Surgery groups of the fundamental groups of hyperplane arrangement complements
Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Ar…
Authors: S. Roushon
SUR GER Y GR OUPS OF TH E FUNDAMENT AL GR OUPS OF HYPER PLANE ARRANGEMENT COMPLEMENTS S. R OUSHON Abstract. Using a r ecent result of Bartels a nd L ¨ uck ([4]) we deduce that the F arrell-Jo nes Fiber ed Iso morphism co njectur e in L h−∞i -theory is true for a n y group which contains a finite index strongly poly-fr ee normal subgroup, in particula r , for the Artin full braid gro ups. As a consequence we explicitly c ompute the surgery gro ups of the Artin pure braid groups. This is obtained as a corollar y to a computation of the surgery gro ups of a more general class of groups, na mely for the fundament al group o f the complement of an y fiber-type hyperpla ne arrangement in C n . 1. Introduction The purp ose of this short note is to compute explic itly t he surgery ( L -)groups o f the Artin pure braid groups ( P B n ). This computation requires the solutions of t w o ot her problems. Firstly , one has to com- pute the low er algebraic K -theory o f the gro up and secondly , to sho w that the classical assem bly map in L -theory is an isomorphism. T his giv es an in terpretation of the surgery g r o ups in terms of a generalized homology theory . F or P B n w e had a lr eady computed t he lo w er algebraic K -theory in [1]. In [9] w e computed it for any subgroup of the Artin full braid group ( B n ), in particular for any subgroup of P B n . Here w e sho w that the classical assem bly map in L -theory is an isomorphism for an y sub- group of B n . The main ingredien ts b ehind the pro o f is the K -theoretic v anishing r esult [[9], theorem 1.1 ] and a recen t result of Bartels and L ¨ uc k ([[4], theorem B]). The later result is used to sho w that the L h−∞i - theory Fib ered Is omorphism conjecture of F arrell and Jones ([[8], § 1.7]) is true for any subgroup of B n . Finally , using the stable homotop y type Date : F ebrua r y 18, 2 011. 2000 Mathematics Su bje ct Classific ation. P rimary: 19G24, 19J25 Secondary : 57R67. Key wor ds and phr ases. assembly map, surger y groups, L - theo ry , Artin braid groups, h yp erplane arra ngement. T o app ear in Ar chiv der Mathematik . 1 2 S. ROUSHON of the corresp onding Eilen b erg-Maclane space of P B n from [22] we do the computation of the surgery groups. Let us first say a few w o r ds ab out the group P B n (and B n ) b efore w e state the computation of its surgery gr o ups. The A rtin ful l br aid gr oup B n is generated b y the sym b ols σ 1 , . . . , σ n with resp ect to t he relations σ i σ j = σ j σ i for | i − j | ≥ 2 a nd σ i σ i − 1 σ i = σ i − 1 σ i σ i − 1 for i ≤ n . See [2] for the o riginal source of this gro up. The map q : B n → S n +1 sending the generator σ i to the transp osition ( i, i + 1) defines a homomorphism on to the symm etric group S n +1 on ( n + 1 )-sym b ols and the k ernel of this homomorphism is defined as the Artin pur e br aid g r oup P B n . See [5] a nd [12] fo r some more information on braid groups. In this pap er w e need a t op ological in terpretation of P B n whic h w e describe below. Let H n b e the h yp erplane arrangement H ij = { ( x 0 , x 1 . . . , x n ) ∈ C n +1 | x i = x j } for i, j = 0 , 1 , . . . , n and i 6 = j in the ( n + 1 )- dimensional complex space C n +1 . The fundamental group of the com- plemen t C n +1 − ∪ i 6 = j H ij is isomorphic to P B n . Note that the group S n +1 acts freely on C n +1 − ∪ i 6 = j H ij b y p erm uting co ordinates. The fundamen t a l group of the quotien t space ( C n +1 − ∪ i 6 = j H ij ) /S n +1 is iso- morphic to B n . Therefore, there is a n exact seque nce of the follo wing t yp e. 1 → P B n → B n → S n +1 → 1 . F urthermore, the homomorphism B n → S n +1 in the ab o v e exact sequence coincides with q w e defined ab ov e. F or this in terpretation of the braid gro ups see [10]. Corollary 1.1. F or al l n ≥ 1 the sur gery gr oups o f the Artin p ur e br aid gr oup P B n ar e c ompute d as fol lows. L i ( P B n ) = Z