TOPOLOGY HYPERPLANE ARRANGEMENTS. Alex Suciu. Northeastern University.

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1 TOPOLOGY OF HYPERPLANE ARRANGEMENTS Alex Suciu Northeastern University A.M.S. Fall Eastern Section Meeting Columbia University, New York, NY November 5,

2 Hyperplane arrangements Arrangement: A collection A = {H 1,...,H n } of hyperplanes, H i =ker α i,inc l. Defining polynomial: Q A = α 1 α n. Intersection lattice: L(A) = { H B H B A}. ordered by reverse inclusion, ranked by codimension Complement: X(A) =C l \ H A H. Example. Boolean arrangement D n in C n. Q Dn = z 1 z n L(D n ): X(D n )=(C ) n π 1 (X(D n )) = Z n subsets of [n] :={1,...,n} complex n-torus Example. Braid arrangement B l in C l. Q Bl = 1 i<j l (z i z j ) L(B l ): X(B l )=F (l, C) π 1 (X(B l )) = P l partition lattice of [l] configuration space of l ordered points in C pure braid group on l strings; in fact: X(B l ) K(P l, 1) 2

3 Types of arrangements Complexified A = A R C (i.e., Q A has real coefficients) Reflection Complexification of reflecting hyperplanes of a Coxeter group. E.g., B l of type A l-1. Simplicial Complexification of real arrangement, all of whose complementary regions are open simplices. Deligne: A simplicial = X(A) K(π, 1). Fiber-type (Falk-Randell) There is a tower of linear fibrations X(A) p l X(A l 1 ) p l 1 X(A 2 ) p 2 X(A 1 )=C with fiber(p i )=C \{d i points}. Thus, X K(G, 1), with G = F dl ρl ρ2 F d1, where ρ i : π 1 (X(A i 1 )) P di Aut(F di ). Terao: A fiber-type L(A) supersolvable Cohen-S.: Explicit ZG-resolution C ( X) Z (using the Fox Jacobians of ρ i ) 3

4 Generic A = {H 1,...,H n } in C l (2 < l < n), s.t. codim H B H = B, for all B Awith B l. E.g.: Boolean Hattori: X(A) S 1 (T n 1 ) (l 1). Thus: π 1 (X) =Z n, π i (X) =0 (if 1 <i<l 1) π l 1 (X) 0. Hypersolvable (Jambu-Papadima) Combinatorial condition, generalization of both supersolvable and generic. Hattori s result generalizes in this context (Papadima-S.) Graphic Each simple graph G with vertices {1,...,m} defines an arrangement in C m : A G = {ker(z i z j ) {i, j} edge in G} G diagram of type A = A G Boolean G = K m = A G = B m G polygon = A G generic every cycle in G has a chord A G fiber-type (Stanley, Fulkerson-Gross) 4

5 Cohomology ring H (X(B l )) = H (P l ): Arnol d H (X(A)): Brieskorn, Orlik-Solomon H (F (l, R k )): F. Cohen H (F (l, S k )): Feichtner-Ziegler, Xicoténcatl H (X(subspace arrangement)): Goresky-MacPherson, Björner-Ziegler, De Concini-Procesi, Yuzvinsky, de Longueville Theorem. (Orlik-Solomon) Let A = {H 1,...,H n } be a hyperplane arrangement. Then L(A) determines the cohomology ring of X = X(A): H (X) = Z n /( e B codim H B A ) H< B where Z n = exterior algebra over Z on generators e i (dual to meridian of H i ) in degree 1, and, for B = {H i1,...,h ir }, e B = e i1 e ir and e B = q ( 1)q 1 e i1 ê iq e ir. H (X) is torsion-free; basis: no broken circuits (nbc). Poincaré polynomial: P (X, t) = µ(y )( t) codim Y where µ : L(A) Z Möbius function. Y L(A) 5

