ar X iv :m at h/ 03 07 29 6v 2 [ m at h. A G ] 2 5 O ct 2 00 4 Topology and combinatorics of real line arrangements Enrique ARTAL BARTOLO, Jorge CARMONA RUBER, José Ignacio COGOLLUDO AGUSTÍN and Miguel MARCO BUZUNÁRIZ Accepted in Compositio Math. Abstract We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in P2. Such pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over Q( √ 5). 1. Introduction In a famous preprint [Ryb98], G. Rybnikov proves the existence of two line arrangements L1,L2 in P2 := CP2 which have the same combinatorics but whose pairs, (P2, ⋃ L1) and (P2, ⋃ L2), are not homeomorphic. Unfortunately, this work has not been published in its final form. This may be due to the difficulty of verifying some of its statements. In another work ([ACCM04]), some parts of Rybnikov’s paper are presented and explained, including a detailed verification of the quoted result. Rybnikov also gives an outline for proving a stronger result: the fundamental groups of P2\⋃ L1 and P2 \ ⋃ L2 are not isomorphic. Providing a complete proof of this result would be most interesting. In the present work we are interested in the topology of complexified real arrangements. Com- plexified real arrangements form a very important class of arrangements where more topological properties are known. In this work we prove the existence of complexified real arrangements with the same combinatorics but different embeddings in P2. It is worth mentioning that Rybnikov’s arrangements do not admit real equations. The counterexamples obtained in this work have an ad- ditional property: they admit conjugated equations on the ring of polynomials over a number field, namely Q( √ 5). The goal of this paper is to show the existence of such a pair of arrangements. In the following outline of the paper, we will stress the similarities and major diferences with Ribnikov’s approach: – The first step towards finding such pairs of arrangements involves finding combinatorics whose moduli space is not connected. As a first approach, we will find ordered combinatorics whose moduli space is not connected (an easier task). We proceed by considering an ordered line arrangement with equations in a finite field and then studying its complex realizations. In Rybnikov’s paper, the MacLane arrangement in F3P 2 was considered. – In a second step we exhibit two ordered line arrangements M1,M2 which have the same ordered combinatorics but whose pairs, (P2, ⋃ M1) and (P2, ⋃ M2), are not homeomorphic by an order-preserving map. In order to prove this result we need to obtain some non-generic braid monodromies. In Rybnikov’s paper, the author constructs an invariant of the isomorphism class of fundamental groups of the complements of lines whose ordered combinatorics induce 2000 Mathematics Subject Classification Primary 32S22, 14N20, 20F36; Secondary 20E18, 32S50, 57M05 Keywords: Line arrangements, braid monodromy. First, third and fourth authors are partially supported by BFM2001-1488-C02-02; second author is partially sup- ported by BFM2001-1488-C02-01 http://arxiv.org/abs/math/0307296v2 http://www.ams.org/msc/ E. Artal, J. Carmona, J.I. Cogolludo and M. Marco the identity on the first homology group. In [ACCM04] we describe an extension of Rybnikov’s invariant which has been implemented using Maple. Using it, we are able to double check Rybnikov’s example. Unfortunately this invariant does not give any valuable information in our example. – Finally, using the aforementioned arrangements M1,M2 we construct a pair of arrangements L1,L2 so that the pairs (P2, ⋃ L1) and (P2, ⋃ L2) are not homeomorphic. This idea is part of Rybnikov’s strategy: the fact that M1,M2 are real arrangements allows us to use simpler arguments. Note that in our case the fundamental groups of P2 \ ⋃ L1 and P2 \ ⋃ L2 have isomorphic profinite completions, but we do not know whether or not they are isomorphic. 