if i ≡ 0 m o d 4 Z n ( n +1) 2 if i ≡ 1 m o d 4 Z 2 if i ≡ 2 m o d 4 Z n ( n +1) 2 2 if i ≡ 3 m o d 4 . Pr o of. This is an immed iate corollary of T heorem 2.2 since the arrange- men t H n is fib er- t yp e and there are n ( n +1) 2 h yp erplanes in H n . W e recall here that there are surgery groups for differen t kinds of surgery pro blems and they app ear in the lit era t ur e with the notations L ∗ i ( − ), where ∗ = h, s, h−∞i or h i i for i ≤ 0. But all of them a re naturally isomorphic for torsion-free groups G if the Whitehead group SURGER Y GROUPS OF BRAID GROUPS 3 W h ( G ), the reduce d pro jectiv e class group ˜ K 0 ( Z G ) and the negativ e K -gro ups K − i ( Z G ) for i ≥ 1 v a nish. See [[1 5], remark 1.21 and prop o- sition 1.23]. Therefore, we use the simplified no tation L i ( − ) in this pap er a s t he g roups w e consider ha v e the required prop erties. W e conclude the in tro duction by men t io ning that in fact w e prov e the Fib ered Isomorphism conjecture in L h−∞i -theory fo r a more general class of groups, namely for any finite extension Γ o f a str ongly p oly-fr e e gr oup ( see [[1], definition 1.1] or D efinition 2.1 b elow ) and deduce the isomorphism of the classical assem bly map in L -theory for any torsion- free subgroup of Γ. As a consequenc e w e compute the surgery groups of the fundamen tal group of a n y fib er-t yp e hy p erplane arrangemen t complemen t in the complex n -space C n (see Theorem 2.2). 2. St a te ments of the Main Theorem and its c onsequences Let us recall the definition of the strongly p oly-free groups. Definition 2.1. ( [1 ] ) A discrete group Γ is called str ongly p oly-fr e e if there exists a finite filtration of Γ by subgroups: 1 = Γ 0 ⊂ Γ 1 ⊂ · · · ⊂ Γ n = Γ such that t he following conditions are satisfied: 1. Γ i is normal in Γ for eac h i 2. Γ i +1 / Γ i is a finitely generated fr ee group 3. for eac h γ ∈ Γ and i there is a compact surface F a nd a dif- feomorphism f : F → F suc h that the induced homomorphism f # on π 1 ( F ) is equal to c γ in O ut ( π 1 ( F )), where c γ is the actio n of γ on Γ i +1 / Γ i b y conjug a tion and π 1 ( F ) is iden tified with Γ i +1 / Γ i via a suitable isomorphism. In suc h a situation w e sa y that the group Γ ha s r an k ≤ n . W e no w state our main theorem. Theorem 2.1. L et Γ b e a finite extension of a str ongly p oly-fr e e gr oup (the fin ite gr oup is the quotient gr oup). Then the Fib er e d Isomorphis m c onje ctur e of F arr el l and Jone s in L h−∞i -the ory is true fo r any sub gr oup of Γ . I n p articular, it is true for any sub gr oup of B n . Although w e prov e Theorem 2.1 for the conjecture in L h−∞i -theory stated in [[8], § 1.7], the pro of go es through, under certain conditions (see [[21 ], 3( b ) of theorem 2 .2]), in a general setup of the conjecture in equiv a rian t homology theory formulated in [3 ] and for a more general class of groups. A corollary of the a b o v e theorem and [[9], theorem 1.1] is the fol- lo wing. This show s that f or any torsion-free subgroup G of Γ the surgery group L i ( G ) is isomorphic to the generalized homolo gy gro up 4 S. ROUSHON H i ( B G, L 0 ). Here L 0 is a 1- connectiv e Ω-sp ectrum with 0 t h space ho- motop y equiv alen t to the classifying space G/T O P . This sp ectrum and the assembly m ap (or universal ho m omorphism ) men tioned in the b elo w statemen t we re originally constructed b y Quinn ([17], [18]) using geometric metho ds. F o r an algebraic treatmen t on this sub ject see [19] and [20 ]. Corollary 2.1. The classic al assembly m ap in sur gery the ory is an iso- morphism for any torsion-fr e e sub gr oup of Γ . T hat is, H i ( B G, L 0 ) → L i ( G ) is an isomorphism for al l i and for a l l torsion-fr e e sub gr oups G of Γ . In p articular, the asse m bly m ap is an isom o rphism for any sub gr oup of B n . Pr o of. Let H be a to rsion-free group so that the following a re satisfied. (1). W h ( H ) = K − i ( Z H ) = ˜ K 0 ( Z H ) = 0 for all i ≥ 1. (2). The Isomorphism conjecture in L h−∞i -theory is true for H . Then it is a kno