%% bibtex-file{
%% author = {Alexander I. Suciu},
%% date = {June 15, 2020},
%% filename = {Suciu.bib},
%% url = {https://web.northeastern.edu/suciu/Suciu.bib},
%% www-home = {https://web.northeastern.edu/suciu/},
%% address = {Department of Mathematics,
%% Northeastern University,
%% 360 Huntington Avenue,
%% Boston, MA 02115,
%% United States of America},
%% telephone = {+1 617 373 5635},
%% fax = {+1 617 373 5658},
%% MR = {168600},
%% genealogy = {37530},
%% ORCID = {0000-0002-5060-7754},
%% ftp-archive = {https://web.northeastern.edu/suciu/papers/},
%% email = {a [dot] suciu [at] northeastern [dot] edu},
%% dates = {1984--},
%% supported = {yes},
%% supported-by = {a [dot] suciu [at] northeastern [dot] edu},
%% abstract = {Bibliography for Alexander I. Suciu}}
% ===================================================
@unpublished {Suciu:su19,
author = {Suciu, Alexander I.},
title = {Cohomology jump loci of $3$-manifolds},
MRCLASS = {55N25, 57M27 (16E45, 55P62, 57M05, 57M25, 57N10)},
keywords = {Characteristic variety, resonance variety, tangent cone, algebraic model, formality,
{A}lexander polynomial, closed $3$-manifold, connected sum, graph manifold, link complement},
abstract = { 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 {A}lexander 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.},
arxiv = {https://arxiv.org/abs/1901.01419},
gsid = {10694629407429950151}
}
@article {Suciu:rps19,
author = {Porter, Richard D. and Suciu, Alexander I.},
title = {Homology, lower central series, and hyperplane arrangements},
keywords = {Lower central series, tower of nilpotent quotients,
cohomology ring, associated graded {L}ie algebra, holonomy {L}ie algebra,
{M}alcev {L}ie algebra, hyperplane arrangement, decomposable arrangement},
abstract = {We explore finitely generated groups by studying the nilpotent
towers and the various {L}ie algebras attached to such groups.
Our main goal is to relate an isomorphism extension problem
in the {P}ostnikov tower to the existence of certain commuting diagrams.
This recasts a result of {G}.~{R}ybnikov 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.
},
JOURNAL = {Eur. J. Math},
FJOURNAL = {European Journal of Mathematics},
VOLUME = {},
YEAR = {},
NUMBER = {},
PAGES = {},
MRCLASS = {20J05 (16S37, 16W70, 17B70, 20F14, 20F18, 20F40, 55P62, 57M05)},
MRNUMBER = {},
MRREVIEWER = {},
ZBLNUMBER = {},
ZBREVIEWER = {},
doi = {10.1007/s40879-019-00392-x},
arxiv = {https://arxiv.org/abs/1906.04885},
gsid = {TuM7UPshZo8C}
}
@article {Suciu:sw19,
author = {Suciu, Alexander I. and Wang, He},
title = {Taylor expansions of groups and filtered-formality},
keywords = {{T}aylor expansion, {H}opf algebra, {C}hen iterated integrals,
{M}alcev {L}ie algebra, filtered-formality, $1$-formality, residually
torsion-free nilpotent group, automorphism groups of free groups.},
abstract = {Let $G$ be a finitely generated group, and let $\Bbbk{G}$ be its group algebra
over a field of characteristic $0$. A {T}aylor expansion is a certain type of map from
$G$ to the degree completion of the associated graded algebra of $\Bbbk{G}$
which generalizes the {M}agnus expansion of a free group.
The group $G$ is said to be filtered-formal if its {M}alcev {L}ie
algebra is isomorphic to the degree completion of its associated
graded {L}ie algebra. We show that $G$ is filtered-formal if and
only if it admits a {T}aylor expansion, and derive some consequences.},
JOURNAL = {Eur. J. Math},
FJOURNAL = {European Journal of Mathematics},
VOLUME = {},
YEAR = {},
NUMBER = {},
PAGES = {},
MRCLASS = {20F40 (16T05, 16W70, 17B70, 20F14, 20J05, 55P62)},
MRNUMBER = {},
MRREVIEWER = {},
ZBLNUMBER = {},
ZBREVIEWER = {},
doi = {10.1007/s40879-019-00389-6},
arxiv = {https://arxiv.org/abs/1905.10355},
gsid = {8145995317007639691}
}
@article {Suciu:su18,
author = {Suciu, Alexander I.},
title = {Poincar\'{e} duality and resonance varieties},
MRCLASS = {55U30, 57P10 (13A02, 13E10, 14M12, 15A75, 57N10)},
keywords = {Graded commutative algebra, resonance variety,
{P}oincar\'e duality algebra, connected sum, alternating 3-form, {P}faffian},
abstract = {We explore the constraints imposed by {P}oincar\'e duality on
the resonance varieties of a graded algebra. For a $3$-dimensional
{P}oincar\'e duality algebra $A$, we obtain a fairly precise geometric
description of the resonance varieties $\mathcal{R}^i_k(A)$.},
JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A.},
FJOURNAL = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics},
VOLUME = {},
YEAR = {},
NUMBER = {},
PAGES = {},
MRCLASS = {55U30, 57P10 (13A02, 13E10, 14M12, 15A75, 57N10)},
MRNUMBER = {},
MRREVIEWER = {},
ZBLNUMBER = {},
ZBREVIEWER = {},
doi = {10.1017/prm.2019.55},
arxiv = {https://arxiv.org/abs/1809.01801},
gsid = {8985420256983291970}
}
@article {Suciu:ks20,
author = {Koberda, Thomas and Suciu, Alexander I.},
title = {Residually finite rationally $p$ groups},
JOURNAL = {Commun. Contemp. Math},
FJOURNAL = {Communications in Contemporary Mathematics},
VOLUME = {22},
YEAR = {2020},
NUMBER = {3},
PAGES = {1950016 (44 pages)},
MRCLASS = {20E26 (14H50, 20F65, 52C35, 55N25, 57M10, 57N10)},
MRNUMBER = {4082220},
MRREVIEWER = {},
ZBLNUMBER = {07190495},
ZBREVIEWER = {},
keywords = {Residually finite rationally $p$ group, graph of groups, $3$-manifold,
plane algebraic curve, boundary manifold, {A}lexander varieties, {BNS} invariant.},
abstract = {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.},
doi = {10.1142/S0219199719500160},
arxiv = {http://arxiv.org/abs/1604.02010},
gsid = {1318505386575449464}
}
@article {Suciu:sw19c,
author = {Suciu, Alexander I. and Wang, He},
title = {Chen ranks and resonance varieties of the upper {M}c{C}ool groups},
JOURNAL = {Adv. in Appl. Math},
FJOURNAL = {Advances in Applied Mathematics},
VOLUME = {110},
YEAR = {2019},
NUMBER = {},
PAGES = {197--234.},
MRCLASS = {20E36 (13P10, 14M12, 17B70, 20F14, 20J05)},
MRNUMBER = {3983125},
MRREVIEWER = {Valeriy Bardakov},
ZBLNUMBER = {07116111},
ZBREVIEWER = {},
keywords = {{M}c{C}ool and upper {M}c{C}ool groups, resonance varieties, resonance scheme,
{C}hen ranks, infinitesimal {A}lexander invariant.},
abstract = {The group of basis-conjugating automorphisms of the free group of rank $n$,
also known as the {M}c{C}ool group or the welded braid group $P\Sigma_n$, contains
a much-studied subgroup, called the upper {M}c{C}ool group $P\Sigma_n^+$. Starting from
the cohomology ring of $P\Sigma_n^+$, we find, by means of a {G}r{\”{o}}bner
basis computation, a simple presentation for the infinitesimal {A}lexander invariant of this
group, from which we determine the resonance varieties and the {C}hen ranks of the
upper {M}c{C}ool groups. These computations reveal that, unlike for the pure braid group
$P_n$ and the full {M}c{C}ool group $P\Sigma_n$, the {C}hen ranks conjecture does not
hold for $P\Sigma_n^+$, for any $n\ge 4$. Consequently, $P\Sigma_n^+$ is not isomorphic
to $P_n$ in that range, thus answering a question of {C}ohen, {P}akianathan, {V}ershinin,
and {W}u. We also determine the scheme structure of the resonance varieties
$\mathcal{R}_1(P\Sigma_n^+)$, and show that these schemes are not reduced for $n\geq 4$.
},
doi = {10.1016/j.aam.2019.07.004},
arxiv = {http://arxiv.org/abs/1804.06006},
gsid = {7653340319534771439}
}
@article {Suciu:sw19b,
author = {Suciu, Alexander I. and Wang, He},
title = {Formality properties of finitely generated groups and {L}ie algebras},
JOURNAL = {Forum Math.},
FJOURNAL = {Forum Mathematicum},
VOLUME = {31},
YEAR = {2019},
NUMBER = {4},
PAGES = {867--905},
MRCLASS = {20F40 (16S37, 17B70, 20F14, 20J05, 55P62, 57M05)},
MRNUMBER = {3975666},
MRREVIEWER = {Takao Satoh},
ZBLNUMBER = {07093392},
ZBREVIEWER = {},
keywords = {Lower central series, derived series, {M}alcev {L}ie algebra,
holonomy {L}ie algebra, {C}hen {L}ie algebra, {K}oszul algebra,
{M}agnus expansion, cohomology ring, $1$-formality,
graded formality, filtered formality, nilpotent group, {S}eifert manifold.},
abstract = {We explore the graded and filtered formality properties of a
finitely-generated group by studying the various {L}ie algebras attached to such
a group, including the associated graded {L}ie algebra, the holonomy {L}ie algebra,
and the {M}alcev {L}ie algebra. We explain how these notions behave with respect to
split injections, coproducts, and direct products, and how they are inherited
by solvable and nilpotent quotients.