2. An arrangement over a finite field In what follows, a brief description of combinatorics and ordered combinatorics of a line arrangement will be given. Definition 2.1. A line combinatorics (projective configuration in [Ryb98]) is a triple (L ,P,∈) where L and P are finite subsets and ∈ is a relation between P and L satisfying: – ∀ℓ, ℓ′ ∈ L , ℓ 6= ℓ′, ∃! p ∈ P such that p ∈ ℓ and p ∈ ℓ′ (in notation p = ℓ ∩ ℓ′). – ∀p ∈ P ∃ ℓ, ℓ′ ∈ L , ℓ 6= ℓ′, such that p = ℓ ∩ ℓ′. Analogously we define an ordered line combinatorics, where the elements of L = {ℓ0, ℓ1, . . . , ℓn} are ordered. If the context is not ambiguous, we will omit both P and ∈. The cardinal #{ℓ | p ∈ ℓ} is denoted as ν(p) and referred to as the multiplicity of p (as usual, points of multiplicity 2 are also called double points and points of multiplicity 3 triple points). The second condition in a line combinatorics (L ,P,∈) implies that ν(p) > 2 ∀p ∈ P. A simple way to obtain line combinatorics is via arrangements in FqP 2 for some finite field Fq. In such cases it is more practical to consider arrangements of points and then dualize. For example, the starting point in [Ryb98] is the MacLane arrangement. This arrangement can be defined as follows. Consider F3P 2 as the union of F2 3 and the line at infinity. Consider also the points of F2 3 \ {0}: the MacLane combinatorics is the abstract line combinatorics corresponding to the dual of this 8 point arrangement. Definition 2.2. Let L be a line combinatorics. An automorphism of L is a permutation of L preserving the incidence relations. The set of such automorphisms is the automorphism group of L . Example 2.3. It is easily seen that the automorphism group of MacLane combinatorics is naturally isomorphic to GL(2; F3). The starting point of this work is a similar idea. Let us consider F4P 2 as the union of F2 4 and the line at infinity L∞. Let us consider A := ( 0 1 1 ζ ) ∈ GL(2; F4), where ζ is any of the elements of F4 \ F2 (and hence the other one is ζ̄ = 1 + ζ = ζ−1 = ζ2). Note that A5 = I2. The set B := L∞ ∪ { Aj ( 1 0 ) ∣ ∣ ∣ ∣ 0 6 j 6 4 } = L∞ ∪ {( 1 0 ) , ( 0 1 ) , ( 1 ζ ) , ( ζ ζ ) , ( ζ 1 )} 2 Topology and combinatorics of real line arrangements contains ten points, the last five of which will be denoted by P1, . . . , P5. Note that each point lies in a different line through the origin. Let us denote by Qj the point at infinity of the line passing through both the origin and Pj , j = 1, . . . , 5. The following is a list of all the lines of F4P 2 that contain at least two points (after each equation, a list of points in B contained in the line is provided): – The line L∞ containing Q1, . . . , Q5. – Lines through Q1: y = 0 (P1), y = z (P2, P5), y = ζz (P3, P4). – Lines through Q2: x = z (P1, P3), x = 0 (P2), x = ζz (P4, P5). – Lines through Q3: y = ζ(x+ z) (P1, P5), y = ζx+ z (P2, P4), y = ζx (P3). – Lines through Q4: y = x+ z (P1, P2), y = x+ ζ̄z (P3, P5), y = x (P4). – Lines through Q5: y = ζ̄(x+ z) (P1, P4), y = ζ̄x+ z (P2, P3), y = ζx (P5). Note that each pair Pi, Pj , 1 6 i < j 6 5 appears exactly once in the above list. The line combinatorics C we are interested in is dual to B. It contains the following ten lines: – L1, . . . , L5 dual to P1, . . . , P5 resp. and – M1, . . . ,M5 dual to Q1, . . . , Q5 resp. The intersection points of these lines are: – O := M1 ∩ · · · ∩M5, dual to the origin ( 0 0 ). – P1,{2,5}, P1,{3,4}, P2,{1,3}, P2,{4,5}, P3,{1,5}, P3,{2,4}, P4,{1,2}, P4,{3,5}, P5,{1,4}, P5,{2,3}, where Pi,{j,k} := Mi ∩ Lj ∩ Lk. – Ri := Mi ∩ Li, i = 1, . . . , 5, dual to the line passing through the origin, Pi and Qi. Note that the Falk-Sturmfels arrangement (see [CS97]) is the result of deleting one of the Mi lines from C . Lemma 2.4. Let ψ ∈ Aut(C ). Then there exists a unique σ ∈ Σ5 such that ψ(Li) = Liσ and ψ(Mi) = Miσ , i = 1, . . . , r. The mapping ψ 7→ σ defines a monomorphism Aut(C ) → Σ5 which will be identified with the inclusion. Proof. First of all note that any ψ ∈ Aut(C ) preserves the multiplicity of any point. Since {O} = {P ∈ C | ν(P ) = 5} one concludes that ψ fixes O and hence should interchange the five lines M1, ...,M5 containing O. Let us denote by σ ∈ Σ5 the permutation satisfying ψ(Mi) = Miσ . Note that each line Mi determines Ri (as the only point of multiplicity 2 contained in Mi) and hence it determines Li (as the other line containing Ri). Therefore ψ(Ri) = Riσ and ψ(Li) = Liσ . Remark 2.5. Because of this identification Aut(C ) can be considered to act on C on the right. Example 2.6. The action of the matrix A induces an automorphism ψA of C given by the 5-cycle (1, 2, 3, 4, 5). Example 2.7. Let us consider the automorphism of F2 4 defined by Bσ where B := ( 1 ζ 0 ζ ) ∈ GL(2; F4), and σ is the Galois involution of F4|F2. The projective completion of this mapping preserves B and induces an automorphism ψ̄B of C determined by the permutation (2, 4, 5, 3). Remark 2.8. Regarding again Aut(C ) as a subgroup of Σ5, it is easily seen that ψ̄−1 B ψAψ̄B = ψ3 A, that is, 〈ψA〉 is a normal subgroup in G := 〈ψA, ψ̄B〉. Since G/〈ψA〉 ∼= 〈ψ̄B〉, the subgroup G is a semidirect product of 〈ψA〉 ∼= Z5 and 〈ψ̄B〉 ∼= Z4 and hence has order 20. 