wn f a ct that for all i , H i ( B H, L 0 ) → L i ( H ) is an isomorphism. F o r a detailed pro of see [[15], theorem 1.28] or [[8], 1.6 .3]. No w the pro of of the Corollary is immediate since (1) is satisfied for G by [[9 ], theorem 1.1 ] and (2) is satisfied b y Theorem 2 .1. The part icular case f o llo ws since P B n is stro ng ly p oly-free and B n is torsion-free (see the discussion after the follo wing Remark). Remark 2.1. Here we recall that the isomorphism of the a b o v e as- sem bly map is expected when the g roup is torsion-free. T he in tegr a l No vik ov conjecture in L -theory states that this assem bly map should b e split injectiv e. Before w e state our main c omputation of the surgery groups w e recall the definition of a fib er-type hy p erplane arra ngemen t f rom [[16], p. 162]. Suc h an arrangemen t A n ⊂ C n , that is t he union of a finite n um b er of a ffine h yp erplanes in C n is called strictly line arly fib er e d if after a suitable linear change of co or dinates, the restriction of t he pro jection of C n − A n to the first ( n − 1 ) co ordinates is a fib er bundle pro jection whose base space is the complemen t of an arrangemen t A n − 1 in C n − 1 and whose fib er is the complex plane min us finitely many p oin ts. By definition t he arrangement 0 in C is fib er-type and A n is defined to b e fib er-typ e if A n is strictly linearly fib ered and A n − 1 is of fib er t yp e. It follo ws by rep eated application of the homoto py exact sequence of a fibration that the complemen t C n − A n is aspherical. And hence π 1 ( C n − A n ) is torsion-free. The h yp erplane arrangemen t H n for P B n as described in the In tro- duction is a n example of a fib er-t yp e arrangemen t. SURGER Y GROUPS OF BRAID GROUPS 5 No w recall from [[9], theorem 5.3 ] that if A is a fib er-ty p e h yp erplane arrangemen t in C n , then the fundamen tal group π 1 ( C n −∪A ) is strongly p oly-free. In particular P B n is also strongly p oly-free. This was pro v ed in [[1], theorem 2.1]. As a consequ ence of Theorem 2.1 w e pro ve the following. Theorem 2.2. L et A = { A 1 , A 2 , . . . , A N } b e a fib er-typ e hyp erplane arr angement in C n , then the sur gery gr oups o f Γ = π 1 ( C n − ∪ N j = 1 A j ) ar e give n by the fol lowing. L i (Γ) = Z if i ≡ 0 mo d 4 Z N if i ≡ 1 mo d 4 Z 2 if i ≡ 2 mo d 4 Z N 2 if i ≡ 3 mo d 4 . 3. The Isomorphism Conjecture and rela te d res ul ts The Isomorphism conjecture of F arrell a nd Jones ([[8 ], § 1.6, § 1.7]) is a fundamen tal conjecture and implies man y w ell-known conjectures in algebra and top ology (see [14] for a quic k in tro duction to the conjecture and its consequence s or see [15]). The statemen t of the conjecture has b een stated in a general setup of equiv ariant homology theory in [3]. W e recall the statemen t b elow. Let H ? ∗ b e an equiv arian t homolog y theory with v alues in R -mo dules for R a commutativ e asso ciative ring with unit. A family of subgroups of a g roup G is defined as a set of subgroups of G whic h is closed under taking subgroups a nd conjug a tions. If C is a class of gr o ups whic h is closed under isomorphisms and taking subgroups then w e denote by C ( G ) the set o f a ll subgroups of G whic h b elong to C . Then C ( G ) is a family of subgroups of G . F or example V C , the class of virtually cyclic groups, is closed under isomorphisms and taking subgroups. By definition a virtual ly cyclic gr oup ha s a cyclic subgroup of finite index. Giv en a group homomorphism φ : G → H a nd a family C of sub- groups of H define φ ∗ C to b e the family o f subgroups { K < G | φ ( K ) ∈ C } of G . Giv en a family C of subgroups of a group G there is a G - CW complex E C ( G ) whic h is unique up to G -equiv alence satisfying the prop ert y that for H ∈ C the fixp oin t set E C ( G ) H is con t r actible and E C ( G ) H = ∅ for H no t in C . Let G b e a g r o up and C b e a family of subgroups of G . Then the Isomorphism c onje ctur e for the pair ( G, C ) states that t he pro jection 6 S. ROUSHON p : E C ( G ) → pt to the p oint pt induces