6 Resonance varieties Let A = H (X(A), C) be the Orlik-Solomon algebra of A. Forλ C n, let e λ = n i=1 λ ie i A 1 = C n. Aomoto complex : 0 A 0 e λ A 1 e λ A 2 e λ A l 0 The resonance varieties of A were defined by Falk as: R k d(a) ={λ C n dim C H k (A,e λ ) d} They are actually subsets of n := {λ n i=1 λ i =0}, and depend only on A (up to linear iso of C n ). Theorem. Each component of Rd k (A) is a linear subspace in C n. Conjectured by Falk. Proved by Cohen-S., Libgober (k = 1); Cohen-Orlik, Libgober (all k). The varieties R d (A) =Rd 1 (A) admit a purely combinatorial description started by Falk, completed by Libgober-Yuzvinsky, using Vinberg s classification of affine Kac-Moody algebras. 6

7 A partition P =(p 1 p q )ofa is neighborly if ( p j I I 1) = I p j, I L 2 (A) It defines a linear subspace of C n : L P = n {λ {I L 2 (A) I p j, j} i I λ i =0}. Theorem. (Libgober-Yuzvinsky) All components L i of R 1 (A) arise from neighborly partitions of sub-arrangements A A. dim L i 2. L i L j = {0} for i j. R d (A) ={0} dim L i d+1 L i. E.g., for each I L 2 (A) with I 3, there is a local component L I = {λ n i=1 λ i =0 and λ i =0 for i/ I} corresponding to the partition (I)ofA I = {H i i I}. Note that dim L I = I 1, and thus L I R I 2 (A). 7

8 Example. (Braid arrangement B = B 4 ) L 156 L 246 L 345 L 123 L ( ) The OS-algebra A = H (X(B), C) has generators e 1,...,e 6, and relations e 2 i = e ie j + e j e i =0 and e 1 e 2 e 1 e 3 + e 2 e 3 =0, e 1 e 5 e 1 e 6 + e 5 e 6 =0, e 2 e 4 e 2 e 6 + e 4 e 6 =0, e 3 e 4 e 3 e 5 + e 4 e 5 =0. The resonance variety R 1 (B) C 6 has 5 components (4 local, and 1 non-local), all 2-dimensional: L 123 = {λ λ 1 + λ 2 + λ 3 = λ 4 = λ 5 = λ 6 =0} L 156 = {λ λ 1 + λ 5 + λ 6 = λ 2 = λ 3 = λ 4 =0} L 246 = {λ λ 2 + λ 4 + λ 6 = λ 1 = λ 3 = λ 5 =0} L 345 = {λ λ 3 + λ 4 + λ 5 = λ 1 = λ 2 = λ 6 =0} L ( ) = {λ λ 1 λ 4 = λ 2 λ 5 = λ 3 λ 6 = λ i =0} 8

9 Characteristic varieties X finite CW-complex, G = π 1 (X). Assume G ab = Z n (with basis t 1,...,t n ). Character variety: Hom(G, C ) = (C ) n algebraic torus, with coordinate ring C[t ±1 1,...,t±1 n ] Characteristic varieties: V k d (X) ={t (C ) n dim C H k (X, C t ) d} where C t is the G-module C with action given by the representation t : G C. Vd k (X) depends only on the homotopy type of X (up to a monomial isomorphism of (C ) n ). Theorem. (Arapura) Suppose X is the complement of a normal-crossing divisor in a compact Kähler manifoldwith b 1 =0. Then the components of V k d (X) are subtori of (C ) n, possibly translatedby roots of 1. Uses Deligne s mixed Hodge structures. Generalizes results of Green-Lazarsfeld, Simpson. 9

10 Characteristic varieties of A = {H 1,...,H n }: V k d (A) :=V k d (X(A)) (C ) n Recall the resonance varieties Rd k(a) Cn. Theorem. (Cohen-S., Cohen-Orlik, Libgober) TC 1 (V k d (A)) = R k d(a) As a consequence, the components of Vd k (A) passing through 1 are combinatorially determined (by L(A)). In general, though, there do exist components that do not pass through 1 (i.e., translated subtori). Question. Are such components combinatorially determined? Remark. Rd k (X) may be defined for arbitrary X. Then (Libgober): TC 1 (V k d (X)) R k d(x). But the inclusion is strict in general, e.g., for link complements (Matei), and even for complements of real subspace arrangements (M.-S.). 10