3 E. Artal, J. Carmona, J.I. Cogolludo and M. Marco Lemma 2.9. Under the above considerations Aut(C ) = G. Proof. It is enough to check that Aut(C ) ⊂ G. For the sake of simplicity, we will denote an automorphism in Aut(C ) and its induced permutation in Σ5 by the same symbol. Let us consider ψ an automorphism fixing at least two lines of type Mi. After conjugation by an element in G, one may assume that these lines are M1, M2. Hence L1 and L2 are also fixed (Lemma 2.4). Using ψ(P1,{2,5}) = P1,{2,5ψ}, ψ(P4,{1,2}) = P4ψ ,{1,2}, and ψ(P2,{1,3}) = P2,{1,3ψ}, the lines L5, L4, and L3 are also respectively fixed, hence ψ is the identity. Let us consider ψ an automorphism fixing only one line of type Mi. Again, after conjugation by an element in G one may suppose such a line is M1. Since ψ̄B fixes M1 and permutes the others, after multiplying ψ by a certain power of ψ̄B one may suppose that M2 is also fixed; then by the previous step ψ ∈ G. Finally, let us consider a general ψ. Since ψA is transitive, after multiplying ψ by a certain power of ψA one may suppose that M1 is fixed; then by the previous step ψ ∈ G. Remark 2.10. Following the notation of the previous examples, the automorphisms ψM defined by a linear transformation M will be called positive, whereas the automorphisms ψ̄M defined by Mσ (σ being the Galois transformation of F4|F2) will be called negative. When the automorphisms are viewed as permutations, the concepts of positive and negative agree with their signature. 3. Complex realization Definition 3.1. Let C be a line combinatorics. An arrangement L of P2 = CP2 is a complex realization of C if its combinatorics agrees with C . An ordered complex realization of an ordered line combinatorics is defined accordingly. Notation 3.2. The space of all complex realizations of a line combinatorics C is denoted by Σ(C ). This is a quasiprojective subvariety of P n(n+3) 2 , where n := #C . If C is ordered, we denote by Σord(C ) ⊂ (P̌2)n the space of all ordered complex realizations of C . There is a natural action of PGL(3; C) on such spaces. This justifies the following definition. Definition 3.3. The moduli space of a combinatorics C is the quotient M (C ) := Σ(C )/PGL(3; C). The ordered moduli space M ord(C ) of an ordered combinatorics C is defined accordingly. Example 3.4. Let us consider the MacLane line combinatorics N . It is well known that #M (N ) = 1 and that #M ord(N ) = 2. One can find representatives having equations in the polynomial ring over the field of cubic roots of unity. Moreover, the MacLane line combinatorics has no real realization. Remark 3.5. Let us consider an ordered line combinatorics C . It is easily seen that Aut(C ) acts on both M (C ) and M ord(C ). Example 3.6. The action of Aut(N ) ∼= GL(2,F3) on the moduli spaces is as follows: matrices of determinant +1 (resp. −1) fix (resp. interchange) the two elements of M ord(N ). Of course complex conjugation also acts on M ord(N ) interchanging the two elements. From the topological point of view, after fixing two representatives N ± of N one has that: – There exists a homeomorphism (P2, ⋃ N +) → (P2, ⋃ N −) preserving orientations on both P2 and the lines. Such a homeomorphism does not respect the ordering. – There exists a homeomorphism (P2, ⋃ N +) → (P2, ⋃ N −) preserving orientations on P2, but not on the lines. Such a homeomorphism respects the ordering. 4 Topology and combinatorics of real line arrangements In what follows, we study the space and moduli of complex realizations for the combinatorics C introduced in §2. Let us fix an ordering in the combinatorics so as to choose coordinates. Let us suppose that a complex realization exists. Up to the action of PGL(3; C) one may assume that: M1 : z = 0, M2 : x = 0, M3 : x = z, M4 : x = αz, M5 : x = βz where α, β ∈ C \ {0, 1}, α 6= β (note that α, β cannot be fixed a priori since they represent cross- ratios, which are projective invariants). Taking the incidence relations into account and by means of the subgroup of projective transformations which fix M1, M2, and M3, one can also assume that L5 : y = 0 and L4 : y = z. Note that the only projective transformation which fixes M1, M2, M3, L4, and