arXiv subjects

Alexander I. Suciu

Publications and source records attributed to Alexander I. Suciu.

At least 37 records · Page 2Linked to original sources

arXiv2020

Cohomology jump loci of 3-manifolds

The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for $X$. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

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arXiv2019

Homology, lower central series, and hyperplane arrangements

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.

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arXiv2019

Taylor expansions of groups and filtered-formality

Let $G$ be a finitely generated group, and let $\Bbbk{G}$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\Bbbk{G}$ which generalizes the Magnus expansion of a free group. The group $G$ is said to be filtered-formal if its Malcev Lie algebra is isomorphic to the degree completion of its associated graded Lie algebra. We show that $G$ is filtered-formal if and only if it admits a Taylor expansion, and derive some consequences.

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arXiv2019

Poincaré duality and resonance varieties

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a 3-dimensional Poincaré duality algebra $A$, we obtain a fairly precise geometric description of the resonance varieties $\mathcal{R}^i_k(A)$.

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arXiv2019

Chen ranks and resonance varieties of the upper McCool groups

The group of basis-conjugating automorphisms of the free group of rank $n$, also known as the McCool group or the welded braid group $PΣ_n$, contains a much-studied subgroup, called the upper McCool group $PΣ_n^+$. Starting from the cohomology ring of $PΣ_n^+$, we find, by means of a Gröbner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group $P_n$ and the full McCool group $PΣ_n$, the Chen ranks conjecture does not hold for $PΣ_n^+$, for any $n\ge 4$. Consequently, $PΣ_n^+$ is not isomorphic to $P_n$ in that range, thus answering a question of Cohen, Pakianathan, Vershinin, and Wu. We also determine the scheme structure of the resonance varieties $\mathcal{R}_1(PΣ_n^+)$, and show that these schemes are not reduced for $n\geq 4$.

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arXiv2019

Naturality properties and comparison results for topological and infinitesimal embedded jump loci

We use augmented commutative differential graded algebra (ACDGA) models to study $G$-representation varieties of fundamental groups $π=π_1(M)$ and their embedded cohomology jump loci, around the trivial representation 1. When the space $M$ admits a finite family of maps, uniformly modeled by ACDGA morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of ${\rm Hom} (π,G)$ and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when $M$ is either a compact Kähler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group $G$ is a non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$.

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arXiv2019

Formality properties of finitely generated groups and Lie algebras

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. Another approach to formality is provided by Taylor expansions from the group to the completion of the associated graded algebra of the group ring. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

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arXiv2019

Residually finite rationally $p$ groups

In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in $3$-manifold topology enjoy this residual property. We then prove a combination theorem for RFR$p$ groups, which we use to study the boundary manifolds of algebraic curves $\mathbb{CP}^2$ and in $\mathbb{C}^2$. We show that boundary manifolds of a large class of curves in $\mathbb{C}^2$ (which includes all line arrangements) have RFR$p$ fundamental groups, whereas boundary manifolds of curves in $\mathbb{CP}^2$ may fail to do so.

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arXiv2018

Cup products, lower central series, and holonomy Lie algebras

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.

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arXiv2018

Infinitesimal finiteness obstructions

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1-minimal models, we also show that a finitely generated group $π$ admits a connected 1-model with finite-dimensional degree 1 piece if and only if the Malcev Lie algebra $\mathfrak{m}(π)$ is the lower central series completion of a finitely presented Lie algebra.

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arXiv2018

Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces

We consider smooth, complex quasi-projective varieties $U$ which admit a compactification with a boundary which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

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arXiv2017

Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan-Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\textrm{SL}_n(\mathbb{C})$ for some $n\ge 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

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arXiv2017

The topology of compact Lie group actions through the lens of finite models

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free $K$-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for $M$ with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of $M$ and of the representation varieties of $π_1(M)$. As an application, we show that compact Sasakian manifolds of dimension $2n+1$ are $(n-1)$-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.

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arXiv2017

Abelian duality and propagation of resonance

We explore the relationship between a certain "abelian duality" property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the "EPY property." The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina-Brady groups. Our approach brings to the fore the relevance of the Cohen-Macaulay condition in this combinatorial context.

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arXiv2017

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial $Δ_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $Δ_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $β_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $\mathcal{A}$ defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of $\mathcal{A}$ to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over $\mathbb{C}$) for matroids, and we estimate the number of essential components in the first complex resonance variety of $\mathcal{A}$. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of ${\rm SL}_2(\mathbb{C})$-representation varieties, which are governed by the Maurer--Cartan equation.

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arXiv2016

Pure virtual braids, resonance, and formality

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this class of groups. In the process, we explore various connections between the Alexander-type invariants of a finitely generated group and several of the graded Lie algebras associated to it, and discuss possible extensions of the resonance-Chen ranks formula in this context.

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arXiv2016

The pure braid groups and their relatives

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as well as their `upper' variants, $wP_n^+$ and $vP_n^+$. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

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arXiv2015

Combinatorial covers and vanishing of cohomology

We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen-Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients, and obtain new generalizations.

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