11 Fundamental groups of arrangements A = {H 1,...,H n } hyperplane arrangement, with complement X, and fundamental group G = π 1 (X). A = {l 1,...,l n } generic 2-section. By Lefschetz-type theorem of Hamm and Lê :π 1 (X) = π 1 (X ). So reduce to the case where A is an arrangement of affine lines in C 2. Let v 1,...,v s be the intersection points of the lines. v q = l i1 l ir I q := {i 1,...,i r } Lattice: L 1 (A) =[n], L 2 (A) ={I 1,...,I s }. Group: G(A) =π 1 (X(A)). Question. Is G(A) combinatorially determined? I.e.: L(A 1 ) = L(A 2 ) = G(A 1 ) = G(A 2 )? According to Rybnikov, the answer is no. 11

12 Presentation for G = G(A) (Van Kampen, Artin/Randell, Salvetti, Arvola/Moishezon, Libgober, E. Hironaka, Cordovil-Fachada, Cohen-S.,... ): G = x 1,...,x n α 1 (x i )=x i,...,α s (x i )=x i (1 i n) where α q = A δ q I q P n acting on F n = x 1,...x n via the Artin representation (A I =full twist on I-strands, and δ q B n can be read from a braided wiring diagram ). Example. l 4 v 3 v 1 v 4 v 2 l 1 l 3 l 2 I 1 I 2 I 3 I α 1 = A 23, α 2 = A A 23 13, α 3 = A 124, α 4 = A

13 G = x1,x 2, x 1 x 2 x 4 = x 4 x 1 x 2 = x 2 x 4 x 1, x 3,x 4 [x 1,x 3 ]=[x 2,x 3 ]=[x 4,x 3 ]=1 = F 2 Z 2 13

14 Characteristic varieties of G (over field K): V d (G, K) ={t Hom(G, K ) dim K H 1 (G, K t ) d} For d<n, we have (E. Hironaka): V d (G, K) ={t (K ) n rank K A G (t) <n d} where A G = J ab G is the Alexander matrix of G, obtained by abelianizing the Fox Jacobian J G = ( r i x j ). Resonance varieties of G (over K): R d (G, K) = { λ H 1 (G, K) subspace W H 1 (G, K), dim W = d +1,λ W =0 } Then (Matei-S.): R d (G, K) ={λ K n rank K A (1) G (λ) <n d} where A (1) G (λ) =(A G ti =1 λ i ) linear terms is the linearizedalexander matrix of G. Remark. If G = π 1 (X(A)), then R d (A) =R d (G, C). But: R d (G, C) modp R d (G, F p ), TC 1 (V d (G, F p )) R d (G, F p ). 14

15 Homology of finite covers Theorem. (Libgober, Sakuma) G f.p.group, G ab = Z n, 1 K G γ Γ 1. IfΓ finite abelian, then: b 1 (K) =n + (corank J ρ γ G 1) 1 ρ Hom(Γ,C ) More generally, let b (q) 1 (G) :=dim K H 1 (G, K) where K is a field of characteristic q. Theorem. (Matei-S.) If Γ finite and q Γ, then: b (q) 1 (K) =b(q) 1 (G)+ n ρ (corank J ρ γ G n ρ) ρ 1 (sum over all non-trivial irreps ρ :Γ GL(n ρ, K), K field of char. q containing all roots of 1 of order exp Γ). 15

16 Corollary. (M.-S.) Let K =ker(γ : G Z N ). Then: b (q) 1 (K) =b(q) 1 (G)+ 1 k N φ(k) depth K (γ N/k ) where depth K (t) :=max{d t V d (G, K)}. Application to homology of Milnor fiber of a (central) arrangement A. Milnor fibration: F (A) X(A) Q A C H (F (A)) studied by Randell, Orlik-Randell, Artal Bartolo, Cohen-S., Denham, Cohen-Orlik,... F (A) isthen-fold cyclic cover of X(A), given by γ : G Z n, γ(x i )=1 Thus: b (q) 1 (F )=n 1+ φ(k) depth K (γ n/k ) 1 k n Question. Is H (F (A)) combinatorially determined? Question. Is H (F (A)) torsion-free? In particular, is b (q) 1 (F )=b 1(F ), for all q n? 16