L5 is the identity. Also note that: – L1 passes through L5 ∩M2 = [0 : 0 : 1] and L4 ∩M3 = [1 : 1 : 1], hence L1 : x = y. – L2 passes through L4 ∩M5 = [β : 1 : 1], L5 ∩M3 = [1 : 0 : 1], and L1 ∩M4 = [α : α : 1], hence: L2 : αx− (α− 1)y − αz = 0, β = 2α − 1 α . – L3 passes through L1 ∩M5 = [2α − 1 : 2α− 1 : α], L4 ∩M2 = [0 : 1 : 1], L5 ∩M4 = [α : 0 : 1], and L2 ∩ M1 = [α − 1 : α : 0]. In principle one should obtain four equations out of these conditions, but only three of them are independent, yielding: L3 : x+ αy − αz = 0, α2 + α− 1 = 0. The family M ord(C ) depends on one parameter α satisfying the above equation. The other root of this equation is γ := −1 − α, which also verifies γ2 + γ − 1 = 0, will be taken as the parameter. Summarizing, one has the following: Proposition 3.7. The space M ord(C ) has two elements. Moreover, representatives can be chosen to have the following equations: M1 : z = 0, M2 : x = 0, M3 : x = z, M4 : x = −(γ + 1)z, M5 : x = (γ + 2)z, L1 : y = x, L2 : y = γ(x− z), L3 : y = γx+ z, L4 : y = z, L5 : y = 0, where γ2 + γ − 1 = 0. Remark 3.8. The previous proposition implies that there are representatives of the two ordered complex realizations of C having real equations conjugated in the ring of polynomials over the number field Q( √ 5). It is easily seen that the positive (resp. negative) automorphisms of C preserve (resp. interchange) the elements in M ord(C ). In particular, #M (C ) = 1. 4. Topology of the realizations The first main result of Rybnikov’s work [Ryb98] reads as follows: let us denote by N ± two repre- sentatives of the elements of M ord(N ). Then there is no isomorphism of the fundamental groups of P2 \⋃ N + and P2 \⋃ N − inducing the identity in the homology groups. In particular, there is is no orientation-preserving homeomorphism between (P2, ⋃ N +) and (P2, ⋃ N −). Analogously, one can consider the two ordered arrangements C± representing the elements of M ord(C ). They are determined by the equations given in Proposition 3.7 and by the choice of γ± := −1± √ 5 2 . Let us denote by L± i and M± i , i = 1, . . . , 5 the lines as in section §2. Affine pictures are shown in Figures 1 and 2, where M± 1 are the corresponding lines at infinity. As pointed out in the introduction, we have completely checked Rybnikov’s computations con- cerning MacLane arrangements [ACCM04]. Rybnikov’s approach does not work with C and we do not know whether or not there exists an isomorphism of the fundamental groups of P2 \ ⋃ C + 5 E. Artal, J. Carmona, J.I. Cogolludo and M. Marco Figure 1. Arrangement C + L+ 4 L+ 5 L+ 3 L+ 1 L+ 2 M+ 4 M+ 2 M+ 3 M+ 5 Figure 2. Arrangement C− L− 4 L− 5 L− 3 L− 1 L− 2 M− 4 M− 2 M− 3 M− 5 and P2 \ ⋃ C− inducing the identity in the homology groups. Nevertheless, we are able to prove that there is no orientation-preserving homeomorphism (P2, ⋃ C +) → (P2, ⋃ C−). The required invariant is a non-generic braid monodromy. Let us introduce some concepts in order to adapt braid monodromy to line arrangements. Definition 4.1. An affine horizontal arrangement L is an affine line arrangement in C2 with no vertical lines. The ramification set AL ⊂ C is the set of x-coordinates of the multiple points of L . Definition 4.2. An ordered affine horizontal arrangement L is an affine horizontal arrangement with an ordering on both L and AL . By means of the usual embedding C2 ⊂ P2, the ordered fibered projective arrangement L ϕ associated with L is obtained by considering: – the projectivization of the lines in L ; – the line at infinity L∞; – the projectivization of the fibers of AL (the vertical lines, pre-images of AL ). Example 4.3. The non-vertical lines in Figures 1 and 2 give two affine horizontal arrangements L ±. With a suitable order, their ordered fibered projective arrangements are C±. Remark 4.4. Given an ordered affine horizontal arrangement L , the mapping P2 \ L ϕ → C \AL , induced by the projection (x, y) 7→ x is a locally trivial fibration. 