For a finitely-presented group, we give an explicit formula for the cup
product in low degrees, and an algorithm for computing the holonomy {L}ie
algebra, using a {M}agnus expansion method. We also give a presentation for the
{C}hen {L}ie algebra of a filtered-formal group, and discuss various approaches to
computing the ranks of the graded objects under consideration. We illustrate
our approach with examples drawn from a variety of group-theoretic and
topological contexts, such as $1$-relator groups, finitely generated torsion-free
nilpotent groups, link groups, and fundamental groups of orientable {S}eifert fibered
manifolds.},
doi= {10.1515/forum-2018-0098},
arxiv = {http://arxiv.org/abs/1504.08294},
gsid = {7852857282063325999}
}
@article {Suciu:ps20,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds},
JOURNAL = {J. Inst. Math. Jussieu},
FJOURNAL = {Journal of the Institute of Mathematics of Jussieu},
YEAR = {2020},
month = {march},
VOLUME = {19},
Number = {2},
Pages = {451--485},
MRCLASS = {14F35, 55N25 (20C15, 55P62)},
MRNUMBER = {4031243},
MRREVIEWER = {},
ZBLNUMBER = {07182067},
ZBREVIEWER = {Ivan Limonchenko},
keywords = {Representation variety, variety of flat connections, cohomology jump loci,
analytic germ, differential graded algebra model, quasi-projective manifold,
admissible map, {D}eligne weight filtration, holonomy {L}ie algebra, semisimple {L}ie algebra},
abstract = {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 {M}organ--{D}upont 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 {B}orel 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.},
doi = {10.1017/S1474748018000063},
arxiv = {http://arxiv.org/abs/1702.05661},
gsid = {qt-6tCTBDsQC}
}
@article {Suciu:ps18a,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {The topology of compact {L}ie group actions through the lens of finite models},
JOURNAL = {Int. Math. Res. Not. IMRN},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2019},
month = {oct},
VOLUME = {20},
Number = {20},
Pages = {6390--6436},
MRCLASS = {55P62 (14F35, 17B70, 53C25, 57S25)},
MRNUMBER = {4031243},
MRREVIEWER = {},
ZBLNUMBER = {},
ZBREVIEWER = {},
MRCLASS = {55P62 (14F35, 17B70, 53C25, 57S25)},
keywords = {Differential graded algebra, formality, holonomy {L}ie algebra, {M}alcev completion, representation variety,
cohomology jump loci, {S}asakian geometry, isolated surface singularity.},
abstract = {Let $M$ be a compact smooth manifold supporting an almost free action by a compact, connected {L}ie group $K$.
Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the
action, we describe a commutative differential graded algebra model for $M$ with commensurate finiteness and partial
formality properties. The existence of this model has various implications on the structure of the cohomology jump loci
of $M$ and of the representation varieties of $\pi_1(M)$. As an application, we show that compact {S}asakian manifolds
of dimension $2n+1$ are $(n-1)$-formal, and that their fundamental groups are filtered-formal.
Further applications to the topological study of weighted-homogeneous isolated surface singularities are also given.},
doi = {10.1093/imrn/rnx294},
url = {http://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnx294/4801177},
arxiv = {http://arxiv.org/abs/1511.08948},
gsid = {12798929603215104996}
}
@article{Suciu:ps19b,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Naturality properties and comparison results for topological and infinitesimal embedded jump loci},
JOURNAL = {Adv. in Math},
FJOURNAL = {Advances in Mathematics},
VOLUME = {350},
YEAR = {2019},
NUMBER = {},
PAGES = {256--303},
MRCLASS = {14B12, 14F35, 55N25, 55P62 (20C15, 57S15)},
MRNUMBER = {3947646},
MRREVIEWER = {Matthias Wendt},
ZBLNUMBER = {07066542},
ZBREVIEWER = {He Wang},
keywords = {Representation variety, flat connection, cohomology jump loci, filtered differential
graded algebra, relative minimal model, mixed {H}odge structure, analytic local ring, {A}rtinian
local ring, differential graded {L}ie algebra, deformation theory, formal spaces and maps,
quasi-compact {K}{\"{a}}hler manifold, hyperplane arrangement, principal bundle.},
abstract = {We use augmented commutative differential graded algebra (ACDGA)
models to study $G$-representation varieties of fundamental groups $\pi=\pi_1(M)$
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 $\mathrm{Hom} (\pi,G)$ 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 {K}{\"{a}}hler
manifold, the complement of a central complex hyperplane arrangement, or the total
space of a principal bundle with formal base space, provided the {L}ie algebra of the
linear algebraic group $G$ is a non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$.},
doi = {10.1016/j.aim.2019.04.058},
arxiv = {http://arxiv.org/abs/1609.02768},
gsid = {8791051831947232049}
}
@article {Suciu:sw19a,
author = {Suciu, Alexander I. and Wang, He},
title = {Cup products, lower central series, and holonomy {L}ie algebras},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {223},
YEAR = {2019},
NUMBER = {8},
PAGES = {3359-3385},
MRCLASS = {20F40, 57M05 (17B70, 20F14, 20J05)},
MRNUMBER = {3947646},
MRREVIEWER = {Jens Harlander},
ZBLNUMBER = {07047899},
ZBREVIEWER = {},
keywords = {Lower central series, derived series, holonomy {L}ie algebra, graded formality,
{M}agnus expansion, cohomology ring, {C}hen {L}ie algebra, link group, one-relator group,
{S}eifert manifold.},
abstract = {We generalize basic results relating the associated graded {L}ie algebra and the
holonomy {L}ie 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 {L}ie algebra,
using a {M}agnus 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 {S}eifert fibered manifolds.},
doi ={10.1016/j.jpaa.2018.11.006},
arxiv = {http://arxiv.org/abs/1701.07768},
gsid = {OkrdTXRNpVkC}
}
@article {Suciu:ps19a,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Infinitesimal finiteness obstructions},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society},
YEAR = {2019},
month = {feb},
VOLUME = {99},
Number = {1},
Pages = {173-193},
MRCLASS = {155P62 (17B01, 20F14, 20J05, 55N25)},
MRNUMBER = {3909253 },
MRREVIEWER = {Ricardo Campos},
ZBLNUMBER = {07047899},
ZBREVIEWER = {Daniel Tanr\'{e}},
keywords = {Differential graded algebra, minimal model, metabelian group, cohomology jump loci,
filtered formal group, {H}all bases, holonomy {L}ie algebra, {M}alcev {L}ie algebra.},
abstract = {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 $\pi$ admits a connected $1$-model with finite-dimensional degree $1$ piece if and only if
the {M}alcev {L}ie algebra $\m(\pi)$ is the lower central series completion of a finitely presented {L}ie algebra.},
doi= {10.1112/jlms.12169},
arxiv = {http://arxiv.org/abs/1711.07085},
gsid = {7653340319534771439}
}
@article{Suciu:ds18,
author = {Denham, Graham and Suciu, Alexander I.},
title = {Local systems on arrangements of smooth, complex algebraic hypersurfaces},
JOURNAL = {Forum Math. Sigma},
FJOURNAL = {Forum of Mathematics. Sigma},
YEAR = {2018},
month = {May},
VOLUME = {6},
Number = {e6},
Pages = {20 pages},
MRCLASS = {55N25 (14M27, 20J05, 32E10, 32S22, 55R80, 55U30, 57M07)},
MRNUMBER = {3810026},
MRREVIEWER = {Shaheen Nazir},
ZBLNUMBER = {06876236},
ZBREVIEWER = {Daniel Juan Pineda},
keywords = {Arrangement of hypersurfaces, {S}tein manifold, projective variety, wonderful model,
cohomology with local systems, duality space, abelian duality space, hyperplane arrangement.},
abstract = {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.},
doi = {10.1017/fms.2018.5},
arxiv = {http://arxiv.org/abs/1706.00956},
gsid = {5789049093052623374}
}
@article{Suciu:mgltscbo18,
author = {Caitlyn L. Mills and Rohan Garg and Joslynn S. Lee and Liang Tian and Alexandru Suciu and
Gene Cooperman and Penny J. Beuning and Mary Jo Ondrechen},
title = {Functional classification of protein structures by local structure matching in graph representation},
journal = {Protein Science},
year={2018},
month={May},
volume = {27},
number = {6},
pages = {1125-1135},
keywords = {protein function annotation, graph representation of active sites for prediction of function (GRASP-Func), structurally
aligned local sites of activity (SALSA), ribulose phosphate binding barrel (RPBB) superfamily,
6-hairpin glycosidase (6-HG) superfamily, Concanavalin A-like lectins/glucanase (CAL/G) superfamily},
doi = {10.1002/pro.3416},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/pro.3416},
abstract = {As a result of high-throughput protein structure initiatives, over 14,400 protein structures have been
solved by structural genomics (SG) centers and
participating research groups. While the totality of SG data represents a tremendous contribution to genomics and structural biology,
reliable functional information for these proteins is generally lacking. Better functional predictions for SG proteins will add substantial
value to the structural information already obtained. Our method described herein, Graph Representation of Active Sites for Prediction
of Function (GRASP-Func), predicts quickly and accurately the biochemical function of proteins by representing residues at the
predicted local active site as graphs rather than in Cartesian coordinates. We compare the GRASP-Func method to our previously
reported method, structurally aligned local sites of activity (SALSA), using the ribulose phosphate binding barrel (RPBB), 6-hairpin
glycosidase (6-HG), and Concanavalin A-like Lectins/Glucanase (CAL/G) superfamilies as test cases. In each of the superfamilies,
SALSA and the much faster method GRASP-Func yield similar correct classification of previously characterized proteins, providing
a validated benchmark for the new method. In addition, we analyzed SG proteins using our SALSA and GRASP-Func methods to
predict function. Forty-one SG proteins in the RPBB superfamily, nine SG proteins in the 6‐HG superfamily, and one SG protein in
the CAL/G superfamily were successfully classified into one of the functional families in their respective superfamily by both methods.