17 Corollary. (M.-S.) K =ker(γ : G Z p ). Then: b (q) 1 (K) =b 1(G)+(p 1) depth K (γ) where K = C if q =0,orK = F q s if q prime, q p. s = ord p (q) =smallest positive integer s.t. p q s 1 F s q = F q (ζ), where ζ is a primitive p-th root of 1 Define: β (q) p,d (G) = {K G [G : K] =p and b (q) 1 (K) =b(q) 1 (G)+(p 1)d } Then: β (q) p,d (G) = Tors p(v d (G, K) \ V d+1 (G, K)) p 1 where for V K n. Tors N (V )={t V t N = 1 and t 1} 17

18 Representations onto finite groups G f.g. group, Γ finite group. δ Γ (G) := Epi(G, Γ)/ Aut(Γ) = {factor groups of G that are isomorphic to Γ} May compute δ Γ (G) when Γ abelian, or a semidirect product of (certain) abelian groups (Matei-S.) Γ abelian Write Γ = p Γ Γ p, where Γ p is a finite abelian p-group. Clearly, δ Γ (G) = p Γ δ Γ p (G). Write Γ p = Z p ν 1 Z p ν k,ν=(ν 1 ν k ). If G ab = Z n, then: δ Γp (G) = p ν (n 1) 2 ν ϕ n (p 1 ) ϕ n k (p 1 ) r 1 ϕ m r (ν)(p 1 ) where: ν = k i=1 ν i, ν = k i=1 (i 1)ν i, m r (ν) = {j ν j = r}, ϕ m (t) = m i=1 (1 ti ). 18

19 Γ=Z s q σ Z p (p q primes, s = ord p (q), σ Aut(Z s q) of order p) δ Z s q σ Z p (G) = p 1 s(q s 1) n d=1 β (q) p,d (G)(qsd 1) Thus, we may compute δ Z s q σ Z p (G) from V d (G, F q s). E.g., for S 3 = Z 3 ( 1) Z 2 and A 4 = Z 2 2 ( ) 01 Z3 : δ S3 (G) = 1 2 δ A4 (G) = 1 3 d 1 d 1 11 Tors 2 (V d (G, F 3 ) \ V d+1 (G, F 3 ))(3 d 1) Tors 3 (V d (G, F 4 ) \ V d+1 (G, F 4 ))(4 d 1) This gives info about a k (G) = {index k subgroups of G}, a k (G) = {index k normal subgroups of G}, e.g.: a 3 = 1 2 (3n 1) + 3δ S3 a 4 = 1 3 (2n+1 1)(2 n 1)+4(δ D8 + δ A4 + δ S4 ) a 3 = 1 2 (3n 1) a 4 = 1 3 (2n+1 1)(2 n 1) a 6 = 1 2 (3n 1)(2 n 1) + δ S3 a 8 = 1 21 (2n+2 1)(2 n+1 1)(2 n 1) + δ D8 + δ Q8 19

20 Congruence covers X finite cell complex, with H 1 (X) = Z n. X N is the regular (Z N ) n -cover of X determined by π 1 (X) ab H 1 (X, Z) modn H 1 (X; Z N ). By Libgober and Sakuma: b 1 (X N )=n + t Tors N (C ) n depth C (t). Theorem. (Sarnak-Adams, Sakuma) The sequence {b 1 (X N )} N N is polynomially periodic, i.e., there is T 1, andpolynomials P 1 (x),...,p T (x), such that b 1 (X N )=P i (N), if N i mod T. Follows from above formula, and Theorem. (Laurent) If V is a subvariety of (C ) n, then Tors N (V )= v i=1 Tors N(S i ), where S i are subtori of (C ) n, possibly translatedby roots of unity. For n 6: b 1 (X N (A)) = P A (N). For n =7: As.t. b 1 (X N (A)) has period T =2. 20

21 Hirzebruch covering surfaces A arrangement of n planes in C 3. A projectivized arrangement of lines in CP 2. X N (A) congruence cover of X(A). X N (A) associated branched cover CP 2. M N (A) minimal desingularization of X N (A). Hirzebruch computed the Chern numbers of M N (A): c 2 1 = ( (3b 2 s 5n+9)N 2 4(b 2 n)n +(b 2 +n+m 2 ) ) N n 3 c 2 = ( (b 2 2n +3)N 2 2(b 2 n)n +(b 2 + s m 2 ) ) N n 3 where m r = {I L 2 (A) I = r}, s = m r, b 2 = r m r(r 1) Sakuma computed the first Betti number of M N (A): b 1 (M N (A)) = depth C (t At ) t Tors N (C ) n where A t = {H i A t(x i ) 1}. Theorem. (Hironaka, Sakuma) The sequence {b 1 (M N (A))} N N is polynomially periodic. For n 7: b 1 (M N (A)) = P A (N). For n =8: As.t. b 1 (M N (A)) has period T =4. 21