6 Topology and combinatorics of real line arrangements Definition 4.5. Let Z be a connected projective manifold and let H be a hypersurface of Z. Let ∗ ∈ Z \H and let K be an irreducible component of H. A homotopy class γ ∈ π1(Z \H; ∗) is called a meridian about K with respect to H if γ has a representative δ satisfying the following properties: (a) there is a smooth complex analytic disk ∆ ⊂ Z transverse to H such that ∆ ∩H = {∗′} ⊂ K (transversality implies that ∗′ is a smooth point of H). (b) there is a path α in Z \H from ∗ to ∗′′ ∈ ∂∆. (c) δ = α ·β ·α−1, where β is the closed path obtained by traveling from ∗′′ along ∂∆ in the positive direction. Definition 4.6. Let A := {a1, . . . , ar} ⊂ C and let ∆ ⊂ C be a disk containing A in its interior. Fix ∗ ∈ ∂∆. A geometric basis of the free group π1(C \A; ∗) is a basis γ1, . . . , γr where each γi is a meridian of a point in A and γr · . . . · γ1 = ∂∆ positively oriented. Remark 4.7. In order to compute braid monodromies we use the reversed lexicographic order on C, i.e., if z,w ∈ C, z 6= w, then z ≻ w if ℜz > ℜw or ℜz = ℜw and ℑz > ℑw. Construction 4.8. Let us construct a special geometric basis of π1(C\A; ∗), where ∗ is a big enough real number and A := {a1, . . . , ar} ⊂ C. Let us order the elements of A such that a1 ≻ · · · ≻ ar. Consider the polygonal path Γ with vertices ∗, a1, . . . , ar. Let us choose a small enough ε > 0 so that the disks ∆i of radius ε centered at a1, . . . , ar are pairwise disjoint. The geometric basis γ1, . . . , γr is defined as follows. Fix i ∈ {1, . . . , r}; let us consider αi a path starting at ∗ and running along Γ outside the interior of the disks ∆1, . . . ,∆i−1 and counterclockwise along the arcs ∂∆1, . . . , ∂∆i−1. This path should stop at the first intersection ∗i of Γ with ∂∆i. Let us denote by δi the loop based at ∗i and going counterclockwise along ∂∆i. Then γi := αi · δi · α− i . This basis will be called a lexicographic basis. Example 4.9. Let us denote by α+ 5 , α+ 3 , α+ 2 , and α+ 4 a lexicographic geometric basis of π1(C \ AL +;R), R ≫ 0; the indices of α+ i refer to the lines they are meridians of. Analogously, α− 3 , α− 4 , α− 5 , and α− 2 will denote a lexicographic geometric basis of π1(C \ AL − ;R). Let us recall that AL ± = {γ± + 2, 1, 0,−(γ± + 1)}. Let Bn be the abstract braid group on n = #C strings given by the standard presentation: 〈σ1, . . . , σn−1 : [σi, σj] = 1, |i− j| > 2, σiσi+1σi = σi+1σiσi+1, i = 1, . . . , n − 2〉 . The pure braid group Pn is the kernel of the natural epimorphism of Bn onto the symmetric group Σn (obtained from the above presentation by adding the relations σ2 i = 1). Let y ⊂ C be a subset of exactly n elements. Let us recall that the braid group By is the set of homotopy classes, relative to {0, 1}, of sets of paths {γ1, . . . , γn}, γj : [0, 1] → C, starting and ending at y and such that ∀t ∈ [0, 1], {γ1(t), . . . , γn(t)} is a set of n distinct points. The elements of By are called braids based at y and are represented as a set of non-intersecting paths in C × [0, 1], as usual. The pure braids correspond to sets of loops and they form the subgroup Py. Products are also defined in the standard way. Analogously, if y1,y2 ⊂ C are subsets of exactly n elements then By2,y1 is the set of braids starting at y1 and ending at y2. The abstract braid group is canonically identified with By0, where y0 := {−1, . . . ,−n}. The groups Py0 and Pn are identified as well. Example 4.10. There is a special element of By2,y1 which is called the (y2,y1)-lexicographic braid. It is easily defined as follows. Let us order y1 and y2 according to the lexicographic order. The braid is defined by taking the linear segments joining the corresponding elements in y1 and y2. The main point of this construction is that given y1 and y2 one can identify By2,y1 with Bn by means of the lexicographic braids associated with (y0,y1) and (y2,y0). 7 E. Artal, J. Carmona, J.I. Cogolludo and M. Marco The braid monodromy of an ordered affine horizontal arrangement L can be defined as follows. Let us choose ∗ ∈ C\AL and let y∗ ⊂ C the set of y-coordinates of the points in L with x-coordinate ∗. Moving around a loop in C \ AL based at ∗ one can obtain a braid in By∗ and conjugating by the (y0,y ∗)-lexicographic braid one obtains a braid in Pn. Definition 4.11. The braid monodromy of an ordered affine horizontal arrangement L is the group homomorphism ρ : π1(C \ AL ; ∗) → Pn defined above. Remark 4.12. The braid monodromy is well defined up to conjugation in Pn. Proposition 4.13. The braid monodromies ρ± of the arrangements L ± are given by: ρ+(α+ 5 ) = ρ−(α− 3 ) =σ2 1σ 2 3 ρ+(α+ 3 ) = ρ−(α− 4 ) =(σ1σ3) ∗ (σ2 2σ 2 4) ρ+(α+ 2 ) = ρ−(α− 5 ) =(σ1σ3σ2σ4) ∗ (σ2 1σ 2 3) ρ+(α+ 4 ) = ρ−(α− 2 ) =(σ1σ3σ2σ4σ1σ3) ∗ (σ2 2σ 2 4) where a ∗ b := aba−1 and the strings in the positive (resp. negative) case correspond to the