , 1 + tq n − d s ) · · · ( P s , 1 + tq n − d h − 1 ) . So w e ha v e redued the ideal I ( n ) in the pro dut stated in the Lemma and it is easy to  he k that all the ideals of the splitting are oprime.  Pr o of of The or em 1.1 . W e an no w pro v e our Theorem using the sp etral sequene desrib ed in the Equations 3.3 and 3.4 . W e in tro due, as in [DCPS01 ℄, the follo wing notation for the generators of the sp etral sequene: w h = 01 h − 2 0 z h = 1 h − 1 0 + ( − 1) h 01 h − 1 b h = 01 h − 2 c h = 1 h − 1 z h ( i ) = i − 1 X j =0 ( − 1) hj w j h z h w i − j − 1 h v h ( i ) = i − 2 X j =0 ( − 1) hj w j h z h w i − j − 2 h b h + ( − 1) h ( i − 1) w i − 1 h c h W e write { m } [ t ± 1 ] for the mo dule R/ ( ϕ m ( q )) . The E 1 -term of the sp etral sequene has a mo dule { m } [ t ± 1 ] in p osition ( s, r ) if and only if one of the follo wing ondition is satised: a) m | n − s − 1 and r = n − s − 2 n − s − 1 m ; b) m | n − s and r = n − s + 1 − 2( n − s m ) . Moreo v er w e ha v e mo dules R in p osition ( n − 1 , 0) and ( n, 0) . W e no w lo ok at the d 1 map b et w een these t w o mo dules. Notie that E n − 1 , 0 1 is generated b y the string 01 n − 1 and E n, 0 1 is generated b y the string 1 n . F urthermore the map d n − 1 , 0 1 : E n − 1 , 0 1 → E n, 0 1 is giv en b y the m ultipliation b y  n n − 1  ′ q ,t = [ n ] q (1 + tq n − 1 ) and is inje- tiv e. It turns out that E n − 1 , 0 2 = 0 and E n, 0 2 = R/ ([ n ] q (1 + tq n − 1 )) . Moreo v er all the follo wing terms E n, 0 j are quotien t of E n, 0 2 . 12 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI Notie that ev ery map b et w een mo dules of kind { m } [ t ± 1 ] and { m ′ } [ t ± 1 ] m ust b e zero if m 6 = m ′ . So w e an study our sp etral sequene onsidering only maps b et w een the same kind of mo dules. First let us onsider an in teger m that do esn't divide n . Sa y that m | n + c with 1 ≤ c < m and set i = n + c m . The mo dules of t yp e { m } [ t ± 1 ] are: E λm − c − 1 ,n + c − λ ( m − 2) − 2 i +1 1 generated b y z m ( i − λ )01 λm − c − 1 E λm − c,n + c − λ ( m − 2) − 2 i +1 1 generated b y v m ( i − λ )01 λm − c for λ = 1 , . . . , i − 1 . Here is a diagram for this ase (w e use the notation h for { m } [ t ± 1 ] ): h d 1 / / h · · · d 1 / / · · · h d 1 / / h 0 ) ) R R R R R R R R R R R R/I The map d 1 : E λm − c − 1 ,n + c − λ ( m − 2) − 2 i +1 1 → E λm − c,n + c − λ ( m − 2) − 2 i +1 1 is giv en b y the m ultipliation b y  λm − c λm − c − 1  ′ q ,t = [ λm − c ] q (1+ tq λm − c − 1 ) . Sine ϕ m ( q ) ∤ [ λm − c ] q the map is injetiv e and in the E 2 -term w e ha v e: E λm − c − 1 ,n + c − λ ( m − 2) − 2 i +1 2 = 0 E λm − c,n + c − λ ( m − 2) − 2 i +1 2 = { m } λm − c − 1 = { m } m − c − 1 for λ = 1 , . . . , i − 1 . The other map w e ha v e to onsider is d n − m,m − 1 m : E n − m,m − 1 m → E n, 0 m . The mo dule E n − m,m − 1 m = { m } m − c − 1 is generated b y 1 m − 1 01 n − m and so the map is the m ultipliation b y  n n − m  ′ q ,t . Sine (1 + tq n − 1 ) divides the o eien t  n n − m  ′ q ,t , the image of the map d n − m,m − 1 m m ust b e on tained in the submo dule (1 + tq n − 1 ) E n, 0 m = (1 + tq n − 1 ) R/ ([ n ] q (1 + tq n − 1 )) that is in the quotien t R/ ([ n ] q ) . Sine ( ϕ m ( q ) , [ n ] q ) = (1) (reall that m do es not divide n ) there an b e no non trivial map b et w een