This improved, faster, validated computational method can yield more reliable predictions of function that can be used for a wide variety
of applications by the community.}
}
@incollection {Suciu:sw17b,
author = {Suciu, Alexander I. and Wang, He},
title = {The pure braid groups and their relatives},
SERIES = {Springer INdAM Series},
FSERIES = {Springer INdAM Series},
BOOKTITLE = {{P}erspectives in {L}ie {T}heory},
VOLUME = {19},
PAGES = {403-426},
PUBLISHER = {Springer},
ADDRESS = {Cham},
YEAR = {2017},
month = {12},
ISBN = {},
EDITOR = {{F}ilippo {C}allegaro and {G}iovanna {C}arnovale and {F}abrizio {C}aselli
and {C}orrado {D}e {C}oncini and {A}lberto {D}e {S}ole},
MRCLASS = {20F40 (16S37, 17B70, 20F14, 20J05, 55P62, 57M05)},
MRNUMBER = {3751136},
MRREVIEWER = {},
ZBLNUMBER = {06854707},
ZBREVIEWER = {},
keywords = {Pure braid groups, welded pure braid group, virtual pure braid groups,
lower central series, {C}hen ranks, resonance varieties, formality.},
abstract = {In this survey, we investigate the resonance varieties,
the lower central series ranks, the {C}hen ranks,
and the 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.},
doi = {10.1007/978-3-319-58971-8_15},
url = {https://link.springer.com/chapter/10.1007/978-3-319-58971-8_15},
arxiv = {http://arxiv.org/abs/1602.05291},
gsid = {11794328476137991467}
}
@article {Suciu:dsy17,
author = {Denham, Graham and Suciu, Alexander I. and Yuzvinsky, Sergey},
title = {Abelian duality and propagation of resonance},
JOURNA = {Selecta Mathematica},
FJOURNAL = {Selecta Math. (N.S.)},
VOLUME = {23},
YEAR = {2017},
month = {oct},
NUMBER = {4},
PAGES = {2331-2367},
MRCLASS = {55N25 (13C14, 20F36, 32S22, 55U30, 57M07)},
MRNUMBER = {3703455},
MRREVIEWER = {He Wang},
ZBLNUMBER = {1381.55005},
ZBREVIEWER = {Alexandru Dimca},
keywords = {Duality space, abelian duality space, characteristic variety,
resonance variety, propagation, {EPY} property, hyperplane arrangement,
toric complex, right-angled {A}rtin group, {B}estvina--{B}rady group,
{C}ohen--{M}acaulay property.},
abstract = {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 {A}rtin groups,
and {B}estvina--{B}rady groups. Our approach brings to the fore the relevance of the
{C}ohen--{M}acaulay condition in this combinatorial context.},
doi = {10.1007/s00029-017-0343-5},
url = {https://link.springer.com/article/10.1007%2Fs00029-017-0343-5},
arxiv = {http://arxiv.org/abs/1512.07702},
gsid = {14575197265802691190}
}
@article {Suciu:sw17a,
author = {Suciu, Alexander I. and Wang, He},
title = {Pure virtual braids, resonance, and formality},
JOURNAL = {Mathematische Zeitschrift},
FJOURNAL = {Math. Zeit.},
VOLUME = {286},
YEAR = {2017},
NUMBER = {3-4},
PAGES = {1495--1524},
MRCLASS = {20F36 (16S37, 20F14, 20F40, 20J05, 55P62, 57M07)},
MRNUMBER = {3671586},
MRREVIEWER = {Matthew C. B. Zaremsky},
ZBLNUMBER = {06780338},
ZBREVIEWER = {},
keywords = {Pure virtual braid groups, lower central series, {C}hen ranks, {A}lexander invariants,
resonance varieties, holonomy {L}ie algebra, $1$-formality, graded-formality, filtered-formality.},
abstract = { We investigate the resonance varieties, lower central series ranks, and {C}hen
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 {A}lexander-type invariants of a finitely generated
group and several of the graded {L}ie algebras associated to it, and discuss possible extensions
of the resonance-{C}hen ranks formula in this context.},
doi = {10.1007/s00209-016-1811-x},
url = {http://link.springer.com/article/10.1007%2Fs00209-016-1811-x},
arxiv = {http://arxiv.org/abs/1602.04273},
gsid = {9884256930794724823}
}
@article {Suciu:ps17a,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society},
VOLUME = {114},
YEAR = {2017},
NUMBER = {6},
PAGES = {961-1004},
MRCLASS = {32S55, 52C35 (05B35, 14C21, 14F35, 32S22, 55N25)},
MRNUMBER = {3661343},
MRREVIEWER = {Shaheen Nazir},
ZBLNUMBER = {1378.32020},
ZBREVIEWER = {Piotr Pokora},
keywords = {Milnor fibration, algebraic monodromy, hyperplane arrangement, simple matroid, resonance variety,
characteristic variety, holonomy Lie algebra, flat connection, multinet, pencil.},
abstract = {A central question in arrangement theory is to determine whether the characteristic polynomial $\Delta_q$
of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the {M}ilnor 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 $\Delta_1$ corresponding
to certain eigenvalues of order a power of a prime $p$ are equal to the {A}omoto--{B}etti numbers $\beta_p(\A)$,
which in turn are extracted from $L(\A)$. 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. Along the way, we
characterize nets supported by arrangements in terms of resonance in characteristic $2$ and $3$, and we obtain
a lower bound for the number of essential components in the first complex resonance variety of $\A$ in terms of
$\beta_3(\mathcal{A})$. Our approach is based on a rather unexpected connection with the geometry of
$\SL_2(\mathbb{C})$-representation varieties, which are governed by the {M}aurer--{C}artan equation.},
doi = {10.1112/plms.12027},
url = {http://dx.doi.org/10.1112/plms.12027},
arxiv = {http://arxiv.org/abs/1401.0868},
gsid = {6991752821135895062}
}
@article {Suciu:su17,
author = {Suciu, Alexandru I},
title = {On the topology of the {M}ilnor fibration of a hyperplane arrangement},
JOURNAL = {Revue Roumaine de Math\'{e}matiques Pures et Appliqu\'{e}es},
FJOURNAL = {Rev. Roumaine Math. Pures Appl.},
VOLUME = {62},
YEAR = {2017},
NUMBER = {1},
PAGES = {191--215},
MRCLASS = {32S55, 52C35 (05B35, 14C21, 14F35, 32S22, 55N25, 57M10)},
MRNUMBER = {3626439},
MRREVIEWER = {Dmitry Kerner},
ZBLNUMBER = {1389.32030},
ZBREVIEWER = {},
keywords = {This note is mostly an expository survey, centered on the topology
of complements of hyperplane arrangements, their {M}ilnor fibrations, and their
boundary structures. An important tool in this study is provided by the degree $1$
resonance and characteristic varieties of the complement, and their tight relationship
with orbifold fibrations and multinets on the underlying matroid. In favorable situations,
this approach leads to a combinatorial formula for the first {B}etti number of the {M}ilnor
fiber and the algebraic monodromy. We also produce a pair of arrangements
for which the respective {M}ilnor fibers have the same {B}etti numbers, yet are
not homotopy equivalent: the difference is picked up by isolated torsion points
in the higher-depth characteristic varieties.},
url = {http://imar.ro/journals/Revue_Mathematique/pdfs/2017/1/10.pdf},
arxiv = {http://arxiv.org/abs/1607.06340},
gsid = {1719921037292242167}
}
@article{Suciu:mpps13,
author = {Macinic, Anca Daniela and Papadima, Stefan and Popescu, Clement Radu and Suciu, Alexander I.},
title = {Flat connections and resonance varieties: from rank one to higher ranks},
JOURNAL = {Transactions of the American Mathematical Society},
FJOURNAL = {Trans. Amer. Math. Soc.},
VOLUME = {369},
YEAR = {2017},
NUMBER = {2},
month = {feb},
PAGES = {1309-1343},
MRCLASS = {55N25, 55P62 (14F35, 20F36, 20J05)},
MRNUMBER = {3572275},
MRREVIEWER = {Dmitry Kerner},
ZBLNUMBER = {1352.55004},
ZBREVIEWER = {Cenap {\"{O}}zel},
keywords = {Resonance variety, characteristic variety, differential graded algebra,
{L}ie algebra, flat connection, quasi-projective manifold, {A}rtin group.},
abstract = {Given a finitely-generated group $\pi$ and a linear algebraic group $G$,
the representation variety ${\rm Hom}(\pi,G)$ has a natural filtration by the
characteristic varieties associated to a rational representation of $G$.
Its algebraic counterpart, the space of $\mathfrak{g}$-valued flat connections
on a commutative, differential graded algebra $(A,d)$ admits a filtration by the
resonance varieties associated to a representation of $\mathfrak{g}$. We establish
here a number of results concerning the structure and qualitative properties
of these embedded resonance varieties, with particular attention to the case when the
rank $1$ resonance variety decomposes as a finite union of linear subspaces.