lines (L+ 1 , L + 3 , L + 2 , L + 4 , L + 5 ), resp. (L− 1 , L − 4 , L − 5 , L − 2 , L − 3 ). Proof. It is straightforward from Figures 1 and 2 and the choice of lexicographic braids. Remark 4.14. Replacing ρ−(α− j ) by τ−1ρ−(α− j )τ , where τ := σ3σ4σ2σ3σ2, the strings of the braids correspond to the same sets of ordered lines. After this change, we will keep the original notation ρ−. Our goal is to adapt [ACC03, Theorem 1] to the case of ordered horizontal line arrangements. The main difference is that the orbit of an ordered arrangement is given by the action of the pure braid group instead of the whole braid group as is required in the unordered case. Let us fix an ordered horizontal line arrangement L . Let us consider its braid monodromy ρ : π1(C \AL ) → Pn. Fixing a geometric basis γ1, . . . , γr of π1(C \ AL ), r = #AL , the r-tuple (ρ(γ1), . . . , ρ(γr)) is said to determine, or represent, the braid monodromy ρ. Recall that ρ was defined up to conjugation in Pn; this comes from the two choices that have been made: the base point ∗ and the geometric basis. Definition 4.15. Two elements of Pr n are said to be equivalent if they can be related by simultaneous conjugation in Pn and Hurwitz moves. Recall that the jth Hurwitz move of (τ1, . . . , τr) ∈ Pr n produces (τ1, . . . , τj−1, τj+1, τj+1 ∗ τj, τj+2, . . . , τr). Remark 4.16. Given τ := (τ1, . . . , τr) ∈ Pr n it is important to consider its pseudo-Coxeter element χ(τ) := τr · . . . · τ1 ∈ Pn. Note that χ(τ) remains invariant by Hurwitz moves. Lemma 4.17. Under the above notation, let (ρ(γ1), . . . , ρ(γr)) represent the braid monodromy of L . Then τ := (τ1, . . . , τr) ∈ Pr n also represents the braid monodromy of L if and only if τ is equivalent to (ρ(γ1), . . . , ρ(γr)). Proof. The ideas involved in the proof of this lemma are classical, see [ACC03]. Theorem 4.18. Let L1,L2 ⊂ P2 be two ordered affine horizontal line arrangements such that their associated fibered projective arrangements have the same combinatorics. Let also F : P2 → P2 be an orientation-preserving homeomorphism such that: (i) F (P ) = P , P being the projection point, (ii) F (L1) = L2 respecting ordering and orientations, and (iii) F respects the vertical lines preserving both ordering and orientations. 8 Topology and combinatorics of real line arrangements Then both ordered affine horizontal line arrangements have equivalent braid monodromies. Proof. The proof essentially follows the one in [ACC03, Theorem 1]. Theorem 4.19. There is no order-preserving homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−). Proof. We break the proof up into several steps. Step 1. There is no homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−) preserving the ordering and the orientation on both the plane and the lines. Proof. This result follows from Theorem 4.18. Recall that ρ− has been replaced by a conjugate (Remark 4.14). Let us suppose that ρ+ and ρ− are equivalent. Let K± denote the image of ρ± in the pure braid group. Let c± := χ(ρ±(α±)) be the pseudo- Coxeter elements of ρ+(α+) := (ρ+(α+ 5 ), ρ+(α+ 3 ), ρ+(α+ 2 ), ρ+(α+ 4 )) and ρ−(α−) := (ρ−(α− 3 ), ρ−(α− 4 ), ρ−(α− 5 ), ρ−(α− 2 )); since c± is invariant by Hurwitz moves, one should have that c+ and c− are conjugate in the pure braid group P5. Replacing ρ− by the action of an appropriate inner automorphism, one may assume that c := c+ = c−. Let Cc be the centralizer of c in P5. Since ρ+(α+) and ρ−(α−) are equivalent, K+ and K− should be conjugate by an element g ∈ P5 such that gχ(ρ+(α+))g−1 = χ(ρ−(α−)), that is, gc = cg, hence g ∈ Cc (note that Hurwitz moves preserve the groups K± and the pseudo-Coxeter elements). Let us consider the Burau representation of B5 into GL(5; Z[t±1]). Replacing t by 2 mod 5 one obtains a representation β : B5 → GL(5; Z/5Z). The subgroup β(P5) has order 58032 · 106. Let us denote C̃c := β(Cc), and K̃± := β(K±). Checking that K̃+ and K̃− are not conjugate in GL(5; Z/5Z) by an element in C̃c is a matter of comparing finite tables; the groups K̃± have order 30000 while #C̃c = 115200. This has been verified using a GAP4 [GAP00] program. Its source can be found in the appendix or downloaded from a public site1. Step 2. There is no homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−) preserving the ordering, the orientations on the plane, and reversing the orientations on the lines. Proof. Composing such a homeomorphism with the complex