Topological Invariance of Intersection Lattices of Arrangements in Cp2

Let s/* = {l\,li, ■■■ , ln} be a line arrangement in CP2 , i.e., a collection of distinct lines in CP2 . Let L(s/ * ) be the set of all intersections of elements of A* partially ordered byX 0) as multiple intersection points (i.e., multiplicity t(x¡) > 3 ). By blowing up CP2 at {xi, ... , xk} , we get a set si* of lines in a blown-up surface CP2. si* is called an associated arrangement in CP2 induced by si*. Each pair of lines of si* intersects at most one point. Let U(si*) be a regular neighborhood of si* and K(si*) = dU(s7*). Thus K(s7*) is a plumbed 3-manifold which is homeomorphic to K(si*), the boundary of a regular neighborhood of si* in CP2. A class of 3-manifolds was classified by Waldhausen [12] in terms of graphs and reduced graph structures of 3-manifolds. We call these 3-manifolds classified in [12] as Waldhausen graph manifolds. Lemma 2. If si* is a nonexceptional arrangement in CP2, then K(si*) is a Waldhausen graph manifold. We define a graph G(si*) of si* as follows. Let each vertex correspond to a line in si* with the weight of the self-intersection number of this line. Let each edge correspond to the intersection point of two lines in si *. We state some definitions and results derived from [12, 13]. Let M and N be compact orientable 3-manifolds. An isomorphism y/ of nx(N) onto n\(M) is said to respect the peripheral structure if for each boundary surface F in TV there is a boundary surface G of M such that y/(i*(n\(F))) c R and R is conjugate in n\(M) to i*(n\(G)) where /» denotes inclusion homomorphism. Theorem 3 (cf. [13, (6.5)]. // M and N are two Waldhausen graph manifolds and y/ is an isomorphism from nx(N) onto n\(M) which respect the peripheral structure and H\ (M) is infinite, then there exists a homeomorphism from N to M which induces y. 90 TAN JIANG AND S. S.-T. YAU Theorem 4 (cf. [12, (9.4)]). Two Waldhausen graph manifolds are homeomorphic if and only if the corresponding graphs are equivalent. Now suppose that six* and si2* are two nonexceptional arrangements in CP2 and M (si*) is homeomorphic to M(si2). In view of Theorem 3 and Lemmas 1 and 2, we have that K(six*) is homeomorphic to K(si2). By Theorem 4 we conclude that there is an isomorphism from L(six) to L(si2). This isomorphism also preserves weights (i.e., self-intersection number). So the main theorem follows from Theorem 5. Let six* and si2 be two arrangements in CP2. By blowing up their multiple points (of multiplicity > 3), we obtain two associated arrangements six* and si2* in some blown-up surfaces CP2. Then there exists an isomorphism from L(si*) onto L(si2) which preserves weights if and only if there is an isomorphism from L(si*) onto L(si2). We next suppose that both si* and si2* are exceptional. Write (1) six* — {Ho , H\, ... , Hp, Hp+\, ... , Hp+q), (2) si2* — {Hq, Gi, ... , Gs, Gs+l, ... , Gs+t} where Hq (respectively Go ) intersects with H\, ... , Hp (respectively G\, ... , Gs) at one point and interacts with Hp+\, ..., Hp+q (respectively, Gs+\, ..., Gs+t) at another point. If M(six) is homeomorphic to M(si2), then the Orlik-Solomon algebras associated to A\ and A2 are isomorphic. It follows that p + q = s + t and pq-st. So L(A\) is isomorphic to L(A2). Finally, we assume that six* is exceptional, but si2* is not. We need to show that M (si*) is not homeomorphic to M(si2). There are four subcases to consider. Case a. six consists of at most three lines. We need to observe only that the first betti number of M (A) is precisely \A\. So we have b\(M(A\)) < 3 < b\(M(A2)), and M(si*) is not homeomorphic to M(si2). Case b. si* is a pencil, and \si*\ > 4. This follows immediately from the following two lemmas. Lemma 6. Let si* be an arrangement in CP2. If si* is not a pencil (i.e., f\si* = 0 ) and \si*\>3, then b3(M(si)), the third betti number of M (si), is nonzero. Lemma 7. Let si* be an arrangement in CP2. If si* is a pencil (i.e., f]si* is a point), then b3(M(si)), the third betti number of M (si*), is zero. Case c. si* consists of a pencil and a line in general position, and \A\\ > 4 (see Figure 1). INTERSECTION LATTICES OF ARRANGEMENTS IN CP2 91