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On the fundamental group of the complement of a complex hyperplane arrangement

We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the complement to a line arrangement of a given combinatorial type with respect to isomorphisms inducing the canonical isomorphism of the first homology groups.


Introduction
Many topological properties of a complex hyperplane arrangement can be expressed in terms of its combinatorial structure (matroid). For example, this is so for the cohomology ring of the complement of a complex hyperplane arrangement [6] and for the Malcev completion of its fundamental group [3]. However, all known algorithms for computing the fundamental group itself use non-matroidal data. (In the case of a complexified real arrangement, the answer can be expressed in terms of the corresponding oriented matroid, which requires more information then the mere dimensions of all intersections of hyperplanes, as for the ordinary matroid.) Due to a classical theorem of Zariski, the fundamental group of the complement of a hypersurface in CP n is isomorphic to the fundamental group of the complement of the hypersurface section in a generic 2-dimensional plane. Hence, in the case under consideration, it suffices to study the fundamental groups of the complements of line arrangements in CP 2 . In this paper we describe the combinatorial structure of line arrangements in terms of incidence structures, which is essentially equivalent to using the matroid language but has the advantage of being more visual.
The aim of this paper is to construct two combinatorially equivalent line arrangements in CP 2 whose complements have non-isomorphic fundamental groups. Our approach is as follows.
Let X be the complement of a line arrangement L, and let C be the incidence structure describing the combinatorics of L. By H we denote a free Abelian group whose basis is indexed by the lines of C . The first homology group H 1 (X, Z) is a free Abelian roup as well, and its basis consists of cycles going around the lines of L in the positive direction. Thus, it is canonically isomorphic to H . Let X ′ be the complement of another line arrangement whose combinatorics is described by the same structure C . In view of the canonical isomorphisms H 1 (X, Z) ∼ = H and H 1 (X ′ , Z) ∼ = H , we have canonical projections π 1 (X) → H and π 1 (X ′ ) → H , whose kernels are the commutator subgroups of π 1 (X) and π 1 (X ′ ).
First, we obtain a necessary condition for the existence of an isomorphism π 1 (X) → π 1 (X ′ ) concordant with these projections, i.e., an isomorphism inducing the identity automorphism of H . To this end, for a given structure C , we construct an invariant, which depends on the group π 1 (X) together with the projection π 1 (X) → H . If this invariant takes different values for π 1 (X) and π 1 (X ′ ), then no isomorphism π 1 (X) → π 1 (X ′ ) of the above type can exist. The invariant depends only on π 1 (X)/γ 4 π 1 (X), where γ k G is the k th term of the lower central series of G.
Next, we consider two complex conjugate realizations of the incidence structure C 8 corresponding to the MacLane matroid [4] (see also [10]) and show that our invariant distinguishes between these realizations. We also prove that, in the case of C 8 , the group of those automorphisms of H which are reductions of automorphisms of the group π 1 (X) is isomorphic to the group of automorphisms of the incidence structure itself. Then, at last, we combine these results to prove that the incidence structure C 13 , which is obtained by gluing together two MacLane structures, has at least two realizations whose complements have non-isomorphic fundamental groups.

Construction of the Distinguishing Invariant
Definition 1. An incidence structure is a triple C = (L, P, ≻), consisting of a set of lines L, a set of points P , and a binary relation ≻ (incidence) between L and P which satisfies the following axioms.
(1) For any two distinct lines l and l ′ , there is a unique point p such that l ≻ p and l ′ ≻ p.
(2) Any point is incident to at least two lines.
If lines l and l ′ and a point p are as in axiom (1), we write p = l ∩l ′ and call p the intersection point of the lines l and l ′ . We say that an incidence structure C = (L, P, ≻) is non-degenerate if |P| > 1, i.e., if not all of the lines meet at the same point. 1 Definition 2. A complex realization of an incidence structure C = (L, P, ≻) is an injective map φ from L to the set of lines in CP 2 and from P to the set of points in CP 2 such that φp ∈ φl if and only if p ≺ l .
Let C = (L, P, ≻) be a finite non-degenerate incidence structure. We order the elements of L as L = {l 0 , l 1 , . . . , l n } and call l 0 the line at infinity. Let P 0 = {p ∈ P | p ≺ l 0 }. Below we describe in abstract terms the presentation by generators and relations for the fundamental group of the complement of a line arrangement in C 2 as proposed in [1] (see also [7]).
Let F be a free group with free generators w 1 , . . . , w n , and let A = {(i, p) ∈ {1, . . . , n} × P 0 | p ≺ l i }. Suppose that, for any pair (i, p) ∈ A, we are given an element w i (p) ∈ F conjugate in F to w i , i.e., such that w i (p) = (g(i, p)) −1 w i g(i, p) for some g(i, p) ∈ F . Thus, the elements w i (p) are determined by some mapping g : A → F .
