Alexandru Ion - Academia.edu (original) (raw)

Papers by Alexandru Ion

Research paper thumbnail of Quasi-Kähler Bestvina-Brady groups

Journal of Algebraic Geometry, 2008

A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each v... more A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The Bestvina-Brady group N Γ is the kernel of the projection G Γ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kähler (respectively, Kähler) groups N Γ . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

Research paper thumbnail of Alexander Invariants of Complex Hyperplane Arrangements

Transactions of The American Mathematical Society, 1997

Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is... more Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, α : Fs → Pn. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

Research paper thumbnail of Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

Transactions of The American Mathematical Society, 2006

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π... more If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H * (X, k), viewed as a module over the exterior algebra E on A:

Research paper thumbnail of Algebraic invariants for Bestvina-Brady groups

Journal of The London Mathematical Society-second Series, 2007

Bestvina-Brady groups arise as kernels of length homomorphisms G Γ → Z from right-angled Artin gr... more Bestvina-Brady groups arise as kernels of length homomorphisms G Γ → Z from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ∆ Γ , we compute several algebraic invariants of such a group N Γ , directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina-Brady groups which are not isomorphic to any Artin group or arrangement group.

Research paper thumbnail of Characteristic varieties of arrangements

Mathematical Proceedings of The Cambridge Philosophical Society, 1999

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

Research paper thumbnail of When does the associated graded Lie algebra of an arrangement group decompose

Commentarii Mathematici Helvetici, 2006

Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H.... more Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.

Research paper thumbnail of Moment angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, ZK ;... more Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, 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 moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.

Research paper thumbnail of Higher homotopy groups of complements of complex hyperplane arrangements

Advances in Mathematics, 2002

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from ... more 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 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.

Research paper thumbnail of Translated tori in the characteristic varieties of complex hyperplane arrangements

Topology and Its Applications, 2002

We give examples of complex hyperplane arrangements for which the top characteristic variety, , c... more We give examples of complex hyperplane arrangements for which the top characteristic variety, , contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus . 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.

Research paper thumbnail of Alexander invariants of complex hyperplane arrangements

Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is... more Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, α : Fs → Pn. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

Research paper thumbnail of Inequivalent frame-spun knots with the same complement

Commentarii Mathematici Helvetici, 1992

Research paper thumbnail of Homotopy Lie algebras, lower central series and the Koszul property

Geometry & Topology, 2004

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational... more Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k -rescaling of the rational cohomology ring of X . Assume H * (X, Q) is a Koszul algebra. Then, the homotopy Lie algebra π * (ΩY ) ⊗ Q equals, up to k -rescaling, the graded rational Lie algebra associated to the lower central series of π 1 (X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H * (X, Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of π 1 (X) and the completion of [ΩS 2k+1 , ΩY ]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

Research paper thumbnail of Homotopy types of complements of 2-arrangements in R 4

Topology, 2000

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtain... more We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.

Research paper thumbnail of Homotopy types of complements of 2-arrangements in R 4

Topology, 2000

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtain... more We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.

Research paper thumbnail of Cohomology rings and nilpotent quotients of real and complex arrangements

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ... more For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H ≤2 (X), to the second nilpotent quotient, G/G 3 . We define invariants of G/G 3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n ≤ 6 planes in R 4 .

Research paper thumbnail of Cohomology rings and nilpotent quotients of real and complex arrangements

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ... more For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H ≤2 (X), to the second nilpotent quotient, G/G 3 . We define invariants of G/G 3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n ≤ 6 planes in R 4 .

Research paper thumbnail of Quasi-Kähler Bestvina-Brady groups

Journal of Algebraic Geometry, 2008

A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each v... more A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The Bestvina-Brady group N Γ is the kernel of the projection G Γ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kähler (respectively, Kähler) groups N Γ . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

Research paper thumbnail of Alexander Invariants of Complex Hyperplane Arrangements

Transactions of The American Mathematical Society, 1997

Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is... more Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, α : Fs → Pn. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

Research paper thumbnail of Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

Transactions of The American Mathematical Society, 2006

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π... more If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H * (X, k), viewed as a module over the exterior algebra E on A:

Research paper thumbnail of Algebraic invariants for Bestvina-Brady groups

Journal of The London Mathematical Society-second Series, 2007

Bestvina-Brady groups arise as kernels of length homomorphisms G Γ → Z from right-angled Artin gr... more Bestvina-Brady groups arise as kernels of length homomorphisms G Γ → Z from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ∆ Γ , we compute several algebraic invariants of such a group N Γ , directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina-Brady groups which are not isomorphic to any Artin group or arrangement group.

Research paper thumbnail of Characteristic varieties of arrangements

Mathematical Proceedings of The Cambridge Philosophical Society, 1999

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

Research paper thumbnail of When does the associated graded Lie algebra of an arrangement group decompose

Commentarii Mathematici Helvetici, 2006

Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H.... more Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.

Research paper thumbnail of Moment angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, ZK ;... more Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, 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 moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.

Research paper thumbnail of Higher homotopy groups of complements of complex hyperplane arrangements

Advances in Mathematics, 2002

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from ... more 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 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.

Research paper thumbnail of Translated tori in the characteristic varieties of complex hyperplane arrangements

Topology and Its Applications, 2002

We give examples of complex hyperplane arrangements for which the top characteristic variety, , c... more We give examples of complex hyperplane arrangements for which the top characteristic variety, , contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus . 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.

Research paper thumbnail of Alexander invariants of complex hyperplane arrangements

Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is... more Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, α : Fs → Pn. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

Research paper thumbnail of Inequivalent frame-spun knots with the same complement

Commentarii Mathematici Helvetici, 1992

Research paper thumbnail of Homotopy Lie algebras, lower central series and the Koszul property

Geometry & Topology, 2004

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational... more Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k -rescaling of the rational cohomology ring of X . Assume H * (X, Q) is a Koszul algebra. Then, the homotopy Lie algebra π * (ΩY ) ⊗ Q equals, up to k -rescaling, the graded rational Lie algebra associated to the lower central series of π 1 (X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H * (X, Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of π 1 (X) and the completion of [ΩS 2k+1 , ΩY ]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

Research paper thumbnail of Homotopy types of complements of 2-arrangements in R 4

Topology, 2000

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtain... more We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.

Research paper thumbnail of Homotopy types of complements of 2-arrangements in R 4

Topology, 2000

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtain... more We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.

Research paper thumbnail of Cohomology rings and nilpotent quotients of real and complex arrangements

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ... more For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H ≤2 (X), to the second nilpotent quotient, G/G 3 . We define invariants of G/G 3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n ≤ 6 planes in R 4 .

Research paper thumbnail of Cohomology rings and nilpotent quotients of real and complex arrangements

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ... more For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H ≤2 (X), to the second nilpotent quotient, G/G 3 . We define invariants of G/G 3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n ≤ 6 planes in R 4 .