The general theory is illustrated in detail in the case when $\pi$ is either an
{A}rtin group, or the fundamental group of a smooth, quasi-projective variety.},
doi = {10.1090/tran/6799},
url = {http://www.ams.org/journals/tran/2017-369-02/S0002-9947-2016-06799-9},
arxiv = {http://arxiv.org/abs/1312.1439},
gsid = {16628903419647233650}
}
@incollection {Suciu:su16,
author = {Suciu, Alexander I.},
title = {Around the tangent cone theorem},
SERIES = {Springer INdAM Series},
FSERIES = {Springer INdAM Series},
BOOKTITLE = {Configurations Spaces: Geometry, Topology and Representation Theory},
VOLUME = {14},
PAGES = {1-39},
PUBLISHER = {Springer},
ADDRESS = {Cham},
YEAR = {2016},
month = {aug},
ISBN = {978-3-319-31580-5},
EDITOR = {{F}ilippo {C}allegaro and {F}rederick {C}ohen and {C}orrado {D}e {C}oncini
and {E}va {M}aria {F}eichtner and {G}iovanni {G}aiffi and {M}ario {S}alvetti},
MRCLASS = {55N25 (14F35, 14M12, 32S22, 55P62, 55R80)},
MRNUMBER = {3615726},
MRREVIEWER = {Zach Teitler},
ZBLNUMBER = {06851477},
ZBREVIEWER = {},
keywords = {Algebraic model, cohomology ring, formality, resonance variety, characteristic variety, tangent cone,
quasi-projective variety, configuration space, hyperplane arrangement, {M}ilnor fiber, elliptic arrangement.},
abstract = {A cornerstone of the theory of cohomology jump loci is the {T}angent {C}one theorem, which
relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this
theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by
fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples
from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their
{M}ilnor fibers, configuration spaces, and elliptic arrangements.},
doi = {dx.doi.org/10.1007/978-3-319-31580-5_1},
url = {http://link.springer.com/chapter/10.1007/978-3-319-31580-5_1},
arxiv = {http://arxiv.org/abs/1502.02279},
gsid = {2555895721687753874}
}
@article {Suciu:dsy16,
author = {Denham, Graham and Suciu, Alexander I. and Yuzvinsky, Sergey},
title = {Combinatorial covers and vanishing of cohomology},
JOURNAL = {Selecta Mathematica},
FJOURNAL = {Selecta Math. (N.S.)},
VOLUME = {22},
YEAR = {2016},
month = {march},
NUMBER = {2},
PAGES = {561-594},
MRCLASS = {55T99 (14F05, 16E65, 20J05, 32S22, 55N25)},
MRNUMBER = {3477330},
MRREVIEWER = {},
ZBLNUMBER = {1407.55008},
ZBREVIEWER = {Michael J. Falk},
keywords = {Combinatorial cover, cohomology with local coefficients,
spectral sequence, hyperplane arrangement, elliptic arrangement, toric
complex, {C}ohen--{M}acaulay property.},
abstract = {We use a {M}ayer--{V}ietoris-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, {C}ohen--{M}acaulay-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.},
doi = {10.1007/s00029-015-0196-8},
url = {http://link.springer.com/article/10.1007/s00029-015-0196-8},
arxiv = {http://arxiv.org/abs/1411.7981},
gsid = {16577660483364198333}
}
@incollection {Suciu:ds15,
author = {Denham, Graham and Suciu, Alexander I.},
title = {Torsion in the homology of {M}ilnor fibers of hyperplane arrangements},
SERIES = {Springer INdAM Series},
FSERIES = {Springer INdAM Series},
BOOKTITLE = {Combinatorial Methods in Topology and Algebra},
VOLUME = {12},
PAGES = {31-36},
PUBLISHER = {Springer},
ADDRESS = {},
YEAR = {2015},
month = {oct},
ISBN = {978-3-319-20154-2},
EDITOR = {Bruno Benedetti and Emanuele Delucchi and Luca Moci},
MRCLASS = {32S55,55N25 (32S22, 57M10, 18D50)},
MRNUMBER = {3467321},
MRREVIEWER = {Kristopher Williams},
ZBLNUMBER = {1339.32013},
ZBREVIEWER = {Piotr Pokora},
keywords = {Hyperplane arrangement, {M}ilnor fibration, characteristic variety, orbifold fibration,
cyclic cover, multiarrangement, multinet, deletion, parallel connection operad, polarization},
abstract = {As is well-known, the homology groups of the complement of a complex hyperplane
arrangement are torsion-free. Nevertheless, as we showed in a recent paper, the homology groups
of the {M}ilnor fiber of such an arrangement can have non-trivial integer torsion. We give here a brief
account of the techniques that go into proving this result, outline some of its applications, and indicate
some further questions that it brings to light.},
doi = {10.1007/978-3-319-20155-9_7},
url = {http://link.springer.com/chapter/10.1007/978-3-319-20155-9_7},
arxiv = {http://arxiv.org/abs/1510.01400},
gsid = {12646041448414403421}
}
@article {Suciu:dps15,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexandru},
title = {Algebraic models, {A}lexander-type invariants, and {G}reen--{L}azarsfeld sets},
JOURNAL = {Bull. Math. Soc. Sci. Math. Roumanie (N.S.)},
FJOURNAL = {Bulletin Math\'ematique de la Soci\'et\'e des Sciences
Math\'ematiques de Roumanie. Nouvelle S\'erie},
VOLUME = {58},
YEAR = {2015},
NUMBER = {3},
PAGES = {257-269},
MRCLASS = {14M12, 55N25 (14F35, 20J05)},
MRNUMBER = {3410255},
MRREVIEWER = {},
ZBLNUMBER = {1389.14020},
ZBREVIEWER = {Juan C. Migliore},
keywords = {Resonance varieties, characteristic varieties, {A}lexander invariants,
completion, {G}ysin models, intersection form.},
abstract = {We relate the geometry of the resonance varieties associated to a commutative
differential graded algebra model of a space to the finiteness properties of the completions
of its {A}lexander-type invariants. We also describe in simple algebraic terms the non-translated
components of the degree-one characteristic varieties for a class of non-proper complex manifolds.},
jstor = {https://www.jstor.org/stable/43678822},
url = {https://ssmr.ro/bulletin/pdf/58-3/articol_4.pdf},
arxiv = {http://arxiv.org/abs/1401.0868},
gsid = {5936112017394152251}
}
@article {Suciu:syz15,
author = {Suciu, Alexander I. and Yang, Yaping and Zhao, Gufang},
title = {Homological finiteness of abelian covers},
JOURNAL = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
FJOURNAL = {Annali della Scuola Normale Superiore di Pisa},
YEAR = {2015},
month = {march},
VOLUME = {14},
Number = {1},
Pages = {101-153},
MRCLASS = {14F35, 55N25 (20J05, 57M07)},
MRNUMBER = {3379489},
MRREVIEWER = {},
ZBLNUMBER = {1349.55003},
ZBREVIEWER = {},
keywords = {Abelian cover, characteristic variety, {D}wyer--{F}ried set, {G}rassmannian},
abstract = {We present a method for deciding when a regular abelian cover
of a finite {CW}-complex has finite {B}etti numbers. To start with, we
describe a natural parameter space for all regular covers of a
finite {CW}-complex $X$, with group of deck transformations a fixed
abelian group $A$, which in the case of free abelian covers
of rank $r$ coincides with the {G}rassmanian of $r$-planes in
$H^1(X,\Q)$. Inside this parameter space, there is a subset
$\Omega_A^i(X)$ consisting of all the covers with finite Betti
numbers up to degree $i$.
Building on work of {D}wyer and {F}ried, we show how to
compute these sets in terms of the jump loci for homology
with coefficients in rank~$1$ local systems on $X$. For
certain spaces, such as smooth, quasi-projective varieties,
the generalized {D}wyer--{F}ried invariants that we introduce
here can be computed in terms of intersections of
algebraic subtori in the character group. For many
spaces of interest, the homological finiteness of
abelian covers can be tested through the corresponding
free abelian covers. Yet in general, abelian covers exhibit
different homological finiteness properties than their free
abelian counterparts.},
doi = {10.2422/2036-2145.201205_008},
url = {http://annaliscienze.sns.it/index.php?page=Article&id=335},
arxiv = {http://arxiv.org/abs/1204.4873},
gsid = {4301353495952639806}
}
@article {Suciu:ps14b,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Non-abelian resonance: product and coproduct formulas},
SERIES = {Springer Proceedings in Mathematics and Statistics},
FSERIES = {Springer Proc. Math. Stat.},
BOOKTITLE = {Bridging Algebra, Geometry, and Topology},
VOLUME = {96},
PAGES = {269--280},
PUBLISHER = {Springer},
ADDRESS = {Cham},
YEAR = {2014},
month = {sept},
ISBN = {978-3-319-09186-0},
EDITOR = {{D}enis {I}badula, {W}im {V}eys},
MRCLASS = {55N25 (18A30)},
MRNUMBER = {3297121},
MRREVIEWER = {Jason Stuart Hanson},
ZBLNUMBER = {06515936},
ZBREVIEWER = {},
keywords = {Resonance variety, differential graded algebra, Lie algebra, product, coproduct.},
abstract = {We investigate the resonance varieties attached to a commutative differential graded algebra and to a
representation of a {L}ie algebra, with emphasis on how these varieties behave under finite products and coproducts.},
doi = {10.1007/978-3-319-09186-0_17},
url = {http://link.springer.com/chapter/10.1007/978-3-319-09186-0_17},
arxiv = {http://arxiv.org/abs/1312.1828},
gsid = {6289342802289422377}
}
@article {Suciu:ps14a,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Jump loci in the equivariant spectral sequence},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {21},
YEAR = {2014},
month = {oct},
Number = {4},
Pages = {863-883},
MRCLASS = {55N25 (14M12, 20J05, 55T99)},
MRNUMBER = {3275650},
MRREVIEWER = {Jorge A. Vargas},
ZBLNUMBER = {1314.55002},
ZBREVIEWER = {Andrzej Weber},
keywords = {Affine algebra, maximal spectrum, homology jump loci, support varieties,
equivariant spectral sequence, resonance variety, characteristic variety,
{A}lexander invariants, completion.},
abstract = {We study the homology jump loci of a chain complex over an affine $\mathbb{k}$-algebra.