an isomorphism H G n ( p ) : H G n ( E C ( G )) ≃ H G n ( pt ) for n ∈ Z . And the Fib er e d Isom orphism c onje ctur e for the pair ( G, C ) states that for an y group homomorphism φ : K → G the Isomorphism con- jecture is true for the pair ( K , φ ∗ C ). In this article w e are concerned with the equiv ariant homology theory arising in L h−∞i -theory and when C = V C and R = Z . This (F ib ered) Isomorphism conjecture is equiv alent to the F arrell-Jones conjectures stated in ( [[8], § 1.7]) [[8 ], § 1.6]. F or details see [[3], § 5 and § 6]. W e sa y that the FICwF L is true for a group G if the Fib ered Isomor- phism conjecture in L h−∞i -theory is true for G ≀ H for an y finite group H . Here G ≀ H denotes the semidirect pro duct G H ≀ H with resp ect to the regular action of H on G H = G × G × · · · × G ( | H | num b er of factors). Also we sa y that the FIC L is true for a group G if the Fib ered Isomorphism conjecture in L h−∞i -theory is true f o r G . Next, w e recall some standard results and some recen t dev elopmen t in this area whic h w e need for the pro of of Theorem 2.1. Also w e pro v e some basic results. Let us start by recalling that the F ib ered Isomorphism conjecture has the her e ditary p r op erty , that is if it is true for a gro up then it is true f o r any of its subgroups. Lemma 3.1. L et G b e a gr oup acting pr op erly disc ontinuously and c o c om p actly by is ometries on a metric sp ac e X . Then for any finite gr oup H the gr oup G ≀ H acts pr op erly dis c ontinuously and c o c omp actly by isometries on the p r o duct m e tric sp a c e X H = X × X × · · · × X ( | H | numb er of factors). Pr o of. This follo ws from the pro of of Serre’s theorem in [[7], theorem 3.1, p.190-191]. An immediate coro lla ry t o the ab ov e Lemma is the following. Re- call that a CA T(0)- space is a connected simply connected metric space whic h is nonp ositiv ely curv ed in the sense of distance comparison. F or example the univ ersal co v er ˜ M of a closed no np ositiv ely curv ed Rie- mannian manifold M with r esp ect t o the lifted metric is CA T(0). F or some mor e inf o rmation o n this sub ject see [6]. Corollary 3.1. I f G acts pr op erly disc ontinuously and c o c omp actly on a C A T (0) -sp ac e, then for any finite gr oup H , G ≀ H also acts p r op erly disc ontinuously and c o c omp actly on a CA T (0) -sp ac e. Pr o of. The pro of is immediate as the pro duct of tw o CA T(0)-spaces is again CA T(0). SURGER Y GROUPS OF BRAID GROUPS 7 A gr o up G is called CA T(0) if it acts prop erly and co compactly by isometries on a CA T(0)- space. Hence if M is as ab ov e then π 1 ( M ) is a CA T(0)-group. Therefore, b y Coro lla ry 3.1 f or an y finite group H , π 1 ( M ) ≀ H is also a CA T(0)-group. Lemma 3.2. The FIC L is true for V 1 × V 2 for a n y two virtual ly cyclic gr oups V 1 and V 2 . Pr o of. Since the FIC L is tr ue for any virtually cyclic group we can assume tha t b oth V 1 and V 2 are infinite. Hence V 1 × V 2 con tains Z × Z (= A , sa y) as a finite index normal subgroup. By the algebraic lemma in [9] V 1 × V 2 is a subgroup of A ≀ H , where H = ( V 1 × V 2 ) / A . Let T b e a flat 2-dimensional torus. Then b y Corollary 3.1 A ≀ H is a CA T(0)- group. Therefore b y [[4], theorem B] the FIC L is true fo r A ≀ H since the CA T(0)-space ˜ T H is finite dimensional. Here ˜ T denotes the unive rsal co v er of T with the lifted metric. Hence FIC L is t rue f o r V 1 × V 2 b y the hereditary prop ert y . Lemma 3.3. L et p : G → Q b e a surje ctive gr oup homomorph ism and assume that the FICwF L is true for Q , for ke r ( p ) and for p − 1 ( C ) for any infinite cyclic sub gr oup C of Q . Then G satisfies the FICwF L . Pr o of. The pro of is immediate using Lemma 3.2 a nd [[21], lemma 3.4 ]. Lemma 3.4. L et G b e isomorphic to one of the fol lowing gr oups. • The fundam ental gr oup of a close d nonp ositively curve d Rie- mannian manifold. • The fundam ental gr oup of a c omp act 3 -manifold M with nonempty b oundary so that ther e is a fib er bund le p r oje ction M → S 1 . • A finitely