the mo dules { m } m − c − 1 and R/ ([ n ] q ) . It follo ws that the dieren tial d n − m,m − 1 m m ust b e zero. As a onsequene the E 2 part desrib ed b efore ollapses to E ∞ and w e ha v e a op y of { m } m − c − 1 as a diret summand of H n − 2 j − 1 ( C n ) for j = 0 , . . . , i − 2 , that is for m ≤ n j +1 . COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 13 No w w e onsider an in teger m that divides n and let i = n m . The mo dules of t yp e { m } [ t ± 1 ] are: E λm − 1 ,n − λ ( m − 2) − 2 i +1 1 generated b y z m ( i − λ )01 λm − 1 for 1 ≤ λ ≤ i − 1 E λm,n − λ ( m − 2) − 2 i +1 1 generated b y v m ( i − λ )01 λm for 0 ≤ λ ≤ i − 1 . The situation is sho wn in the next diagram ( h = { m } [ t ± 1 ] ): h d m − 1 ( ( R R R R R R R R R R R h 0 / / h d m − 1 ) ) S S S S S S S S S S S · · · 0 / / · · · d m − 1 ) ) S S S S S S S S S S S h 0 / / h The map d 1 : E λm − 1 ,n − λ ( m − 2) − 2 i +1 1 → E λm,n − λ ( m − 2) − 2 i +1 1 is giv en b y the m ultipliation b y  λm λm − 1  ′ q ,t = [ λm ] q (1 + tq λm − 1 ) , but in this ase the o eien t is zero in the mo dule { m } [ t ± 1 ] b eause ϕ m ( q ) | [ λm ] q and so w e ha v e that E 1 = · · · = E m − 1 . So w e ha v e to onsider the map d λm,n − λ ( m − 2) − 2 i +1 m − 1 : E λm,n − λ ( m − 2) − 2 i +1 m − 1 → E ( λ +1) m − 1 ,n − ( λ +1)( m − 2) − 2 i +1 1 for λ = 0 , . . . , i − 2 . This map orresp onds to the m ultipliation b y  ( λ + 1) m − 1 λm  ′ q ,t =  ( λ + 1) m − 1 λm  q ( λ +1) m − 1 Y j = λm +1 (1 + tq j − 1 ) . It is easy to see that the p olynomial  ( λ + 1) m − 1 λm  q is prime with the torsion ϕ m ( q ) and so the map d λm,n − λ ( m − 2) − 2 i +1 m − 1 is injetiv e and the ok ernel is isomorphi to R ,   ϕ m ( q ) , ( λ +1) m − 1 Y j = λm +1 (1 + tq j − 1 )   ≃ M 0 ≤ k ≤ m − 2 { m } k . As a onsequene w e ha v e that E λm − 1 ,n − λ ( m − 2) − 2 i +1 m = L 0 ≤ k ≤ m − 2 { m } k for 1 ≤ λ ≤ i − 1 E λm,n − λ ( m − 2) − 2 i +1 m = 0 for 0 ≤ λ ≤ i − 2 . and all these mo dules ollapse to E ∞ . This means that w e an nd ϕ m ( q ) - torsion only in H n − 2 j ( C n ) and for j ≥ 1 the summand is giv en b y M 0 ≤ k ≤ m − 2 { m } k 14 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI for d ≤ n j +1 . W e still ha v e to onsider all the terms E n − m,m − 1 m = { m } [ t ± 1 ] for m | n . Here the maps w e ha v e to lo ok at are the follo wing: d n − m,m − 1 m : E n − m,m − 1 m → E n, 0 m . These maps orresp ond to m ultipliation b y the p olynomials  n n − m  ′ q ,t . Moreo v er reall that E n, 0 1 = R ,  n n − 1  ′ q ,t ! . W e an no w use Lemma 3.1 to sa y that all the maps d n − m,m − 1 m are injetiv e and Lemma 3.2 to sa y that E n, 0 n +1 = E n, 0 ∞ = M m | n, 0 ≤ k ≤ d − 2 { m } k ⊕ { 1 } n − 1 . Sine E n, 0 ∞ = H n ( C n ) , this omplete the pro of of the Theorem.  3.2. Other omputations. W e ma y onsider the ohomology of G B n o v er the mo dule Q [ t ± 1 ] , where the ation is trivial for the generators ǫ 1 , . . . , ǫ n − 1 and ( − t ) -m ultipliation for the last generator ǫ n . This ohomology is om- puted b y the omplex C ∗ n of Setion 3 where w e sp eialize q to − 1 . So w e ma y use similar ltration and asso iated sp etral sequene. W e used this argumen t in [CMS ℄. Here w e briey indiate a dieren t and more onise metho d, using the results of Theorem 1.1 . W e ha v e: Theorem 3.3. H k ( G B n , Q [ t ± 1 ]) = Q [ t ± 1 ] / (1 + t ) 1 ≤ k ≤ n − 1 H n ( G B n , Q [ t ± 1 ]) = Q [ t ± 1 ] / (1 + t ) for o dd n H n ( G B n , Q [ t ± 1 ]) = Q [ t ± 1 ] / (1 − t 2 ) for even n . Sketh of pr o of. Consider the short exat sequene: 0 → Q [ q ± 1 , t ± 1 ] 1+ q − → Q [ q ± 1 , t ± 1 ] → Q [ t ± 1 ] → 0 and the indued long exat sequene for ohomology · · · → H i ( G B n , Q [ q ± 1 ,t ± 1 ]) 1+ q − → H i ( G B n , Q [ q ± 1 ,t ± 1 ]) → H i ( G B n , Q [ t ± 1 ]) → · · · . The result is no w a straigh tforw ard onsequene of Theorem 1.1.  4. More onsequenes By means of Shapiro's lemma (see for instane [ Bro82 ℄), the inlusions in tro dued in Setion 2.2 an b e exploited to link the ohomology of the Artin group of t yp e ˜ A n − 1 , A n to the ohomology of G B n . COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 15 4.1. Cohomology of G ˜ A n − 1 . Let M b e an y domain and let q b e a unit of M . W e indiate b y M q the ring M with the G ˜ A n − 1 -mo dule struture where the ation of the standard generators is giv en b y ( − q ) -m ultipliation. Prop osition 4.1. W e have H ∗ ( G ˜ A n − 1 , M q ) ∼ = H ∗ ( G B n , M [ t ± 1 ] q ,t ) H ∗ ( G ˜ A n − 1 , M q ) ∼ = H ∗ ( G B n , M [[ t ± 1 ]] q ,t ) wher e the ation of G B n on M [ t ± 1 ] q ,t (and on M [[ t ± 1 ]] q ,t ) is given by ( − q ) - multipli ation for the gener ators ǫ 1 , . . . , ǫ n − 1 and ( − t ) -multipli ation for the last gener ator ¯ ǫ n . Pr o of. Applying Shapiro's lemma to the inlusion ˜ A n − 1 < G B n , one obtains: H ∗ ( G ˜ A n − 1 , M q ) ∼ = H ∗ ( G B n , Ind G B n G ˜ A n − 1 M q ) H ∗ ( G ˜ A n − 1 , M q ) ∼ = H ∗ ( G B n , Coind G B n G ˜ A n − 1 M q ) . By Corollary 2.3, an y elemen t of Ind G B n G ˜ A n − 1 M q := Z [ G B n ] ⊗ G ˜ A n − 1 M q an b e represen ted as a sum of elemen ts of the form τ α ⊗ q m . No w, w e ha v e an isomorphism of Z [ G B n ] -mo dules Z [ G B n ] ⊗ G ˜ A n − 1 M q → M [ t ± 1 ] q ,t dened b y sending τ α ⊗ q m 7→ ( − 1) nα t α q ( n − 1) α + m and the result follo ws. In ohomology w e ha v e similarly: Coind G B n G ˜ A n − 1 M q := Hom G ˜ A n − 1 ( Z [ G B n ] , M q ) ∼ = M [[ t ± 1 ]] q ,t .  By Prop ositions 4.1, in order to determine the ohomology H ∗ ( G ˜ A n − 1 , M q ) it is neessary to kno w the ohomology of G B n with v alues in the mo dule M [[ t ± 1 ]] of Lauren t series in the v ariable t . The latter is link ed to the ohomology with v alues in the mo dule of Lauren t p olynomials b y: Prop osition 4.2 (Degree shift) . H ∗ ( G B n , M [[ t ± 1 ]] q ,t ) ∼ = H ∗ +1 ( G B n , M [ t ± 1 ] q ,t ) .  This result w as obtained in [Cal05 ℄ in a sligh tly w eak er form, but it is p ossible to extend it to our ase with little eort. Let from no w on M = Q [ q ± 1 ] . In this ase w e ha v e M [ t ± 1 ] q ,t = R q ,t , so w e obtain the ohomology of the Artin group of ane t yp e ˜ A n − 1 with M q − o eien ts b y means of Theorem 1.1 . In a similar w a y w e get the rational ohomology of G ˜ A n − 1 : Prop osition 4.3. W e have H ∗ ( G ˜ A n − 1 , Q ) ∼ = H ∗ ( G B n , Q [ t ± 1 ]) H ∗ ( G ˜ A n − 1 , Q ) ∼ = H ∗ ( G B n , Q [[ t ± 1 ]]) 16 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI wher e the ation of G B n on Q [ t ± 1 ] (and on Q [[ t ± 1 ]] ) is trivial for the gener- ators ǫ 1 , . . . , ǫ n − 1 and ( − t ) -multipli ation for the last gener ator ǫ n . T o obtain the rational ohomology of G ˜ A n − 1 w e ma y apply Prop osition 4.2 together with Theorem 3.3. 