When the chain complex is the first page of the equivariant spectral sequence associated to a regular
abelian cover of a finite-type CW-complex, we relate those jump loci to the resonance varieties
associated to the cohomology ring of the space. As an application, we show that vanishing resonance
implies a certain finiteness property for the completed {A}lexander invariants of the space.
We also show that, generically, a connected, finite-dimensional commutative graded algebra
has vanishing resonance.},
doi = {10.4310/MRL.2014.v21.n4.a13},
url = {http://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0021/0004/a013/},
arxiv = {http://arxiv.org/abs/1302.4075},
gsid = {15981737636076869149}
}
@article{Suciu:ais14b,
author = {Suciu, Alexander I.},
title = {Hyperplane arrangements and {M}ilnor fibrations},
JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
FJOURNAL = {Annales de la Facult\'{e} des Sciences de Toulouse. Math\'{e}matiques},
VOLUME = {23},
YEAR = {2014},
month = {apr},
Number = {2},
Pages = {417-481},
MRCLASS = {32S55, 57M10, (05B35, 14F35, 32S22, 55N25)},
MRNUMBER = {3205599},
MRREVIEWER = {Michael J. Falk},
ZBLNUMBER = {06297898},
ZBREVIEWER = {Jan Stevens},
keywords = {Hyperplane arrangement, {M}ilnor fibration,
boundary manifold, graph manifold, characteristic variety,
resonance variety, multinet, {A}lexander polynomial, formality.},
abstract = {There are several topological spaces associated to a complex
hyperplane arrangement: the complement and its boundary manifold, as well
as the {M}ilnor fiber and its own boundary. All these spaces are related in
various ways, primarily by a set of interlocking fibrations. We use cohomology
with coefficients in rank $1$ local systems on the complement of the arrangement
to gain information on the homology of the other three spaces, and on the
monodromy operators of the various fibrations.},
doi = {10.5802/afst.1412},
url = {http://afst.cedram.org/afst-bin/item?id=AFST_2014_6_23_2_417_0},
arxiv = {http://arxiv.org/abs/1301.4851},
gsid = {16320965327284366316}
}
@article {Suciu:fs14,
author = {Friedl, Stefan and Suciu, Alexander I.},
title = {K{\"{a}}hler groups, quasi-projective groups, and 3-manifold groups},
JOURNAL = {J. Lond. Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society},
VOLUME = {89},
YEAR = {2014},
month = {feb},
Number = {1},
Pages = {151-168},
MRCLASS = {20F34, 32J27, 57N10, (14F35, 55N25, 57M25)},
MRNUMBER = {3174738},
MRREVIEWER = {Pierre Py},
ZBLNUMBER = {1354.20023},
ZBREVIEWER = {},
keywords = {$3$-manifold, graph manifold, {K}{\"{a}}hler manifold, quasi-projective variety,
fundamental group, {A}lexander polynomial, characteristic varieties, {T}hurston norm.},
abstract = {We prove two results relating $3$-manifold groups to fundamental groups occurring in
complex geometry. Let $N$ be a compact, connected, orientable $3$-manifold. If $N$ has non-empty,
toroidal boundary, and $\pi_1(N)$ is a {K}{\"{a}}hler group, then $N$ is the product of a torus with an interval.
On the other hand, if $N$ has either empty or toroidal boundary, and $\pi_1(N)$ is a quasi-projective
group, then all the prime components of $N$ are graph manifolds.},
doi = {10.1112/jlms/jdt051},
url = {http://jlms.oxfordjournals.org/content/89/1/151},
arxiv = {http://arxiv.org/abs/1212.3022},
gsid = {3220684310602343222}
}
@article {Suciu:ds14,
author = {Denham, Graham and Suciu, Alexander I.},
title = {Multinets, parallel connections, and Milnor fibrations of arrangements},
JOURNAL = {Proc. Lond. Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society},
VOLUME = {108},
YEAR = {2014},
month = {june},
Number = {6},
Pages = {1435-1470},
MRCLASS = {32S55,55N25 (32S22, 57M10, 18D50)},
MRNUMBER = {3218315},
MRREVIEWER = {Guangfeng Jiang},
ZBLNUMBER = {06322149},
ZBREVIEWER = {},
keywords = {Hyperplane arrangement, {M}ilnor fibration, characteristic variety, orbifold fibration,
cyclic cover, multiarrangement, multinet, deletion, parallel connection operad, polarization},
abstract = {The characteristic varieties of a space are the jump loci for homology of
rank $1$ local systems. The way in which the geometry of these varieties may vary with the
characteristic of the ground field is reflected in the homology of finite cyclic covers.
We exploit this phenomenon to detect torsion in the homology of {M}ilnor fibers of projective
hypersurfaces. One tool we use is the interpretation of the degree $1$ characteristic varieties
of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the
corresponding matroid. Another tool is a polarization construction, based on the parallel
connection operad for matroids. Our main result gives a combinatorial machine for producing
arrangements whose {M}ilnor fibers have torsion in homology.},
doi = {10.1112/plms/pdt058},
url = {http://plms.oxfordjournals.org/content/108/6/1435},
arxiv = {http://arxiv.org/abs/1209.3414},
gsid = {9278734647758585811},
}
@article {Suciu:ps15b,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Vanishing resonance and representations of {L}ie algebras},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f{\"u}r die Reine und Angewandte Mathematik [Crelle's Journal]},
VOLUME = {706},
YEAR = {2015},
month = {sept},
NUMBER = {},
PAGES = {83–101},
MRCLASS = {17B10, 20J05 (20E36, 57M07)},
MRNUMBER = {3393364},
MRREVIEWER = {},
ZBLNUMBER = {06490573},
ZBREVIEWER = {},
keywords = {{K}oszul module, resonance variety, root system, weights,
{A}lexander invariant, {T}orelli group},
abstract = {We explore a relationship between the classical representation
theory of a complex, semisimple Lie algebra $\mathfrak{g}$ and the
resonance varieties $\mathcal{R}(V,K)\subset V^*$ attached to irreducible
$\mathfrak{g}$-modules $V$ and submodules $K\subset V\wedge V$.
In the process, we give a precise roots-and-weights criterion insuring
the vanishing of these varieties, or, equivalently, the finiteness
of certain modules $\mathcal{W}(V,K)$ over the symmetric algebra
on $V$. In the case when $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$,
our approach sheds new light on the modules studied by {W}eyman
and {E}isenbud in the context of {G}reen's conjecture on free resolutions
of canonical curves. In the case when $\mathfrak{g}=\mathfrak{sl}_n(\mathbb{C})$
or $\mathfrak{sp}_{2g}(\mathbb{C})$, our approach yields a unified proof
of two vanishing results for the resonance varieties of the (outer) {T}orelli
groups of surface groups, results which arose in recent work by {D}imca,
{H}ain, and the authors on homological finiteness in the {J}ohnson filtration
of mapping class groups and automorphism groups of free groups.},
doi = {10.1515/crelle-2013-0073},
url = {http://www.degruyter.com/view/j/crelle.2015.2015.issue-706/crelle-2013-0073/crelle-2013-0073.xml},
arxiv = {http://arxiv.org/abs/1207.2038},
gsid = {7778481324580566017}
}
@article {Suciu:ais14a,
author = {Suciu, Alexander I.},
title = {Characteristic varieties and {B}etti numbers of free abelian covers},
JOURNAL = {Int. Math. Res. Not. IMRN},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2014},
month = {feb},
VOLUME = {2014},
Number = {4},
Pages = {1063-1124},
MRCLASS = {14F35, 55N25 (20J05, 32S22, 57M07, 57M12)},
MRNUMBER = {3168402},
MRREVIEWER = {Nansen Petrosyan},
ZBLNUMBER = {1352.57004},
ZBREVIEWER = {Alexandru Dimca},
keywords = {Free abelian cover, characteristic variety, exponential tangent cone,
{D}wyer--{F}ried set, special {S}chubert variety, translated subtorus,
{K}{\"{a}}hler manifold, quasi-{K}{\"{a}}hler manifold, hyperplane arrangement,
property ${\rm FP}_n$.},
abstract = {The regular $\Z^r$-covers of a finite cell complex $X$ are parameterized
by the {G}rassmannian of $r$-planes in $H^1(X,\mathbb{Q})$. Moving about this variety,
and recording when the {B}etti numbers $b_1,\dots, b_i$ of the corresponding covers are
finite carves out certain subsets $\Omega^i_r(X)$ of the {G}rassmannian.
We present here a method, essentially going back to {D}wyer and {F}ried, for computing these
sets in terms of the jump loci for homology with coefficients in rank~$1$ local systems
on $X$. Using the exponential tangent cones to these jump loci, we show that each
$\Omega$-invariant is contained in the complement of a union of Schubert varieties
associated to an arrangement of linear subspaces in $H^1(X,\mathbb{Q})$.
