Alexander I. Suciu

Characteristic varieties of arrangements

The k th Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, V k ([Ascr ]), of the algebraic torus ([Copf ]*) n . In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V k ([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ] 1 k ( A ), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [ 1 ], we conclude that all irreducible components of V k ([Ascr ]) which pass through the identity element of ([Copf ]*) n are combinatorially determined, and that [Rscr ] 1 k ( A ) is the union of a subspace arrangement in [Copf ] n , thereby resolving a conjecture of Falk [ 11 ]. We use these results to study the reflection arrangements associated to monomial groups.

Topology and geometry of cohomology jump loci

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

Translated tori in the characteristic varieties of complex hyperplane arrangements

Abstract We give examples of complex hyperplane arrangements A for which the top characteristic variety, V 1 ( A ) , contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus ( C ∗ ) ∣ A ∣ . These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik–Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.

Homology of iterated semidirect products of free groups

Abstract Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of G . This resolution is used to define representations of groups which act compatibly on G , generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for computing the homology of the Milnor fiber of a fiber-type hyperplane arrangement, and more generally, the homology of the complement of such an arrangement with coefficients in an arbitrary local system.

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of GΓ, directly from the graph Γ.

Chen Lie algebras

The Chen groups of a finitely presented group G are the lower central series quotients of its maximal metabelian quotient G/G″. The direct sum of the Chen groups is a graded Lie algebra, with bracket induced by the group commutator. If G is the fundamental group of a formal space, we give an analog of a basic result of Sullivan by showing that the rational Chen Lie algebra of G is isomorphic to the rational holonomy Lie algebra of G modulo the second derived subalgebra. Following an idea of Massey, we point out a connection between the Alexander invariant of a group G defined by commutator-relators and its integral holonomy Lie algebra. As an application, we determine the Chen Lie algebras of several classes of geometrically defined groups, including surface-like groups, fundamental groups of certain link complements in S 3 , and fundamental groups of complements of hyperplane arrangements in ℂ l . For link groups, we sharpen Massey and Traldis solution of the Murasugi conjecture. For arrangement groups, we prove that the rational Chen Lie algebra is combinatorially determined.

Alexander Polynomials: Essential Variables and Multiplicities

We explore the codimension-one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in homology 3-spheres, these bounds become equalities, and our formula shows explicitly how the Alexander polynomial determines all the characteristic varieties.

Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements

Abstract 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 π1-module presentation of πp, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π1-coinvariants of πp. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π2 and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, the algorithm for computing π2 is purely combinatorial. The Fitting varieties associated to π2 may distinguish the homotopy 2-types of arrangement complements with the same π1, and the same Betti numbers in low degrees.

ALEXANDER INVARIANTS OF COMPLEX HYPERPLANE ARRANGEMENTS

LetA be an arrangement of n complex hyperplanes. The funda- mental group of the complement of A is determined by a braid monodromy homomorphism, fi : Fs ! Pn. Using the Gassner representation of the pure braid group, we flnd an explicit presentation for the Alexander invariant ofA. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups ofA. We also provide a combinatorial criterion for when these lower bounds are attained.

The spectral sequence of an equivariant chain complex and homology with local coefficients

We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d 1 differential in terms of the coalgebra structure of H * (X, k) and the kπ 1 (X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a prime-power order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H* (X, k), thereby generalizing a result of Cohen and Orlik.