Let φ be a complex realization of the incidence structure C , and let X = CP 2 \ l∈L φl . Then, according to [1], π 1 (X) ∼ = G(R(g)) for some particular choice of g . Under this isomorphism, the image of w i corresponds to a small loop passing around φl i in positive direction (the lines in C 2 have canonical coorientation).
Consider the lower central series (γ k F ) of the group F ( . It is well known that the groups gr k F = γ k F/γ k+1 F are free Abelian groups of finite rank. Moreover, gr F = ∞ k=1 gr k F is naturally isomorphic (as a graded Abelian group) to the free Lie algebra L over Z with n generators x 1 , . . . , x n which are the images of w 1 , . . . , w n in F/γ 2 F ; L = ∞ k=1 L k is graded by the degree of monomials (see, e.g., [5], Chap. 5). Under this isomorphism, the Lie algebra commutator in L corresponds to the group commutator in F . Hereafter we identify gr k F with L k . Let H = L 1 = F/γ 2 F ; this is a free Abelian group with basis x 1 , . . . , x n . If G = π 1 (X) for some complex realization of C , then we have H ∼ = H 1 (X, Z), and the basis (x i ) is dual to the basis of H 1 (X, Z) formed by the standard generators of the Orlik-Solomon ring [6].
The group R is generated by the elements r(i, p) (which are commutators) and by all elements of the form [w ±1 . . ]] (where the brackets denote the group commutator). Therefore, R ⊂ γ 2 F . It is readily seen that the images of the elements of R in L 2 arer(i, p) = [x i , j:p≺l j x j ] (where the brackets denote the Lie commutator); thus, they depend only on C .
In what follows, we need a broader class of groups G. Let R = {r(i, p) | (i, p) ∈ A, i = min{j | l j ≻ p} } be a set consisting of arbitrary elements of γ 2 F with the only condition that the images of r(i, p) in L 2 are the samer(i, p) as above. We say that such a set R is admissible for C . As above, we set G = G(R) = F/ R . Since R ⊂ γ 2 F , the canonical projection F → H is factored through the uniquely determined surjective homomorphism G → H . The kernel of this homomorphism is the commutator subgroup of G. We are interested in necessary conditions under which two groups of the form G(R) (determined by different admissible sets R) are isomorphic in such a way that the isomorphism is concordant with the canonical projections G(R) → H specified above.
The lower central series of F determines the filtration ( R ∩ γ k F ) of the subgroup R ⊂ F ; the corresponding filtration on the quotient group G = F/ R coincides with its lower central series (γ k G). By gr k R and gr k G denote the graded quotients of these filtrations. By the definition of the admissible set R, the Abelian group R 2 = gr 2 R ⊆ L 2 is generated by the elementsr(i, p) = [x i , j:p≺l j x j ]; thus, it is uniquely determined by C . Let us show that R 3 = gr 3 R ⊆ L 3 is also uniquely determined by C .
Proof. First, note that the elementsr(i, p) are linearly independent in L 2 . Among all elements of the form [w j , r(i, P )] we choose u 1 , . . . , u m so that their images form a basis in [H, R 2 ]. By analogy with the proof of P. Hall's basis theorem (see [5], Theorem 5.13A), it is easy to show that any element of R can be uniquely expressed in the form r∈R r cr · m i=1 u d i i · z , where c r , d i ∈ Z and z ∈ R ∩ γ 4 F (the order of the elements r ∈ R in the product r∈R r cr is fixed in advance). Since the images of the elements r ∈ R in gr 2 R are linearly independent, each We set P 2 = L 2 /R 2 and P 3 = L 3 /R 3 . A presentation of a group G by generators and relations as G(R) for some admissible set R determines an isomorphism G/γ 2 G = gr 1 G ∼ = H . It is readily seen that, given such an isomorphism, isomorphisms γ 2 G/γ 3 G = gr 2 G ∼ = P 2 and γ 3 G/γ 4 G = gr 3 G ∼ = P 3 are uniquely determined. In this sense, the groups γ 2 G/γ 3 G and γ 3 G/γ 4 G are canonically determined by the incidence structure C .
The group G/γ 3 G is an extension of the group H (which is free Abelian) by the trivial Hmodule P 2 . Such extensions are parameterized by the group H 2 (H, P 2 ) ∼ = Hom(Λ 2 H, P 2 ). But since Λ 2 H ∼ = L 2 , we have a natural projection χ 2 : Λ 2 H → P 2 . It is easy to see that χ 2 is exactly the element of H 2 (H, P 2 ) corresponding to the group extension 1 → P 2 → G/γ 3 G → H → 1. Hence G/γ 3 G is canonically determined by the incidence structure up to an automorphism trivial on H and P 2 . The group of such automorphisms is isomorphic to Hom(H, P 2 ).