The theory can be made very explicit in the case when the characteristic varieties of
$X$ are unions of translated tori. But even in this setting, the $\Omega$-invariants
are not necessarily open, not even when $X$ is a smooth complex projective variety.
As an application, we discuss the geometric finiteness properties of some classes
of groups.},
doi = {10.1093/imrn/rns246},
url = {http://imrn.oxfordjournals.org/content/2014/4/1063},
arxiv = {http://arxiv.org/abs/1111.5803},
gsid = {4037310650460849423}
}
@article {Suciu:ps12,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Homological finiteness in the {J}ohnson filtration of
the automorphism group of a free group},
JOURNAL = {J. Topol.},
FJOURNAL = {Journal of Topology},
VOLUME = {5},
YEAR = {2012},
NUMBER = {4},
PAGES = {909-944},
MRCLASS = {20E36, 20J05 (20F14, 20G05, 55N25)},
MRNUMBER = {3001315},
MRREVIEWER = {Thomas Koberda},
ZBLNUMBER = {1268.20037},
ZBREVIEWER = {Stratos Prassidis},
keywords = {Automorphism group of free group, {T}orelli group, {J}ohnson filtration,
{J}ohnson homomorphism, resonance variety, characteristic variety, {A}lexander invariant},
abstract = {We examine the {J}ohnson filtration of the (outer) automorphism group of a finitely
generated group. In the case of a free group, we find a surprising result: the first {B}etti
number of the second subgroup in the {J}ohnson filtration is finite. Moreover, the corresponding
{A}lexander invariant is a non-trivial module over the Laurent polynomial ring. In the process,
we show that the first resonance variety of the outer {T}orelli group of a free group
is trivial. We also establish a general relationship between the {A}lexander invariant
and its infinitesimal counterpart.},
doi = {10.1112/jtopol/jts023},
url = {http://jtopol.oxfordjournals.org/content/early/2012/09/19/jtopol.jts023.abstract},
arxiv = {http://arxiv.org/abs/1011.5292},
gsid = {759587536925193451}
}
@article{Suciu:syz13,
author = {Suciu, Alexander I. and Yang, Yaping and Zhao, Gufang},
title = {Intersections of translated algebraic subtori},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {217},
YEAR = {2013},
NUMBER = {3},
PAGES = {481-494},
MRCLASS = {20G20 (18B35, 20E15, 55N25)},
MRNUMBER = {2974227},
MRREVIEWER = {Marius Tarnauceanu},
ZBLNUMBER = {1278.20062},
ZBREVIEWER = {Leonid N. Vaserstein},
keywords = {Complex algebraic torus, {P}ontrjagin duality,
lattice of subgroups, primitive subgroup, translated algebraic
subgroup, determinant group, characteristic variety,
fibered category},
abstract = {We exploit the classical correspondence
between finitely generated abelian groups and abelian complex
algebraic reductive groups to study the intersection theory of
translated subgroups in an abelian complex algebraic reductive
group, with special emphasis on intersections of (torsion)
translated subtori in an algebraic torus.},
doi = {10.1016/j.jpaa.2012.06.025},
url = {http://www.sciencedirect.com/science/article/pii/S0022404912002186},
arxiv = {http://arxiv.org/abs/1109.1023},
gsid = {10943220494671999175},
}
@incollection {Suciu:ais12b,
author = {Suciu, Alexander I.},
title = {Geometric and homological finiteness in free abelian covers},
SERIES = {Publications of the Scuola Normale Superiore, CRM Series},
FSERIES = {},
BOOKTITLE = {Configuration Spaces: Geometry, Topology and Combinatorics},
VOLUME = {14},
PAGES = {461--501},
PUBLISHER = {Edizioni della Normale},
ADDRESS = {Pisa},
YEAR = {2012},
ISBN = {978-8876424304},
EDITOR = {{A}nders {B}j{\"{o}}rner and {F}red {C}ohen and {C}orrado {D}e Concini
and {C}laudo {P}rocesi and {M}ario {S}alvetti},
MRCLASS = {20J05 (20F36, 32S22, 55N25, 57M07)},
MRNUMBER = {3203652},
MRREVIEWER = {},
ZBLNUMBER = {1273.14109},
ZBREVIEWER = {},
keywords = {{B}ieri--{N}eumann--{S}trebel--{R}enz invariant, free abelian cover,
{D}wyer--{F}ried invariant, characteristic variety, exponential tangent cone,
resonance variety, toric complex, quasi-projective variety, configuration space,
hyperplane arrangement},
abstract = {We describe some of the connections between the
{B}ieri--{N}eumann--{S}trebel--{R}enz invariants, the {D}wyer--{F}ried invariants,
and the cohomology support loci of a space $X$. Under suitable hypotheses, the
geometric and homological finiteness properties of regular, free abelian covers
of $X$ can be expressed in terms of the resonance varieties, extracted from the
cohomology ring of $X$. In general, though, translated components in the
characteristic varieties affect the answer. We illustrate this theory in
the setting of toric complexes, as well as smooth, complex projective
and quasi-projective varieties, with special emphasis on configuration
spaces of {R}iemann surfaces, and complements of hyperplane arrangements.},
doi = {10.1007/978-88-7642-431-1_21},
url = {http://link.springer.com/chapter/10.1007%2F978-88-7642-431-1_21},
arxiv = {http://arxiv.org/abs/1112.0948},
gsid = {8933560789269450749}
}
@incollection {Suciu:ais12a,
author = {Suciu, Alexander I.},
title = {Resonance varieties and {D}wyer-{F}ried invariants},
SERIES = {Adv. Stud. Pure Math.},
FSERIES = {Advanced Studies in Pure Mathematics},
BOOKTITLE = {Arrangements of Hyperplanes---Sapporo 2009},
VOLUME = {62},
PAGES = {359-398},
PUBLISHER = {Kinokuniya},
ADDRESS = {Tokyo},
YEAR = {2012},
ISBN = {9784931469679},
EDITOR = {Hiroaki Terao and Sergey Yuzvinsky},
MRCLASS = {20J05, 55N25 (14F35, 32S22, 55R80, 57M07)},
MRNUMBER = {2933803},
MRREVIEWER = {Graham Denham},
ZBLNUMBER = {1273.14110},
ZBREVIEWER = {Djordje Baralic},
keywords = {Free abelian cover, characteristic variety, resonance variety, tangent cone,
{D}wyer--{F}ried set, special {S}chubert variety, toric complex, {K}{\"{a}}hler manifold,
hyperplane arrangement},
abstract = {The {D}wyer--{F}ried invariants of a finite cell complex $X$ are the subsets
$\Omega^i_r(X)$ of the {G}rassmannian of $r$-planes in $H^1(X,\mathbb{Q})$ which parametrize the
regular $\mathbb{Z}^r$-covers of $X$ having finite {B}etti numbers up to degree $i$. In previous
work, we showed that each $\Omega$-invariant is contained in the complement of a union
of {S}chubert varieties associated to a certain subspace arrangement in $H^1(X,\mathbb{Q})$.
Here, we identify a class of spaces for which this inclusion holds as equality.
For such \enquote{straight} spaces $X$, all the data required to compute the $\Omega$-invariants
can be extracted from the resonance varieties associated to the cohomology ring $H^*(X,\mathbb{Q})$.
In general, though, translated components in the characteristic varieties affect the answer.},
doi = {10.2969/aspm/06210359},
arxiv = {http://arxiv.org/abs/1111.4534},
gsid = {13261373504940347779}
}
@article {Suciu:ds12,
TITLE = {Mini-workshop: {C}ohomology {R}ings and {F}undamental {G}roups
of {H}yperplane {A}rrangements, {W}onderful
{C}ompactifications, and {R}eal {T}oric {V}arieties},
NOTE = {Abstracts from the mini-workshop held September 30--October 6,
2012, Organized by Graham C. Denham and Alexander I. Suciu},
JOURNAL = {Oberwolfach Rep.},
FJOURNAL = {Oberwolfach Reports},
VOLUME = {9},
YEAR = {2012},
NUMBER = {4},
PAGES = {2939--2983},
ISSN = {1660-8933},
MRCLASS = {14M25 (14T05, 52B20, 52C35, 55P62, 57-06)},
MRNUMBER = {3156741},
doi = {10.4171/OWR/2012/49},
URL = {http://dx.doi.org/10.4171/OWR/2012/49}
}
@article {Suciu:mz11,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Quasi-{K}{\"{a}}hler groups, 3-manifold groups, and formality},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {268},
YEAR = {2011},
NUMBER = {1},
PAGES = {169--186},
MRCLASS = {14F35, 20F34, 57N10 (55N25, 55P62)},
MRNUMBER = {2805428},
MRREVIEWER = {Andrzej Kozlowski},
ZBLNUMBER = {1228.14018},
ZBREVIEWER = {Aleksandr G. Aleksandrov},
keywords = {Quasi-{K}{\"{a}}hler manifold, $3$-manifold, cut number, isolated surface
singularity, $1$-formal group, cohomology ring, characteristic variety, resonance variety},
abstract = {In this note, we address the following question: Which $1$-formal groups
occur as fundamental groups of both quasi-{K}{\"{a}}hler manifolds and closed, connected,
orientable $3$-manifolds. We classify all such groups, at the level of {M}alcev completions,
and compute their coranks. Dropping the assumption on realizability by $3$-manifolds,
we show that the corank equals the isotropy index of the cup-product map in degree
one. Finally, we examine the formality properties of smooth affine surfaces and
quasi-homogeneous isolated surface singularities. In the latter case, we describe
explicitly the positive-dimensional components of the first characteristic
variety for the associated singularity link.},
arxiv = {http://arxiv.org/abs/0810.2158},
URL = {http://www.springerlink.com/content/q475773366186q08/},
gsid = {3855685062799657709},
DOI = {10.1007/s00209-010-0664-y}
}
@incollection {Suciu:conm11,
author = {Suciu, Alexander I.},
title = {Fundamental groups, {A}lexander invariants, and cohomology jumping loci},
BOOKTITLE = {Topology of algebraic varieties and singularities},
SERIES = {Contemp. Math.},
FSERIES = {Contemporary Mathematics},
VOLUME = {538},
PAGES = {179--223},
EDITOR = {Jos\'{e} Ignacio Cogolludo-Agust\'{\i}n and Eriko Hironaka},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2011},
ISBN = {978-0821848906},
MRCLASS = {20F34 (20J05, 32S22, 57M12)},
MRNUMBER = {2777821 (2012b:20092)},
MRREVIEWER = {F. E. A. Johnson},
ZBLNUMBER = {1214.14017},
ZBREVIEWER = {},
keywords = {Fundamental group, {A}lexander polynomial, characteristic variety,
resonance variety, abelian cover, formality, {B}ieri--{N}eumann--{S}trebel--{R}enz invariant,
right-angled {A}rtin group, {K}{\"{a}}hler manifold, quasi-{K}{\"{a}}hler manifold, hyperplane
arrangement, {M}ilnor fibration, boundary manifold},
abstract = {We survey the cohomology jumping loci and the {A}lexander-type
invariants associated to a space, or to its fundamental group. Though most
of the material is expository, we provide new examples and applications,
which in turn raise several questions and conjectures.
