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Topology and geometry of cohomology jump loci
 DUKE MATHEMATICAL JOURNAL
, 2009
"... Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk a ..."
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Cited by 30 (21 self)
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Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic, if the group is 1formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1formality 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 quasiprojective 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.
Momentangle complexes, monomial ideals, and Massey products
 Pure and Applied Math. Quarterly
"... Abstract. Associated to every finite simplicial complex K there is a “momentangle” finite CWcomplex, ZK; if K is a triangulation of a sphere, ZK is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Mas ..."
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Cited by 22 (6 self)
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Abstract. Associated to every finite simplicial complex K there is a “momentangle” finite CWcomplex, ZK; if K is a triangulation of a sphere, ZK is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a momentangle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of nonformal manifolds and subspace arrangements are given. Contents
Bieri–Neumann–StrebelRenz invariants and homology jumping loci
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2010
"... We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential ta ..."
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Cited by 19 (18 self)
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We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as Kähler and quasiKähler manifolds. Contents
Toric complexes and Artin kernels
 ADVANCES IN MATHEMATICS
, 2009
"... Abstract. A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the rightangled Artin group GL. Given an epimorphism χ: GL → Z, let T χ L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of ..."
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Cited by 14 (11 self)
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Abstract. A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the rightangled Artin group GL. Given an epimorphism χ: GL → Z, let T χ L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of T χ L with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for H≤r(T χ L; k) to have trivial Zaction, allowing us to compute the truncated cohomology ring, H ≤r (T χ L; k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Zaction, and establish the 1formality of these (not necessarily finitely presentable) groups. Contents
The polyhedral product functor: a method of computation for momentangle complexes, arrangements and related spaces
, 2008
"... This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the ..."
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Cited by 13 (2 self)
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This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in GoreskyMacPherson [20], Hochster[22], Baskakov [3], Panov [36], and BuchstaberPanov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized momentangle complex applies for any homology theory. This decomposition gives an additive decomposition for the StanleyReisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of DenhamSuciu in [16].
QuasiKähler BestvinaBrady groups
 J. ALGEBRAIC GEOMETRY
, 2006
"... A finite simple graph Γ determines a rightangled Artin group GΓ, with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The BestvinaBrady group NΓ is the kernel of the projection GΓ → Z, which sends each generator v to 1. We esta ..."
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Cited by 10 (7 self)
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A finite simple graph Γ determines a rightangled Artin group GΓ, with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The BestvinaBrady group NΓ is the kernel of the projection GΓ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasiKähler (respectively, Kähler) groups NΓ. This yields examples of quasiprojective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.
Formality, Alexander invariants, and a question of Serre
, 2005
"... We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the Iadic completion of the Alexander invariant of a 1formal group G is determined solely by the cupproduct map in low degrees. It follows that the germs at the origin of the chara ..."
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Cited by 10 (4 self)
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We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the Iadic completion of the Alexander invariant of a 1formal group G is determined solely by the cupproduct map in low degrees. It follows that the germs at the origin of the characteristic and resonance varieties of G are analytically isomorphic; in particular, the tangent cone to Vd(G) at 1 equals Rd(G). This provides new and powerful obstructions to 1formality. A detailed analysis of the irreducible components of the first resonance variety yields even stronger obstructions to realizing a 1formal group as the fundamental group of a smooth, complex quasiprojective 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.
Algebraic invariants for BestvinaBrady groups
, 2006
"... BestvinaBrady groups arise as kernels of length homomorphisms GΓ → Z from rightangled Artin groups to the integers. Under some connectivity assumptions on the flag complex ∆Γ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we g ..."
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Cited by 9 (6 self)
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BestvinaBrady groups arise as kernels of length homomorphisms GΓ → Z from rightangled Artin groups to the integers. Under some connectivity assumptions on the flag complex ∆Γ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we give examples of BestvinaBrady groups which are not isomorphic to any Artin group or arrangement group.
On the profinite topology of rightangled Artin groups
 J.of Algebra
"... Abstract. In the present work, we give necessary and sufficient conditions on the graph of a rightangled Artin group that determine whether the group is subgroup separable or not. Also we show that rightangled Artin groups are residually torsionfree nilpotent. Moreover, we investigate the profini ..."
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Cited by 7 (0 self)
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Abstract. In the present work, we give necessary and sufficient conditions on the graph of a rightangled Artin group that determine whether the group is subgroup separable or not. Also we show that rightangled Artin groups are residually torsionfree nilpotent. Moreover, we investigate the profinite topology of F2 × F2 and of the group L in [18], which are the only obstructions for the subgroup separability of the rightangled Artin groups. We show that the profinite topology of the above groups is strongly connected with the profinite topology of F2. 1.
GEOMETRIC AND ALGEBRAIC ASPECTS OF 1FORMALITY
 BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES SCIENCES MATHÉMATIQUES DE ROUMANIE
, 2009
"... Formality is a topological property, defined in terms of Sullivan’s model for a space. In the simplyconnected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting, the weaker 1formality property allows one to reconstruct the ..."
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Cited by 6 (6 self)
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Formality is a topological property, defined in terms of Sullivan’s model for a space. In the simplyconnected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting, the weaker 1formality property allows one to reconstruct the rational prounipotent completion of the fundamental group, solely from the cup products of degree 1 cohomology classes. In this note, we survey various facets of formality, with emphasis on the geometric and algebraic implications of 1formality, and its relations to the cohomology jump loci and the Bieri–Neumann–Strebel invariant. We also produce examples of 4manifolds W such that, for every compact Kähler manifold M, the product M × W has the rational homotopy type of a Kähler manifold, yet M × W admits no Kähler metric.