4.2. Cohomology of G A n with o eien t in the T ong-Y ang-Ma rep- resen tation. The T ong-Y ang-Ma represen tation is an ( n + 1) -dimensional represen tation of the lassial braid group G A n diso v ered in [TYM96℄. Be- lo w w e just reall it, referring to [Sys01℄ for a disussion of its relev ane in braid group represen tation theory . Denition 4.4. Let V b e the free Q [ u ± 1 ] -mo dule of rank n + 1 . The T ong- Y ang-Ma represen tation is the represen tation ρ : G A n → Gl Q [ u ± 1 ] ( V ) dened w.r.t. the basis e 1 , . . . , e n +1 of V b y: ρ ( σ i ) =     I i − 1 0 1 u 0 I n − i     where I j denote the j -dimensional iden tit y matrix and all other en tries are zero. Notie that the image of the pure braid group under the T ong-Y ang- Ma represen tation is ab elian; hene this represen tation fators through the extende d Coxeter gr oup presen ted in [Tit66 ℄. Prop osition 4.5. W e have H ∗ ( G B n , M [ t ± 1 ] q ,t ) ∼ = H ∗ ( G A n , M q ⊗ V ) H ∗ ( G B n , M [ t ± 1 ] q ,t ) ∼ = H ∗ ( G A n , M q ⊗ V ) wher e e ah gener ator of G A n ats on M q ⊗ V by ( − q ) -multipli ation on the rst fator and by the T ong-Y ang-Ma r epr esentation the se  ond fator. Sketh of pr o of. F or the statemen t in homology , b y Shapiro's lemma, it is enough to sho w that Ind G A n G B n M [ t ± 1 ] q ,t ∼ = M q ⊗ V . Notie that [ G A n : G B n ] = n + 1 and let  ho ose as oset represen tativ es for G A n /G B n the elemen ts α i = ( σ i σ i +1 · · · σ n − 1 ) σ n ( σ i σ i +1 · · · σ n − 1 ) − 1 for 1 ≤ i ≤ n − 1 , α n = σ n , α n +1 = e . Then b y denition of indued represen tation, there is an isomorphism of left G A n -mo dules, Ind G A n G B n M [ t ± 1 ] q ,t = n +1 M i =1 M [ t ± 1 ] e i where the ation is on the r.h.s. is as follo ws. F or an elemen t x ∈ G A n , write xα k = α k ′ x ′ with x ′ ∈ G B n . Then x ats on an elemen t r · e k ∈ L n +1 i =1 M [ t ± 1 ] e i as x ( r · e k ) = ( x ′ r ) · e k ′ . COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 17 Computing expliitly this ation for the standard generators of G A n , w e an write the represen tation in the follo wing matrix form: σ i 7→     − q I i − 1 0 − q q − 1 t 0 − q I n − i     for 1 ≤ i ≤ n − 1 , whereas σ n 7→   − q I n − 1 0 1 − t 0   . Conjugating b y U = Diag(1 , 1 , . . . , 1 , − q − 1 ) and setting u = − q − 2 t , one ob- tains the desired result. Finally , sine [ G A n : G B n ] = n + 1 < ∞ , the indued and oindued repre- sen tation are isomorphi; so the analogous statemen t in ohomology follo ws.  In partiular the ohomology of G B n determined in Theorem 1.1 is iso- morphi to the ohomology of G A n with o eien t in the T ong-Y ang-Ma represen tation t wisted b y an ab elian represen tation. By means of Shapiro's lemma, w e ma y as w ell determine the ohomology of G A n with o eien t in the T ong-Y ang-Ma represen tation. Indeed: Prop osition 4.6. W e have H ∗ ( G B n , Q [ t ± 1 ]) ∼ = H ∗ ( G A n , V ) H ∗ ( G B n , Q [ t ± 1 ]) ∼ = H ∗ ( G A n , V ) wher e V is the r epr esentation of G A n dene d in 4.4. As a onsequene w e ha v e Corollary 4.7. L et V b e the ( n + 1) -dimensional r epr esentation of the br aid gr oup Br n +1 dene d in 4.4. Then the  ohomolo gy H ∗ (Br n +1 , V ) is given as in The or em 3.3. R emark 4.8 . In partiular the homology of G ˜ A n − 1 with trivial o eien ts is isomorphi to the homology of G A n with o eien ts in the T ong-Y ang-Ma represen tation. 