gene r ate d virtual ly fr e e gr o up. Then the FICwF L is true for G . Pr o of. Since FIC L is true fo r all finite groups we can assume t ha t G is infinite. Let M b e a closed no np ositiv ely curv ed Riemannian manifold so tha t π 1 ( M ) ≃ G . Then by Corollary 3 .1 for an y finite group H , G ≀ H is a CA T(0)-group and hence the FIC L is true for G ≀ H by [[4], theorem B] since the CA T(0)-space ˜ M H is finite dimensional. This completes the pro of o f the first item. No w w e giv e the pro of for the second item, then the third one will follo w using the hereditary prop ert y of the Fib ered Isomorphism con- jecture. Let S b e a fib er of the fib er bundle M → S 1 with mono drom y diffeo- morphism f : S → S . Then M is diffeomorphic to the mapping to r us of 8 S. ROUSHON f . Therefore M is a compact Haken 3-manifold (that is an irreducible 3-manifold whic h has a π 1 -injectiv e em b edded surface, see [1 1 ]) with b oundary comp o nen ts of zero Euler characteristic. W e now apply [[13], theorem 3.2 and 3.3] to get a complete nonp ositiv ely curv ed Riemann- ian metric in the interior of N so that near the b oundary the metric is the pro duct flat metric, that is eac h end is isometric to X × [0 , ∞ ) for some flat 2-manifold X . Therefore if w e tak e the double D of M , w e get a closed nonp ositiv ely curve d Riemannian manifold. Hence b y the first item FICwF L is true for π 1 ( D ) and consequen tly for π 1 ( M ) a lso b y the hereditary pro p ert y . This completes the pro of of the Lemma. 4. Isomorphism of the Assembl y map and c omput a tion of the surger y groups In this section we give the pro ofs of Theorems 2.1 a nd 2.2. Pr o of o f Th e or em 2 . 1 . Let G b e a finite index strongly p oly-free normal subgroup of Γ. Then by the alg ebraic lemma in [9] Γ can b e em b edded as a subgroup in G ≀ (Γ / G ). Therefore using the hereditary prop ert y it is enough to pro v e the FICwF L for an y strongly po ly free group. Hence w e can assume that Γ is strongly p oly-free. The pro o f is by induction o n the rank of Γ and the framew ork o f the pro of is same as that of the pro of of [[21 ], 3( b ) of theorem 2.2]. If the rank is ≤ 1 then Γ is a finitely generated free group and hence the theorem fo llo ws fr o m the third item in Lemma 3.4. Therefore assume that the rank of Γ ≤ k and that the F ICwF L is true for a ll strongly p oly-f r ee gr o ups Γ of rank ≤ k − 1 . Let 1 = Γ 0 < Γ 1 < · · · < Γ k = Γ b e a filtration of Γ. Consider the follo wing exact sequenc e. 1 → Γ 1 → Γ → Γ / Γ 1 → 1 . Let q : Γ → Γ / Γ 1 b e the ab o v e pro jection. The follow ing assertions are easy to v erify . • Γ / Γ 1 is strongly p oly-free and has rank ≤ k − 1 . • q − 1 ( C ) is a finitely generated free gro up or isomorphic to the fundamen t a l group of a compact Hak en 3-manifold M with nonempt y b oundary so that there is a fiber bundle pro jection M → S 1 , where C is either the trivial group or an infinite cyclic subgroup of Γ / Γ 1 resp ectiv ely . No w w e can apply the induction h yp ot hesis, Lemma 3.3, and Lemma 3.4 to complete the pro of of the Theorem. SURGER Y GROUPS OF BRAID GROUPS 9 The particular case for B n follo ws as P B n is strongly p oly-free ([[1], theorem 2.1]) and is an index ( n + 1)! nor ma l subgroup of B n . T o prov e Theorem 2.2 w e need the following lemma regarding the top ology of an arbitra r y hyperplane ar r a ngemen t complemen t in the complex n -space C n . Lemma 4.1. The first susp en sion Σ( C n − ∪ N j = 1 A j ) of the c omplement of a hyp erplane arr angemen t A = { A 1 , A 2 , . . . , A N } in C n is homo topi- c al ly e quivalent to the we dge of spher es ∨ N j = 1 S j wher e S j is home omor- phic to the 2 -s p her e S 2 for j = 1 , 2 , . . . , N . Pr o of. Let A = { A 1 , A 2 , . . . , A N } b e an arrangemen t b y linear sub- spaces of C n . Then in [[22], (2) of prop osition 8] it is pro v ed t ha t Σ( C n − ∪ N j = 1 A j ) is homotopically equiv alent to the fo llowing space. Σ( ∨ p ∈ P ( S 2 n − d ( p ) − 1 − ∆ P
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