22 Lower central series quotients G f.g. group. Lower central series: G = γ 1 G γ 2 G, where γ k+1 G =[γ k G, G] LCS quotients: gr k G = γ k G/γ k+1 G (f.g. abelian) Chen groups: gr k (G/G ). φ k (G) =rank(gr k G) θ k (G) =rank(gr k G/G ) Clearly, φ 1 = θ 1, φ 2 = θ 2, φ 3 = θ 3, φ k θ k. E.g.: φ k (F n )=w k (n) := 1 k µ(d)n k/d (Witt) θ k (F n )= ( n+k 2 k d k ) (k 1), for k 2 (Murasugi, Massey-Traldi) Massey: gr k (G/G )=I k 2 B/I k 1 B B = G /G : Alexander invariant (Z[G/G ]-module), I = ker ɛ: augmentation ideal. 22

23 Hence, if G/G = Z n : k 0 θ k+2 t k =Hilb(gr B) A presentation for gr B = k 0 gr k B (as module over gr Z[G/G ] = Z[λ 1,...,λ n ]) can be obtained from a presentation for B via a Gröbner basis algorithm. Let A be arrangement of n hyperplanes, G = G(A). Falk: φ k = φ k (G) combinatorially determined. Cohen-S.: Presentation for the Alexander invariant B = G /G. This gives algorithm for computing the Chen groups of G(A). E.g., for P n = n 1 i=1 F i: θ k (P n )=(k 1) ( ) n+1 4, for k 3. It follows that θ k (P n ) θ k (Π n ), where Π n = n 1 i=1 F i. On the other hand, φ k (P n )=φ k (Π n ), by Theorem. (LCS formula of Falk andrandell) If A fiber-type, with exponents d 1,...,d l : (1 t k ) φ k = P (X, t) = k 1 l (1 d i t) i=1 i.e., φ k (G) = l i=1 φ k(f di ). 23

24 Kohno: First proved LCS formula for A = B l. Shelton-Yuzvinsky: Consequence of Koszul duality. Papadima-Yuz.: Extend LCS formula to formal rational K(G, 1) spaces. Also, if A arrangement in C 3 : LCS formula holds A fiber-type. Jambu-Papadima: Extend LCS formula to hypersolvable arrangements. Let R 1 (A) = v i=1 L i, with L i linear subspaces of C n. Let h r = {L i dim L i = r}. Conjecture. The Chen groups of G = G(A) are free abelian, of rank θ k (G) = r 2 h r θ k (F r ), for k 4. (This would imply that the Chen groups of an arrangement are combinatorially determined.) Conjecture. If φ 4 = θ 4, then gr k G is free abelian, of rank φ k (G) = r 2 h r φ k (F r ), for k 4. 24

25 Resonance varieties & nilpotent quotients Let X be a finite CW-complex, with H (X) torsionfree, and H (X) generated in degree 1. E.g.: X = X(A), or X = X(link in S 3 with lk i,j = ±1). Let G = π 1 (X). Theorem. (Matei-S.) If X, X as above, then: H 2 (X) = H 2 (X ) G/γ 3 G = G /γ 3 G. For p prime, d 0, define: { ν p,d (G/γ 3 G)= K G/γ 3 G Then: [G/γ 3 G : K] =p and dim Fp (Tors H 1 (K)) F p = d } ν p,d (G/γ 3 G)= (R p,d(g, F p ) \ R p,d+1 (G, F p )) p 1 One may define higher-order resonance varieties, S (k) d (G, K), using higher-order Massey products <x,λ,...,λ>,or higher-order truncations of A G (D. Matei, Ph.D. thesis). One may also define ν p,d (G/γ k+2 G), as above. An analogue of the framed formula remains to be found. 25