conjugation (which preserves the ori- entation of the plane, but reverses the orientations on the lines) one obtains a homeomorphism that is ruled out by Step 1. For the remaining cases the following lemma is required. Lemma 4.20. Fix an orientation on S3 and consider ϕ : S3 → S3 a homeomorphism,K1,K2, L1, L2 ⊂ S3 oriented knots such that K1 ∩K2 = ∅ and ϕ(Ki) = Li (disregarding orientations), i = 1, 2. Let us set ϑ and εi = ±1, i = 1, 2, such that: ϑ := { 1 if ϕ preserves orientation −1 otherwise, Li = εiKi. Then, if LS3 is the linking number, LS3(L1, L2) = ϑε1ε2LS3(K1,K2). Remark 4.21. Note also that, from the well-known relationship between linking and intersection numbers, if two lines intersect at one point and one considers the oriented knots they define on the oriented boundary of a small ball centered at the intersection point, then their linking number equals 1. 1http://riemann.unizar.es/geotop/pub/gap-programs/lines/monodromy groups 9 http://riemann.unizar.es/geotop/pub/gap-programs/lines/monodromy_groups E. Artal, J. Carmona, J.I. Cogolludo and M. Marco Step 3. There is no homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−) preserving the ordering, the orientation on the plane, and the orientation on at least one line. Proof. By Step 1 there should be at least one line whose orientation is reversed and by hypothesis, at least one line whose orientation is preserved. Both lines should of course intersect at one point. Let us consider K and K ′ their links at the intersection point. According to the notation of Lemma 4.20, the hypotheses imply that ϑ = 1 and εε′ = −1, and hence LS3(L,L′) = −1 where L and L′ are the links of two lines around their intersection point. This contradicts Remark 4.21. Step 4. There is no homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−) preserving the ordering and reversing the orientation on the plane. Proof. Since #C + > 2, there exist at least two lines whose orientations are simultaneously either preserved or reversed. Consider their links at the intersection point. According to the notation of Lemma 4.20, the hypotheses imply that ϑ = −1 and εε′ = 1, which again contradicts Remark 4.21. This concludes the result. 5. A counterexample Since the aforementioned line combinatorics N and C have a great symmetry, they verify the equalities #M (N ) = #M (C ) = 1 and hence will not provide pairs of arrangements having the same combinatorics but different topologies. In [Ryb98] the problem is solved by juxtaposition of two arrangements with combinatorics of type N . In this work we will follow a simpler approach than can be applied in many different ways. For the sake of brevity, we describe only one example. Let H ± := C± ∪ {N±} be two new arrangements, where N± is the line joining the points L± 3 ∩L± 5 ∩M± 4 = [1 : 0 : −γ±] and L± 2 ∩M± 2 = [0 : 1 : −(γ± + 1)]. In particular, N± : γ±x+ (γ± + 1)y + z = 0. Note that these two arrangements have the same combinatorics, say H , since their equations are conjugate in Q( √ 5). The new abstract line of H will be denoted by N . Proposition 5.1. The automorphism group of H is trivial. Proof. Let ψ ∈ Aut(H ). Note that ψ(N) 6= Mj, j = 1, . . . , 5, since N does not pass through the unique point of multiplicity 5. Also, note that N contains 3 double points and each Lj contains at most 2 double points. Hence, ψ(N) = N , ψ(C ) = C and therefore ψ induces an automorphism of C . By simple combinatorial arguments, ψ(M4) = M4 and ψ(M2) = M2, therefore ψ = 1H . Corollary 5.2. #M (H ) = 2 and H ± are representatives of the two elements. Theorem 5.3. There is no homeomorphism between (P2, ⋃ H +) and (P2, ⋃ H −). Proof. By Proposition 5.1, such a homeomorphism would induce the identity on H , hence having to sendN+ to N−. Therefore, it would define an order-preserving homeomorphism between (P2, ⋃ C +) and (P2, ⋃ C−). The result follows from Theorem 4.19. Remark 5.4. An important feature of this pair of arrangements is the fact that they are defined by conjugate equations in Q( √ 5). This fact implies that they cannot be distinguished by alge- braic methods. Note for example that the fundamental groups of P2 \ ⋃ H + and P2 \ ⋃ H − have isomorphic profinite completions. 