The jump loci of a space $X$ come in two basic flavors: the characteristic
varieties, or, the support loci for homology with coefficients in rank $1$ local
systems, and the resonance varieties, or, the support loci for the homology of
the cochain complexes arising from multiplication by degree $1$ classes in the
cohomology ring of $X$. The geometry of these varieties is intimately related
to the formality, (quasi-) projectivity, and homological finiteness properties
of $\pi_1(X)$.
We illustrate this approach with various applications to the study of hyperplane
arrangements, Milnor fibrations, $3$-manifolds, and right-angled Artin groups.},
arxiv = {http://arxiv.org/abs/0910.1559},
URL = {http://www.math.neu.edu/~suciu/papers/libsurvey.pdf},
gsid = {11027901707018836620},
DOI = {10.1090/conm/538/10600}
}
@article {Suciu:forum10,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Algebraic monodromy and obstructions to formality},
JOURNAL = {Forum Math.},
FJOURNAL = {Forum Mathematicum},
VOLUME = {22},
YEAR = {2010},
NUMBER = {5},
PAGES = {973-983},
MRCLASS = {20J05, 57M07 (20F34, 55P62)},
MRNUMBER = {2719766 (2011j:57004)},
MRREVIEWER = {Masaki Kameko},
ZBLNUMBER = {1229.57002},
ZBREVIEWER = {Samuel Smith},
keywords = {Fibration, monodromy, formality, cohomology jumping loci, link, singularity},
abstract = {Given a fibration over the circle, we relate the eigenspace
decomposition of the algebraic monodromy, the homological finiteness properties
of the fiber, and the formality properties of the total space. In the process,
we prove a more general result about iterated group extensions. As an application,
we obtain new criteria for formality of spaces, and $1$-formality of groups,
illustrated by bundle constructions and various examples from low-dimensional
topology and singularity theory.},
arxiv = {http://arxiv.org/abs/0901.0105},
URL = {http://www.reference-global.com/doi/abs/10.1515/FORUM.2010.052},
gsid = {9921396940464328718},
DOI = {10.1515/forum.2010.052}
}
@article {Suciu:plms10,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Bieri-{N}eumann-{S}trebel-{R}enz invariants and homology jumping loci},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society},
VOLUME = {100},
YEAR = {2010},
NUMBER = {3},
PAGES = {795-834},
MRCLASS = {20J05 (55N25, 14F35, 20F36, 20F65)},
MRNUMBER = {MR2640291 (2011i:55006)},
MRREVIEWER = {Brita E. A. Nucinkis},
ZBLNUMBER = {05708721},
ZBREVIEWER = {},
keywords = {Characteristic variety, {A}lexander variety, resonance
variety, exponential tangent cone, homology of free abelian covers,
{B}ieri-{N}eumann-{S}trebel-{R}enz invariant, {N}ovikov homology,
valuation, algebraic integer, right-angled {A}rtin group, {A}rtin
kernel, {K}{\"{a}}hler manifold, quasi-{K}{\"{a}}hler manifold},
abstract = {We investigate the relationship between the geometric
{B}ieri-{N}eumann-{S}trebel-{R}enz 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 {A}rtin kernels, as well as {K}{\"{a}}hler and
quasi-{K}{\"{a}}hler manifolds.},
arxiv = {http://arxiv.org/abs/0812.2660},
URL = {http://plms.oxfordjournals.org/cgi/content/abstract/pdp045},
gsid = {14133650059252773982},
DOI = {10.1112/plms/pdp045}
}
@article {Suciu:tams10,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {The spectral sequence of an equivariant chain complex
and homology with local coefficients},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {362},
YEAR = {2010},
NUMBER = {5},
PAGES = {2685-2721},
MRCLASS = {55T99 (20J05, 55N25, 57M05)},
MRNUMBER = {2584616 (2011b:55017)},
MRREVIEWER = {Masaki Kameko},
ZBLNUMBER = {1195.55005},
ZBREVIEWER = {Haruo Minami},
keywords = {Equivariant chain complex, $I$-adic filtration, spectral
sequence, twisted homology, minimal cell complex, {A}omoto complex, {B}etti numbers},
abstract = {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 $\k\pi_1(X)$-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 {G}alois 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 {A}omoto complex,
arising from the cup-product structure of $H^*(X,\k)$,
thereby generalizing a result of {C}ohen and {O}rlik.},
arxiv = {http://arxiv.org/abs/0706.4262},
URL = {http://www.ams.org/tran/2010-362-05/S0002-9947-09-05041-7/},
DOI = {10.1090/S0002-9947-09-05041-7},
gsid = {14206702977556627289}
}
@book {Suciu:alss10,
TITLE = {Arrangements, local systems and singularities},
EDITOR = {El Zein, Fouad and Suciu, Alexander I. and Tosun, Meral and
Uluda{\u{g}}, A. Muhammed and Yuzvinsky, Sergey},
NOTE = {Lecture notes from the CIMPA Summer School held at
Galatasaray University, Istanbul, June 11-22, 2007},
PUBLISHER = {Birk{\"{a}}user},
ADDRESS = {Basel, Boston, Berlin},
SERIES = {Progress in Mathematics},
YEAR = {2010},
PAGES = {i-x and 1-319},
ISBN = {978-3034602082},
MRCLASS = {32-06 (14N20 32S22 52C35)},
MRNUMBER = {3074961},
ZBLNUMBER = {06208938},
DOI = {10.1007/978-3-0346-0209-9},
URL = {http://www.springerlink.com/content/978-3-0346-0208-2},
gsid = {14789552917727087670}
}
@article {Suciu:bmssmr09,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Geometric and algebraic aspects of $1$-formality},
JOURNAL = {Bull. Math. Soc. Sci. Math. Roumanie (N.S.)},
FJOURNAL = {Bulletin Math\'ematique de la Soci\'et\'e des Sciences
Math\'ematiques de Roumanie. Nouvelle S\'erie},
VOLUME = {52 (100)},
YEAR = {2009},
NUMBER = {3},
PAGES = {355-375},
MRCLASS = {55P62 (57M07, 14F35, 20J05, 55N25)},
MRNUMBER = {2554494 (2010k:55018)},
MRREVIEWER = {John F. Oprea},
ZBLNUMBER = {1199.55010 },
ZBREVIEWER = {Corina Mohorianu},
Keywords = {Formality, fundamental group, cohomology jumping loci,
holonomy {L}ie algebra, {B}ieri--{N}eumann--{S}trebel invariant, {M}alcev completion,
lower central series, {K}{\"{a}}hler manifold, quasi-{K}{\"{a}}hler manifold,
{M}ilnor fiber, hyperplane arrangement, {A}rtin group, {B}estvina--{B}rady group,
pencil, fibration, monodromy},
abstract = {Formality is a topological property, defined in terms
of {S}ullivan's model for a space. In the simply-connected
setting, a space is formal if its rational homotopy type is
determined by the rational cohomology ring. In the general
setting, the weaker $1$-formality property allows one to
reconstruct the rational pro-unipotent 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
$1$-formality, and its relations to the cohomology jump loci
and the {B}ieri--{N}eumann--{S}trebel invariant. We also
produce examples of $4$-manifolds $W$ such that, for every
compact {K}{\"{a}}hler manifold $M$, the product $M\times W$
has the rational homotopy type of a {K}{\"{a}}hler manifold,
yet $M\times W$ admits no {K}{\"{a}}hler metric.},
jstor = {https://www.jstor.org/stable/43679144},
url = {https://www.rms.unibuc.ro/bulletin/pdf/52-3/Papadima.pdf},
arxiv = {http://arxiv.org/abs/0903.2307},
gsid = {4615266568370548024}
}
@article {Suciu:duke09,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Topology and geometry of cohomology jump loci},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {148},
YEAR = {2009},
NUMBER = {3},
PAGES = {405-457},
MRCLASS = {14F35 (20F14, 55N25, 14M12, 20F36, 55P62)},
MRNUMBER = {2527322 (2011b:14047)},
MRREVIEWER = {},
ZBLNUMBER = {1222.14035},
ZBREVIEWER = {Keith Johnson},
keywords = {Characteristic variety, resonance variety, $1$-formal group,
holonomy {L}ie algebra, {M}alcev completion, {A}lexander invariant, tangent cone,
smooth quasi-projective variety, arrangement, configuration space, {A}rtin group},
abstract = {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}. {S}erre.
