We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Z\pi_1-module presentation of \pi_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the \pi_1-coinvariants of \pi_p. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of \pi_2, and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, we obtain information on \pi_2, directly from the graph. The \pi_1-coinvariants of \pi_2 may distinguish the homotopy 2-types of arrangement complements with the same \pi_1, and the same Betti numbers in low degrees.

Let X ⊂ C n+1 (resp. V ⊂ P n+1) be an algebraic hypersurface and set MX = C n+1 \ X (resp. MV = P n+1 \ V) where we suppose n> 0. The study of the topology of X, V and of their complements MX, MV is a classical subject going back to Zariski. In a sequence of papers Libgober has introduced and studied the Alexander invariants associated to X, V,

Abstract. We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups. 1.

Abstract. For line arrangements in P 2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above particular case. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action. We detect the nodal arrangements of smooth irreducible plane curves, by using the mixed Hodge structure on the cohomology of their complements. Contents

Abstract. Let V be a degree d, reduced hypersurface in CP n+1, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP n+1 −V ∪H, and let U c be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U c; Q) of the hypersurface complement and show that, if i ≤ n, these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials. Their zeros are roots of unity of order d and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair (CP n+1, V). As an application, we give obstructions on the eigenvalues of monodromy operators associated to the Milnor fibre of a projective hypersurface arrangement.

Abstract. Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of p-cells equals the p-th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-Mac Lane spaces. 1.

Abstract. We show that the π-equivariant chain complex (π = π1(M(A))), C• ( ˜ X), associated to a Morse-theoretic minimal CW-structure X on the complement M(A) of an arrangement A, is independent of X. The same holds for all scalar extensions, C• ( ˜ X) ⊗Zπ KZ, K a field, where X is an arbitrary minimal CW-structure on a space M. When A is a section of another arrangement Â, we show that the divisibility properties of the first Betti number of the Milnor fiber of A obstruct the homotopy realization of M(A) as a subcomplex of a minimal structure on M ( Â). If  is aspherical and A is a sufficiently generic section of Â, then H∗(M(A); L) may be described in terms of π, L and χ(M(A)), for an arbitrary local system L; explicit computations may be done, when  is fiber-type. In this case, explicit KZpresentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of M(A), πp, may also be obtained. For nonresonant abelian scalar extensions, the CZ-rank of πp ⊗Zπ CZ is combinatorially determined. Contents

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Abstract. Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra gA is defined to be the Lie algebra of primitives of the Yoneda algebra, ExtA(k, k). Under certain homological assumptions on A and its quadratic closure, we express gA as a semi-direct product of the well-understood holonomy Lie algebra hA with a certain hA-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincaré polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras. 1. Definitions and statements of results 1.1. Holonomy and homotopy Lie algebras. Let A be a graded, graded-commutative algebra over a field k, with graded piece Ak, k ≥ 0. We will assume that A is locally finite, connected, and generated in degree 1. In other words, A = T(V)/I, where V is a finite-dimensional k-vector space, T(V) = ⊕ k≥0 V ⊗k is the tensor algebra on V, and I is a two-sided ideal, generated in degrees 2 and higher. To such an algebra A, one naturally associates two graded Lie algebras over k (see for instance [12]). Definition 1.1. The holonomy Lie algebra hA is the quotient of the free Lie algebra on the dual of A1, modulo the ideal generated by the image of the transpose of the multiplication map µ: A1 ∧ A1 → A2: (1) hA = Lie(A ∗ 1) / ideal (im(µ ∗ : A ∗ 2 → A ∗ 1 ∧ A ∗ 1)). Note that hA depends only on the quadratic closure of A: if we put A = T(V)/(I2), then hA = h A Definition 1.2. The homotopy Lie algebra gA is the graded Lie algebra of primitive elements in the Yoneda algebra of A: (2) gA = Prim(ExtA(k, k)). In other words, the universal enveloping algebra of the homotopy Lie algebra is the Yoneda algebra: (3) U(gA) = ExtA(k, k).

Let A = {H1,..., Hd} be an affine essential hyperplane arrangement in C n+1, see [OT1], [OT2] for general facts on arrangements. We set as usual M = M(A) = C n+1 \X, X being the union of all the hyperplanes in A. One