The group M = γ 2 G/γ 4 G is Abelian, but it has non-trivial H -module structure. Arguments similar to those used for G/γ 3 G show that this module is also canonically determined by C up to an action of the Abelian group Hom(P 2 , P 3 ). Now, consider the group extension 1 → M → G/γ 4 G → H → 1. It corresponds to a cohomology class χ 3 ∈ H 2 (H, M). From the short exact sequence of H -modules 0 → P 3 → M → P 2 → 0 we obtain the long exact sequence
Recall that the H -module M is determined by C only up to an action of the Abelian group Hom(P 2 , P 3 ); hence only the orbitχ 3 of χ 3 under this action has an invariant sense. The corresponding set of orbits Y = β −1 (χ 2 )/ Hom(P 2 , P 3 ) is a principle homogeneous space over the group T = Hom(R 2 , P 3 )/δ Hom(H, P 2 ), whereδ is the composition of δ with the natural projection Hom(Λ 2 H, P 3 ) → Hom(R 2 , P 3 ). We see that Y , T , and the action of T on Y are determined by C canonically; thus, for any admissible R, the elementχ 3 (R) ∈ Y is well defined.
Assume now that we have two admissible sets R and R ′ . Consider the corresponding groups The groups G/γ 2 G and G ′ /γ 2 G ′ are both canonically isomorphic to H . Thus, we have a canonical isomorphism λ 1 : G/γ 2 G → G ′ /γ 2 G ′ . As mentioned above, there always exists an isomorphism Proof. As shown above, the Lie algebras gr(G/γ 4 G) and gr(G ′ /γ 4 G ′ ) are canonically isomorphic and are generated by G/γ 2 G and G ′ /γ 2 G ′ , respectively. Therefore, if λ 3 exists, then gr λ 3 is uniquely determined and coincides with this canonical isomorphism. Hence κ(R, R ′ ) = 0.
In order to use the invariant provided by this theorem, we must describe P 2 and P 3 . In the general case, they can be described as follows.
Let us identify L 2 with Λ 2 H and denote the basis of H * dual to the basis (x i ) of H by (x * i ).
hence P 2 is a free Abelian group, and P * 2 ∼ = R ⊥ 2 . Therefore, we may regard the forms ω ijk and ω ij defined above as elements of P * 2 .
We have the exact sequence of Abelian groups I do not claim that these elements generate R ⊥ 3 -as we shall see in the next section, this is not so,-but I cannot give any general description for the remaining generators of R ⊥ 3 . Translating the definition of κ(R, R ′ ) from the homological language, in which it is given above, we obtain the following explicit algorithm for computing this invariant.
To each elementr(i, p) ∈R we assign the element t(i, p) = (r ′ (i, p)) −1 r(i, p) ∈ F . Since r(i, p), r ′ (i, p) ∈ γ 2 F and their images in gr 2 F coincide (and are equal tor(i, p)), we have t(i, p) ∈ γ 3 F . We define a homomorphism R 2 → P 3 by its action on the basisR as follows: each elementr(i, p) goes to the image of t(i, p) in P 3 = (gr 3 F )/R 3 . The image of this homomorphism in T = Hom(R 2 , P 3 )/δ Hom(H, P 2 ) is κ(R, R ′ ).
Consider the Abelian group A = H A . For a ∈ A, we defineτ a ∈ Hom(R 2 , P 3 ) by setting Let τ a =τ a +δ Hom(H, P 2 ).
Proof. The required assertion follows from the algorithm for computing κ(R, R ′ ).
By U denote the subgroup of A generated by them.
Proof. This inclusion is verified by direct computation.
Let B be the subgroup of A consisting of those functions which do not depend on p.
Proof. Let a ∈ B , a(i, p) = n j=1 k ij x j . Consider the homomorphism f ∈ Hom(H, We haveτ a =δ(f ). Let W = A/(U + B). The homomorphism τ is the composition of the projection π : A → W and a homomorphismτ : W → T . I do not know whetherτ is always injective, but this is so in the case that we consider in the next section.

Computations for the MacLane Incidence Structure
The MacLane matroid ML 8 can be defined as the affine plane over F 3 minus one element. The lines of the corresponding incidence structure are in one-to-one correspondence with the elements of F 2 3 \ {(0, 0)}, and three lines of C 8 meet at one point if and only if the corresponding elements belong to one affine line (in the sense of F 2 3 ). It follows that the automorphism group of C 8 is isomorphic to GL 2 (F 3 ).