5. Rela ted topologial onstr utions W e refer to [CMS ℄ for the few  hanges whi h ha v e to b e done to the onstrution giv en in [Sal94 ℄ (see also [Sal87 ℄) for non-nite t yp e Artin groups (but still nitely generated). W e obtain a nite CW-omplex X W , expliitly desrib ed, whi h is a deformation retrat of the orbit sp a e of the Artin group. The latter is dened as the quotien t spae M ( A ) W := M ( A ) /W . where 18 F. CALLEGAR O, D. MOR ONI, AND M. SAL VETTI M ( A ) := [ U 0 + i R n ] \ [ H ∈A H C U 0 ⊂ R n b eing the in terior part of the Tits  one of W , while A is the h yp er- plane arrangemen t of W . The asso iated Artin group G W is the fundamen tal group of the orbit spae (see [Bou68 , Vin71, Bri71, D  un83 , vdL83℄). The simplest w a y to realize X W is b y taking one p oin t x 0 inside a hamb er C 0 and, for an y maximal subset J ⊂ S su h that the parab oli subgroup W J is nite, onstrut a | J | -ell (a p olyhedron) in U 0 as the on v ex h ull of the W J -orbit of x 0 in R n . So, w e obtain a nite ell omplex whi h is the union of (in general, dieren t dimensional) p olyhedra. Next, there are iden tiations on the faes of these p olyhedra, whi h are the same as desrib ed in [ Sal94 ℄ for the nite ase. The resulting quotien t spae is a CW-omplex X W whi h has a | J | -ell for ea h J ⊂ S su h that W J is nite. W e sho w an example in the ase ˜ A 2 (g. 4). Figure 4. the sp a e K ( G ˜ A 2 , 1) is given as union of 3 exagons with e dges glue d a  or ding to the arr ows (ther e ar e: 1 0- el l, 3 1- el ls, 3 2- el ls in the quotient). R emark 5.1 . When W is an ane group, the orbit spae is kno wn to b e a K ( π , 1) for t yp es ˜ A n , ˜ C n (see [Ok o79 , CP03 ℄) and reen tly for t yp e ˜ B n ([CMS1 ℄); see [CD95 ℄ for further lasses. R emark 5.2 . The standard presen tation for G W is quite easy to deriv e from the top ologial desription of X W ; w e ma y th us reo v er V an del Lek's result [vdL83 ℄. It follo ws Prop osition 5.3. L et K f in W := { J ⊂ S : | W J | < ∞} with the natur al strutur e of simpliial  omplex. Then the Euler har ateristi of the orbit sp a e (so, of the gr oup G W when suh sp a e is of typ e K ( π , 1) )) e quals χ ( K f in W ) . COHOMOLOGY OF AFFINE AR TIN GR OUPS AND APPLICA TIONS 19 If W is ane of r ank n + 1 we have χ ( M ( A ) W ) = χ ( K f in W ) = 1 − χ ( S n − 1 ) = ( − 1) n If W is t wo − dimensional (so, al l 3-subsets of S gener ate an innite gr oup) of r ank n then χ ( M ( A ) W ) = 1 − n + m wher e m is the numb er of p airs in S having nite weight ( m = n ( n − 1) 2 if ther e ar e no ∞ -e dges in the Coxeter gr aph). Pr o of. The rst t w o statemen ts w ere already remark ed in [CMS ℄. The last one is lear.  R emark 5.4 . The ohomology of the orbit spae in ase ˜ A n with trivial o eien ts is dedued from Corollary 4.3 and from Theorem 3.3; that with lo al o eien ts in the G ˜ A n -mo dule Q [ q ± 1 ] is dedued from Theorem 1.1 . Referenes [All02℄ D. Allo  k, Br aid pitur es for Artin gr oups , T rans. Amer. Math. So . 354 (2002), no. 9, 34553474 (eletroni). MR MR1911508 (2003f:20053) [Arn68℄ V. I. 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