26 Higher homotopy groups Let G be a graph, A = A G, X = X(A). Assume G has no 3-cycles. Then π 1 (X) =Z n, where n = edges. Let S be the set of 4-cycles. Proposition. (Papadima-S.) If S =, then π 2 (X) =0. Otherwise, the Zπ 1 -module π 2 (X) is combinatorially determined (by the graph G), and π 2 (X) π1 = Z[S] 0. Let V d (π 2 )=V ( d π 2 ) (C ) n. These varieties may distinguish π 2 s with the same coinvariants. Example. e 6 e 1 e 1 e 7 e 2 e 7 e 2 e 9 e e 9 e 8 8 e 6 e3 e3 e 5 e 5 e 4 The graphs G 1 and G 2 have no 3-cycles. Each graph has exactly two 4-cycles. The complements X i = X(A Gi )have: π 1 = Z 9, (π 2 ) π1 = Z 2, b 2 =36, b 3 =82. V 1 (π 2 (X 1 )) has 2 components, V 1 (π 2 (X 1 )) has 3. Hence: e 4 π 2 (X 1 ) = π 2 (X 2 ) (as Zπ 1 -modules). 26

27 Example (Braid arrangement) C 156 C 246 C 345 Π C 123 Q = xyz(x y)(x z)(y z). n =6, s =7, m 2 =3, m 3 =4. P (X, t) =(1 + t)(1 + 2t)(1 + 3t). G = P 4 = F 3 F 2 F 1. V 1 (G, K) =C 124 C 135 C 236 C 456 Π, where Π=C ( ) = {(s, t, (st) 1,s,t,(st) 1 ) s, t K }. V 2 (G, K) ={1} β (q) p,1 = ν p,1 =5(p + 1). b 1 (X N )=5N b 1 (M N )=5(N 1)(N 2) c 2 1(M N )=5N 3 (N 2) 2, c 2 (M N )=N 3 (2N 2 10N+15). δ S3 = 15, δ A4 = 20, a 2 = 63, a 3 = 364, a 3 = 409. φ 1 =6,φ 2 =4,φ 3 = 10, φ 4 = 21, φ k = w k (2) + w k (3). θ 1 =6,θ 2 =4,θ 3 = 10, θ 4 = 15, θ k =5(k 1). 27

28 Example (Non-Fano plane) ρ Π 1 Π 2 Π 3 Q = xyz(x y)(x z)(y z)(x + y z). n =7,s =9,m 2 =3,m 3 =6. P (X, t) =(1+t)(1+3t) 2. V 1 (G, K) has nine 2-dim components. V 2 (G, K) =Π 1 Π 2 Π 3 = {1,ρ}, where ρ 2 = 1 R 1 (G, F 2 ) has 3-dim component Υ TC 1 (V 1 (G, F 2 )). R 2 (G, F 2 ) has 1-dim component Υ TC 1 (V 2 (G, F 2 )). β (q) p,1 = ν p,1 =9(p + 1), except for: β (q) 2,1 = ν 2,1 = 24, β (q) 2,2 = ν 2,2 =1. 9N 2 3 if N even, b 1 (X N )= 9N 2 2 if N odd. b 1 (M N )=9(N 1)(N 2). δ S3 = 28, δ A4 = 36, a 2 = 127, a 3 =1, 093, a 3 =1, 177. φ 1 =7,φ 2 =6,φ 3 = 17, φ 4 = 42, φ 5 = 123, φ 6 = 341, φ 7 =1, 041. θ 1 =7,θ 2 =6,θ 3 = 17, θ 4 = 27, θ k =9(k 1). Υ Υ 28

29 Example (Deleted B 3 arrangement) ρ + ρ Ω Π 1 Π 2 Π 3 Π 4 t 1 1 t t 2 t 2 t 1 t 1 Q = xyz(x y)(x z)(y z)(x y z)(x y + z). n =8, s =11, m 2 =4, m 3 =6, m 4 =1. V 1 (G, K) has a 1-dim component which does not pass through 1 (unless char K =2): Ω={(t, t 1, t 1,t,t 2, 1,t 2, 1) t K } Π 5 29