It is easy to show that any complex realization of C 8 is projectively equivalent to a realization of the form where ω is a root of the polynomial x 2 + x + 1. The points φp for p ∈ P are defined as the intersection points of the corresponding lines.
By Theorem 2.2 and Proposition 2.4, it suffices to show that τ a 0 = 0 in T . Proof. First, we computeW = A/U . Since each generator a (α) i,p of U vanishes at (j, q) with q = p, we have the decompositions A = p∈P 0 A p , U = p∈P 0 U p , andW = p∈P 0W p , where A p , U p , andW p are defined in the obvious way. For each p, the computation is quite simple. All of theW p turn out to be free Abelian groups; thus,W is free Abelian as well, and hence its dualW * is naturally isomorphic to U ⊥ ⊂ A * .
Let e ij (p) ∈ A * be defined by (e ij (p), a) = (x * j , a(i, p)) for a ∈ A, where (x Consider the homomorphism t :W → Z/3Z defined by It is not hard to check that t| B = 0 and t(a 0 ) = 1. Hence the image of a 0 in W =W /B is nonzero. Proof. To prove the required assertion, we first study P 3 . It turns out that R ⊥ 3 has a basis consisting of the elements for l i ≻ l j ∩ l k ∈ P 0 with i < j , i < k , and the additional elements Moreover, we have (R ⊥ 3 ) ⊥ = R 3 ; hence P 3 is a free Abelian group and P * 3 ∼ = R ⊥ 3 . Therefore, it is enough to show that U ⊥ = Imτ * .
We identify (Hom(R 2 , P 3 )) * with R 2 ⊗ P * 3 . It is easy to check that C 13 = (L,P, ≻). We denote the equivalence class of l ′ i (i = 0, 1, 2) byl i and the equivalence class of p ′ 012 byp 012 . Let ψ be a projective transformation of CP 2 such that ψφ + l i = φ + l i (= φ − l i ) for i = 0, 1, 2. This condition means that ψ acts as (z 0 : z 1 : z 2 ) → (z 0 : z 1 : az 0 + bz 1 + cz 2 ), where a, b, c ∈ C, c = 0. Assume that none of the lines ψφ ± l i , i = 3, . . . , 7, passes through any of the intersection points of the lines φ + l j , j = 0, 1, . . . , 7. In this case we say that ψ is generic. Let us construct realizations φ ++ and φ +− of C 13 by gluing together the realizations φ + and φ − of C 8 as follows: we set φ +±l i = φ + l i for i = 0, 1, 2, φ +± l ′ i = φ + l i and φ +± l ′′ i = ψφ ± l i for i = 3, . . . , 7. For generic ψ , the intersection points of complex lines φ +±l (wherel ∈L) are in one-to-one correspondence with the elements ofP , hence we obtain realizations of C 13 . Let X +± ψ = CP 2 \ φ +± ψ l . Note that the space of generic transformations ψ is connected, and hence the fundamental groups G ++ = π 1 (X ++ ψ ) and G +− = π 1 (X +− ψ ) are independent of ψ up to an isomorphism preserving the canonical basis in G +± /γ 2 G +± . Therefore, we can omit the index ψ . Proof. LetĜ be one of the groups G ++ and G +− , and letx 1 , . , x ′′ 7 denote the images of the standard generators ofĜ inĤ =Ĝ/γ 2Ĝ . We setx As in the previous section, let G = π 1 (X + ). There exist concordant projections π ′ :Ĝ → G, π ′′ :Ĝ → G and injections ν ′ : G →Ĝ, ν ′′ : G →Ĝ such that the following relations hold modulo the second term of the lower central series: Proof. The required projections are defined in an obvious way, by removing the lines ψφ ± l i for i = 3, . . . , 7 (or the lines φ + l i for i = 3, . . . , 7). To construct the injections, we choose a generic projective transformation ψ so that the lines ψφ ± l i for i = 3, . . . , 7 do not intersect some open ball containing all intersection points of the affine lines φ + l i for i = 1, . . . , 7, and the lines φ + l i for i = 3, . . . , 7 do not intersect some open ball containing all intersection points of affine lines ψφ ± l i for i = 1, . . . , 7. It is readily seen that the complements of lines φ + l i and ψφ ± l i in these open balls are homeomorphic to X + and X ± , respectively.
The same is true for any isomorphism G ++ → G +− .
Proof. The required assertion follows from Theorem 3.5 and Lemma 4.2.
Lemma 4.5. There exists no automorphism η : Proof. The required assertion follows from Theorem 3.5 and Lemma 4.2.
The assertion of the theorem readily follows from Lemmas 4.3, 4.4, and 4.5.