10 Topology and combinatorics of real line arrangements 6. Appendix In this appendix, we provide the source (with comments) of the short GAP4 program mentioned in Step 1 of Theorem 4.19. # n is the number of strings. n:=5;; # The braid group in n strings is constructed by generators and relations. f:=FreeGroup(n-1); x:=GeneratorsOfGroup(f); rel:=[]; for i in [1..n-3] do for j in [i+2..n-1] do Add(rel,Comm(x[i],x[j])); od; od; for i in[1..n-2] do Add(rel,x[i]*x[i+1]*x[i]/x[i+1]/x[i]/x[i+1]); od; g:=f/rel; y:=GeneratorsOfGroup(g); # The following is a system of generators of the pure braid group. sub:=[y[1]^2,y[2]^2,y[1]*y[2]^2*y[1],y[3]^2,y[2]*y[3]^2*y[2], y[1]*y[2]*y[3]^2*y[2]*y[1], y[4]^2,y[3]*y[4]^2*y[3], y[2]*y[3]*y[4]^2*y[3]*y[2], y[1]*y[2]*y[3]*y[4]^2*y[3]*y[2]*y[1]];; h:=Subgroup(g,sub);; # The generators of K+ will be defined, as well as the pseudo-Coxeter element and the braid used in Remark 4.14 to obtain K− are given. sub1:=[y[1]^2*y[3]^2,(y[2]^2*y[4]^2)^(y[1]^-1*y[3]^-1), (y[1]^2*y[3]^2)^(y[2]^-1*y[4]^-1*y[1]^-1*y[3]^-1), (y[2]^2*y[4]^2)̂ (y[1]^-1*y[3]^-1*y[2]^-1*y[4]^-1*y[1]^-1*y[3]^-1)];; del:=sub1[4]*sub1[3]*sub1[2]*sub1[1];; cj:=y[3]*y[4]*y[2]*y[3]*y[2]; k1:=Subgroup(g,sub1);; # The function burau is used to construct the composition of the Burau representation with the mapping Z[t±1] → Fm such that t is sent to the nth power of a generator of the invertible elements. The output of this function is a list of two elements containing the representation itself and the image of the pure braid group. burau:=function(n,m) local uno,cero,z,gg,hom,hh,s,R; R:=GF(m); s:=Z(m)^n;; uno:=Z(m)^0; cero:=0*Z(m); z:=[[[uno-s,s,cero,cero],[uno,cero,cero,cero], [cero,cero,uno,cero],[cero,cero,cero,uno]], [[uno,cero,cero,cero],[cero,uno-s,s,cero], [cero,uno,cero,cero],[cero,cero,cero,uno]], [[uno,cero,cero,cero],[cero,uno,cero,cero], 11 Topology and combinatorics of real line arrangements [cero,cero,uno-s,s],[cero,cero,uno,cero]], [[uno,cero,cero,-uno],[cero,uno,cero,-uno], [cero,cero,uno,-uno],[cero,cero,cero,-s]]]; gg:=Group(z); hom:=GroupHomomorphismByImages(g,gg,y,z); hh:=Image(hom,h); return [hom,hh]; end; # The representation β is set in hom. aux:=burau(1,5); hom:=aux[1]; hh:=aux[2];; # del1 and del2 are the images of the pseudo-Coxeter elements. del1:=Image(hom,del); cj1:=Image(hom,cj); del2:=del1^cj1; # The image nv of a pure braid such that del1=del2^nv is obtained. We compute the group K̃+; the group K̃− is obtained after a conjugation in order to have the equality of pseudo-Coxeter elements. nv:=RepresentativeAction(hh,del2,del1);; kk1:=Image(hom,k1); kk2:=kk1^(cj1*nv);; # The centralizer of the pseudo-Coxeter element is computed and K̃± are verified to not be conju- gated by an element in this centralizer. hh1:=Centralizer(hh,del1); IsConjugate(hh1,kk1,kk2); References ACC03 E. Artal, J. Carmona, and J.I. Cogolludo, Braid monodromy and topology of plane curves, Duke Math. J. 118 (2003), no. 2, 261–278. MR1980995(2004k:14015) ACCM04 E. Artal, J. Carmona, J.I. Cogolludo, and M. Marco, Alexander invariants and Rybnikov’s ar- rangement, Preprint available at arXiv:math.AG/0403543, 2004. CS97 D.C. Cohen and A.I. Suciu, The braid monodromy of plane algebraic curves and hyperplane ar- rangements, Comment. Math. Helv. 72 (1997), no. 2, 285–315. MR1470093(98f:52012) GAP00 The GAP Group, Aachen, St Andrews, GAP – Groups, Algorithms, and Programming, Version 4.2, 2000, http://www.gap-system.org. Ryb98 G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, Preprint available at arXiv:math.AG/9805056, 1998. Enrique ARTAL BARTOLO artal@unizar.es Departamento de Matemáticas, Campus Plaza de San Francisco s/n, E-50009 Zaragoza SPAIN Jorge CARMONA RUBER jcarmona@sip.ucm.es Departamento de Sistemas informáticos y programación, Universidad Complutense, Ciudad Universitaria s/n E-28040 Madrid SPAIN José Ignacio COGOLLUDO AGUSTÍN jicogo@unizar.es Departamento de Matemáticas, Campus Plaza de San Francisco s/n, E-50009 Zaragoza SPAIN 12 http://www.ams.org/mathscinet-getitem?mr=MR1980995(2004k:14015) http://arXiv.org/abs/math.AG/0403543 http://www.ams.org/mathscinet-getitem?mr=MR1470093(98f:52012) http://www.gap-system.org http://arXiv.org/abs/math.AG/9805056 mailto:artal@unizar.es mailto:jcarmona@sip.ucm.es mailto:jicogo@unizar.es Topology and combinatorics of real line arrangements Miguel MARCO BUZUNÁRIZ mmarco@unizar.es Departamento de Matemáticas, Campus Plaza de San Francisco s/n, E-50009 Zaragoza SPAIN 13 mailto:mmarco@unizar.es Introduction An arrangement over a finite field Complex realization Topology of the realizations A counterexample Appendix