30 ν p,1 =11(p + 1), ν p,2 = p3 1 p 1, β(q) p,d = ν p,d, except: β (q) 2,1 =27, β(q) 2,2 =9,β(2) 3,1 =45. Distribution of index 3, normal subgroups in G: K ab Z 8 Z 8 Z 2 2 Z 10 Z 12 K 1, ( ) ( ) K = ker(γ : G Z 3 ), γ =(ω, ω 2,ω 2,ω,ω 2, 1,ω,1) (Z 3 ) 8. Since γ/ V 1 (G, C), but γ Ω V 1 (G, F 2 (ω)): b 1 (X N )= b 1 (K) =8, b (2) 1 (K)=8+(3 1) 1=10. 2N 3 +11N 2 + N 9 2N 3 +11N 2 5 if N even, if N odd, b 1 (M N )= (N 1)(2N 2 +9N 24) + N 2 if N 0 mod 4 (N 1)(2N 2 +9N 24) + 1 (N 2) if N 2 mod 4 2 (N 1)(2N 2 +9N 24) if N odd. δ S3 = 63, δ A4 = 110, a 2 = 255, a 3 =3, 280, a 3 =3, 469. φ 1 =8,φ 2 =9,φ 3 = 28, φ 4 = 78, φ k = w k (3) + w k (4). θ 1 =8,θ 2 =9,θ 3 = 28, θ 4 = 48, θ k =(k + 12)(k 1). 30

31 Example (MacLane arrangement) Q = xyz(y x)(z x)(z+ωy)(z+ω 2 x+ωy)(z x ω 2 y). n =8,s = 12, m 2 =4,m 3 =8. P (X, t) =(1+t)(1+7t +13t 2 ). V d (G, K) has only local components R 1 (G, F 3 ) has one non-local, 2-dim component: Ξ={(λ µ, λ, µ, λ, µ, λ + µ, λ µ, µ λ) λ, µ F 3 } β (q) p,1 =8(p+1) and ν p,1 =8(p+1), except for ν 3,1 = 36. b 1 (X N )=8N 2. b 1 (M N )=8(N 1)(N 2). δ S3 = 24, δ A4 = 32, a 2 = 255, a 3 =3, 280, a 3 =3, 352. φ 1 =8,φ 2 =8,φ 3 = 21, φ 4 = 42, φ 5 = 87, φ 6 = 105. θ 1 =8,θ 2 =8,θ 3 = 21, θ 4 = 24, θ k =8(k 1). Note: There is torsion in the LCS quotients of G. E.g.: gr 5 G = Z 87 Z 4 2 Z 3 Ξ 31

32 Example (Ziegler arrangements). A 1 1 A 1 2 n = 13, s = 31, m 2 = 20, m 3 =9,m 5 =1,m 7 =1. P (X i,t)=(1+t)(1+6t) 2. φ 1 = 13, φ 2 = 30, φ 3 = 140, φ k =2w k (6). θ k = (k 1)(k4 +10k 3 +47k 2 +86k+696) 24. R d (G 1, K) and R d (G 2, K) are (abstractly) isomorphic. Hence, ν p,d (G 1 )=ν p,d (G 2 ). a 2 =8, 191, a 3 = 797, 161, a 3 = 820,

33 V 1 (G 1, K) has τ 1 = 3 translated subtori V 1 (G 2, K) has τ 2 = 2 translated subtori β (q) 2 = (69, 4, 15, 0, 63) β (q) p = (27(p +1), 0, p4 1 p 1, 0, p6 1 p 1 ) except if p =3,q =2,d =1 Thus: δ Γ (G 1 )=δ Γ (G 2 ), if Γ = Z s q σ Z p = A 4. But: Hence: β (2) 3,1 (G 1) =111, β (2) 3,1 (G 2) =110. δ A4 (G 1 ) =124, 435, δ A4 (G 2 ) =124, 434. b 1 (X N (A i )) = 5N 6 +3N 4 +27N 2 + τ i (N 2) 26 if N even 5N 6 +3N 4 +27N 2 22 if N odd f(n)+τ i (N 2) if N 0 mod 4 b 1 (M N (A i )) = f(n)+ τ i 2 (N 2) if N 2 mod 4 f(n) if N odd, where f(n) =(N 1)(5N 5 2N 4 + N 3 4N 2 +23N 58) 33