arXiv subjects

Alexander I. Suciu

Publications and source records attributed to Alexander I. Suciu.

At least 19 recordsLinked to original sources

arXiv2026

Profinite completions and cohomology jump loci

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$, under two hypotheses treated separately: that the loci are finite unions of torsion-translated subtori, which holds for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$ and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. We show also that $\widehat G$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements $\mathcal{A}$, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show that the profinite completion of the arrangement group $G(\mathcal{A})$ is not combinatorially determined, and does not determine $G(\mathcal{A})$.

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arXiv2026

Koszul modules, holonomy Lie algebras, and resonance of groups and CDGAs

We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The central mechanism is Koszul linearization, which replaces nonlinear equivariant constructions with functorial algebraic objects defined from a CDGA. To a connected CDGA $(A,d)$ with finite-dimensional $H^1(A)$ we associate Koszul modules $\mathcal{B}_i(A)$ over the symmetric algebra on $H_1(A)$. We prove that the first Koszul module $\mathcal{B}_1(A)$ is isomorphic to the infinitesimal Alexander invariant of the holonomy Lie algebra $\mathfrak{h}(A)$, yielding explicit formulas for holonomy Chen ranks. We establish a tangent cone theorem for resonance varieties, showing that cohomology controls their first-order behavior at the origin. For finitely generated groups admitting 1-finite 1-models, we prove that classical Alexander invariants agree with Koszul invariants after completion and passage to associated graded objects, and that Chen ranks are determined by the model. Applications include Chen rank computations for 2-step nilpotent Lie algebras and pure elliptic braid groups, partial formality results for Sasakian manifolds, and a general framework for detecting non-formality of spaces, groups, and maps.

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arXiv2026

Twisted homology jump loci, twisted Alexander polynomials, and $Σ$-invariants

We introduce the twisted homology jump loci of a space $X$: the jump loci for homology with coefficients in rank-one local systems, twisted by a fixed finite-dimensional representation $σ$ of $π_1(X)$. These loci refine the classical characteristic varieties, and their defining equations in degree one are the twisted Alexander polynomials of knot theory. Our main theorem is that their tropicalizations bound from above the Bieri--Neumann--Strebel--Renz (BNSR) $Σ$-invariants of $X$. Twisting gains real ground. The resulting bound is strictly stronger than the untwisted tropical bounds obtained from the usual characteristic varieties: for a one-relator group whose $Σ^1$ was computed by Brown, the untwisted bound excludes only two directions in $H^1(G;\mathbb{R})$, whereas the twisted bound determines $Σ^1(G)$ exactly. For a compact orientable $3$-manifold $M$ with toroidal or empty boundary, the twisted bound is sharp: the union of the twisted tropical varieties over all finite-image integral representations of $π_1(M)$ computes $Σ^1(π_1(M))$, and hence recovers the fibered faces of the Thurston norm ball. Sharpness genuinely requires twisting: a non-fibered class enters the tropical variety through the vanishing of a twisted Alexander polynomial along it, and the untwisted polynomial need not vanish. For a compact Kähler manifold $X$, we prove that the first twisted Alexander polynomial $Δ^σ(X)$ is either $0$ or $1$, for every representation $σ$ over every field, and that $Σ^1(π_1(X))$ is controlled by the hyperbolic orbifold fibrations of $X$ for every $σ$. The obstruction to Kählerianity that comes out of this is strictly finer than its untwisted counterpart: we exhibit groups $G$ with $Δ(G)=1$ but $Δ^σ(G)\ne 1$, which the twisted test excludes from being Kähler and the classical one does not.

math.AG↗

arXiv2026

Highly connected non-formal Milnor fibers via polyhedral products

We show that the realization theorem of Fernández de Bobadilla, which identifies the Milnor fiber of a weighted-homogeneous polynomial with the complement of a germ of analytic set, can be combined with the systematic Massey product constructions of Grbić-Linton for moment-angle complexes $\mathcal{Z}_K = \mathcal{Z}_K(D^2, S^1)$ to produce weighted-homogeneous polynomials whose Milnor fibers are arbitrarily highly connected and non-formal. The original application of this strategy, due to Fernández de Bobadilla, used the Denham-Suciu classification of lowest-degree triple Massey products and yielded only 2-connected non-formal Milnor fibers. The Grbić-Linton framework, which constructs non-trivial $n$-fold Massey products in $H^*(\mathcal{Z}_K;\mathbb{Q})$ for arbitrary $n$ and in arbitrary cohomological degrees, removes this connectivity restriction entirely.

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arXiv2026

Groups associated to 1-minimal models for binomial $\cup_1$-algebras

We give an explicit, cochain-level algebraic model for the pronilpotent completion of a group with finitely generated first cohomology. To each binomial $\cup_1$-dga $(A,d_A)$ over $R=\mathbb{Z}$ or $\mathbb{F}_p$ ($p$ prime) -- a differential graded algebra endowed with a Steenrod $\cup_1$-product and a compatible binomial operation -- we associate a pronilpotent group $G(A)$ that depends only on the 1-quasi-isomorphism type of $A$, provided $H^0(A)=R$ and $H^1(A)$ is a finitely generated free $R$-module. This group arises functorially from the 1-minimal model of $A$, which is unique up to isomorphism. When $A=C^*(X;R)$ is the cochain algebra of a connected CW-complex $X$ with $H^1(X;R)$ finitely generated, the group $G(A)$ recovers the Bousfield--Kan $R$-completion of $π_1(X)$ when $R=\mathbb{F}_p$, and its pro-torsion-free-nilpotent completion when $R=\mathbb{Z}$. Moreover, the group $G(A)$ comes equipped with a natural inverse system $\{G_n(A)\}_{n\ge 1}$ whose structure maps $G_{n+1}(A)\to G_n(A)$ are surjective. If $A=C^*(X;R)$, then $G_n(A)$ is the quotient of $π_1(X)$ by the $(n+1)$th term of the fastest descending central series whose successive quotients are free $R$-modules. We give a purely algebraic necessary and sufficient criterion that, given an isomorphism $G_n(A)\cong G_n(B)$, determines whether $G_{n+1}(A)\cong G_{n+1}(B)$, and we illustrate the use of this criterion with examples distinguishing spaces with isomorphic cohomology rings.

math.GR↗

arXiv2026

Lower central series and split extensions

Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series (LCS), Stallings' rational and mod-$q$ versions, and Zassenhaus' mod-$p$ version of the LCS. We treat them as part of a general construction of the $\mathcal P$-LCS, for a property $\mathcal P$ of filtrations. We describe these $N$-series and the associated Lie algebras in the case when $G$ splits as a semi-direct product, in terms of the relevant data for the factors and the monodromy action. This allows us to generalize the well-known theorem of Falk-Randell regarding the LCS of split extensions to other versions of the LCS. In particular, we generalize the mod-$q$ version of Bellingeri-Gervais to any integer $q$, and we prove analogous results for the rational LCS and Zassenhaus' mod-$p$ LCS. We then use the same tools to study residual properties of semi-direct products, and how they interact with residual properties of the factors. We also give a new proof of a classical theorem of Gruenberg. Finally, we apply our results to surface braid groups, which naturally split as semi-direct products, allowing us to recover and generalize known results about the residual nilpotency of groups of pure braids on surfaces.

math.GR↗

arXiv2026

Hyperplane arrangements with non-formal Milnor fibers

Each complex hyperplane arrangement $\mathcal{A}$ gives rise to a Milnor fibration of its complement. Building on work of Zuber, we give a combinatorial sufficient condition for the Milnor fiber $F(\mathcal{A})$ to be non-$1$-formal, expressed in terms of the multinet structure on $\mathcal{A}$, and use it to produce an infinite family of monomial arrangements $\mathcal{A}(3k,3k,3)$ with non-formal Milnor fibers. We also review the relevant background on cohomology jump loci, formality, and the topology of Milnor fibers of arrangements.

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arXiv2025

The effective Chen ranks conjecture

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. We give applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of every arrangement whose projective resonance is reduced. Finally, we formulate a sharp generic vanishing conjecture for Koszul modules and present a parallel between this statement and the Prym--Green Conjecture on syzygies of general Prym canonical curves.

math.AG↗

arXiv2025

On the topology and combinatorics of decomposable arrangements

A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

math.GR↗

arXiv2025

Cup-one algebras and 1-minimal models

In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. In this paper we show that binomial cup-one algebras capture homotopy 1-type. In particular, given such an $R$-dga, $(A,d_A)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{F}_p$ (for $p$ a prime), with $H^0(A)=R$ and with $H^1(A)$ a finitely generated, free $R$-module, we show that $A$ admits a functorially defined 1-minimal model, $ρ\colon (\mathcal{M}(A),d)\to (A,d_A)$, which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, whose continuous cohomology is isomorphic to that of $\mathcal{M}(A)$. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic torsion-free integral cohomology rings. Moreover, we show that there is an equivalence of categories between isomorphism classes of finitely-generated, torsion-free-nilpotent groups and isomorphism classes of finitely generated 1-minimal models over the integers.

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arXiv2024

Reduced resonance schemes and Chen ranks

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

math.AG↗

arXiv2024

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle Δ\rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $Δ$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

math.AC↗

arXiv2023

Cohomology, Bocksteins, and resonance varieties in characteristic 2

We use the action of the Bockstein homomorphism on the cohomology ring $H^*(X,\mathbb{Z}_2)$ of a finite-type CW-complex $X$ in order to define the resonance varieties of $X$ in characteristic 2. Much of the theory is done in the more general framework of the Maurer-Cartan sets and the resonance varieties attached to a finite-type commutative differential graded algebra. We illustrate these concepts with examples mainly drawn from closed manifolds, where Poincaré duality over $\mathbb{Z}_2$ has strong implications on the nature of the resonance varieties.

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arXiv2022

Alexander invariants and cohomology jump loci in group extensions

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form $1\to K\to G\to Q\to 1$, where $Q$ is an abelian group acting trivially on $H_1(K;\mathbb{Z})$, with suitable modifications in the rational and mod-$p$ settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups $K$ and $G$. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

math.GR↗

arXiv2022

Formality and finiteness in rational homotopy theory

We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1-formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3-dimensional manifolds.

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arXiv2021

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Motivated by the construction of Steenrod cup-$i$ products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.

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arXiv2021

Sigma-invariants and tropical varieties

The Bieri-Neumann-Strebel-Renz invariants $Σ^q(X,\mathbb{Z})\subset H^1(X,\mathbb{R})$ of a connected, finite-type CW-complex $X$ are the vanishing loci for Novikov-Sikorav homology in degrees up to $q$, while the characteristic varieties $\mathcal{V}^q(X) \subset H^1(X,\mathbb{C}^{\times})$ are the nonvanishing loci for homology with coefficients in rank 1 local systems in degree $q$. We show that each BNSR invariant $Σ^q(X,\mathbb{Z})$ is contained in the complement of the tropical variety associated to the algebraic variety $\mathcal{V}^{\le q}(X)$, and provide applications to several classes of groups and spaces.

math.AG↗

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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