Applications to arrangements, configuration spaces, coproducts of groups, and {A}rtin
groups are given.},
arxiv = {http://arxiv.org/abs/0902.1250},
URL = {http://projecteuclid.org/euclid.dmj/1245350753},
DOI = {10.1215/00127094-2009-030},
gsid = {14935695469801667148}
}
@article {Suciu:jems09,
author = {Dimca, Alexandru and Suciu, Alexander I.},
title = {Which 3-manifold groups are {K}{\"{a}}hler groups?},
JOURNAL = {J. Eur. Math. Soc. (JEMS)},
FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
VOLUME = {11},
YEAR = {2009},
NUMBER = {3},
PAGES = {521-528},
MRCLASS = {20F34 (32J27, 57N10)},
MRNUMBER = {2505439 (2011f:32041)},
MRREVIEWER = {},
ZBLNUMBER = {1217.57011},
ZBREVIEWER = {Qilin Yang},
keywords = {{K}{\"{a}}hler manifold, $3$-manifold, fundamental group,
cohomology ring, resonance variety, isotropic subspace},
abstract = {The question in the title, first raised by {G}oldman and {D}onaldson, was
partially answered by {R}eznikov. We give a complete answer, as follows: if $G$ can
be realized as both the fundamental group of a closed $3$-manifold and of a compact
{K}{\"{a}}hler manifold, then $G$ must be finite, and thus belongs to the well-known
list of finite subgroups of ${\rm O}(4)$, acting freely on $S^3$.},
arxiv = {http://arxiv.org/abs/0709.4350},
URL = {http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=11&iss=3&rank=3},
DOI = {10.4171/JEMS/158},
gsid = {1350185589996748279}
}
@article {Suciu:crelle09,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Non-finiteness properties of fundamental groups of smooth
projective varieties},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f{\"u}r die Reine und Angewandte Mathematik [Crelle's Journal]},
VOLUME = {629},
YEAR = {2009},
NUMBER = {},
PAGES = {89-105},
MRCLASS = {14F35 (57M07, 14H30, 20J05)},
MRNUMBER = {2527414},
MRREVIEWER = {},
ZBLNUMBER = {1170.14017},
ZBREVIEWER = {Roberto Pignatelli},
abstract = {For each integer $n>1$, we construct an irreducible,
smooth, complex projective variety $M$ of dimension $n$, whose fundamental
group has infinitely generated homology in degree $n+1$ and whose universal
cover is a {S}tein manifold, homotopy equivalent to an infinite bouquet of
$n$-dimensional spheres. This non-finiteness phenomenon is also reflected
in the fact that the homotopy group $pi_n(M)$, viewed as a module over
$\matbbb{Z}\pi_1(M)$, is free of infinite rank. As a result, we give a
negative answer to a question of {K}oll{\'a}r on the existence of quasi-projective
classifying spaces (up to commensurability) for the fundamental groups of
smooth projective varieties. To obtain our examples, we develop a complex
analog of a method in geometric group theory due to {B}estvina and {B}rady.},
arxiv = {http://arxiv.org/abs/math.AG/0609456},
URL = {http://www.reference-global.com/doi/abs/10.1515/CRELLE.2009.027},
DOI = {10.1515/crelle.2009.027},
gsid = {15117354820439181771}
}
@article {Suciu:adv09,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Toric complexes and {A}rtin kernels},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {220},
YEAR = {2009},
month = {jan},
NUMBER = {2},
PAGES = {441-477},
MRCLASS = {57M07 (20F36 55N25 55P62)},
MRNUMBER = {2466422 (2010h:57007)},
MRREVIEWER = {Michael J. Falk},
ZBLNUMBER = {1208.57002},
ZBREVIEWER = {Michael J. Falk},
Keywords = {toric complex, right-angled {A}rtin group, {A}rtin kernel,
{B}estvina-{B}rady group, cohomology ring, {S}tanley-{R}eisner ring,
cohomology jumping loci, monodromy action, holonomy {L}ie algebra,
{M}alcev {L}ie algebra, formality},
abstract = {A simplicial complex $L$ on $n$ vertices determines
a subcomplex $T_L$ of the $n$-torus, with fundamental
group the right-angled {A}rtin group $G_{L}$. Given an
epimorphism $\chi\colon G_{L}\to \Z$, let $T_L^{\chi}$
be the corresponding cover, with fundamental group
the {A}rtin kernel $N_{\chi}$. We compute the cohomology
jumping loci of the toric complex $T_L$, as well as the
homology groups of $T_L^{\chi}$ with coefficients in a
field $\k$, viewed as modules over the group algebra $\k\Z$.
We give combinatorial conditions for $H_{\le r}(T_L^{\chi};\k)$
to have trivial $\Z$-action, allowing us to compute
the truncated cohomology ring, $H^{\le r}(T_L^{\chi};\k)$.
We also determine several {L}ie algebras associated to
{A}rtin kernels, under certain triviality assumptions on the
monodromy $\Z$-action, and establish the $1$-formality
of these (not necessarily finitely presentable) groups.},
arxiv = {http://arxiv.org/abs/0801.3626},
URL = {http://www.sciencedirect.com/science/article/pii/S0001870808002685},
DOI = {10.1016/j.aim.2008.09.008},
gsid = {5663151474938366467}
}
@incollection {Suciu:gtm08,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {The boundary manifold of a complex line arrangement},
BOOKTITLE = {Groups, homotopy and configuration spaces},
SERIES = {Geom. Topol. Monogr.},
YEAR = {2008},
month = {},
VOLUME = {13},
PAGES = {105-146},
PUBLISHER = {Geom. Topol. Publ., Coventry},
MRCLASS = {32S22, 57M27},
MRNUMBER = {2508203 (2010c:32051)},
MRREVIEWER = {Henry K. Schenck},
ZBLNUMBER = {1137.32013},
ZBREVIEWER = {},
keywords = {line arrangement, graph manifold, fundamental group, twisted {A}lexander
polynomial, {BNS} invariant, cohomology ring, holonomy {L}ie algebra,
characteristic variety, resonance variety, tangent cone, formality},
abstract = {We study the topology of the boundary manifold of a line arrangement in
$\mathbb{CP}^2$, with emphasis on the fundamental group $G$ and associated invariants.
We determine the {A}lexander polynomial $\Delta(G)$, and more generally, the twisted
{A}lexander polynomial associated to the abelianization of $G$ and an arbitrary complex
representation. We give an explicit description of the unit ball in the {A}lexander norm,
and use it to analyze certain {B}ieri--{N}eumann--{S}trebel invariants of $G$. From the
{A}lexander polynomial, we also obtain a complete description of the first characteristic
variety of $G$. Comparing this with the corresponding resonance variety of the cohomology
ring of $G$ enables us to characterize those arrangements for which the boundary manifold
is formal.},
gsid = {16537190844876752437},
arxiv = {http://arxiv.org/abs/math.GT/0607274},
URL = {http://www.msp.warwick.ac.uk/gtm/2008/13/p005.xhtml},
DOI = {10.2140/gtm.2008.13.105}
}
@article {Suciu:imrn08,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Alexander polynomials: {E}ssential variables and multiplicities},
JOURNAL = {Int. Math. Res. Not. IMRN},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2008},
month = {},
Number = {3},
Pages = {Art. ID rnm119, 36 pp.},
MRCLASS = {32S22 (20F65 55R80 57M27)},
MRNUMBER = {2416998 (2009i:32036)},
MRREVIEWER = {Michael J. Falk},
ZBLNUMBER = {1156.32018},
ZBREVIEWER = {Claus Ernst},
keywords = {characteristic varieties, {A}lexander polynomial, almost principal ideal,
multiplicity, twisted {B}etti number, i-projective group, boundary manifold, {S}eifert link},
abstract = {We explore the codimension-one strata in the degree-one cohomology
jumping loci of a finitely generated group, through the prism of the multivariable
{A}lexander 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
{B}etti ranks of a group, in terms of multiplicities constructed from the {A}lexander
polynomial. For {S}eifert links in homology $3$-spheres, these bounds become equalities,
and our formula shows explicitly how the {A}lexander polynomial determines all the
characteristic varieties.},
gsid = {17336954368824909249},
arxiv = {http://arxiv.org/abs/0706.2499},
URL = {http://imrn.oxfordjournals.org/cgi/content/abstract/2008/rnm119/rnm119},
DOI = {10.1093/imrn/rnm119}
}
@article {Suciu:jag08,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Quasi-{K}{\"{a}}hler {B}estvina-{B}rady groups},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {17},
YEAR = {2008},
month = {},
NUMBER = {1},
PAGES = {185-197},
MRCLASS = {20F65, 14F35},
MRNUMBER = {2357684 (2008i:20052)},
MRREVIEWER = {Eddy Godelle},
ZBLNUMBER = {1176.20037},
ZBREVIEWER = {},
keywords = {fundamental groups, quasi-{K}{\"{a}}hler groups, compact {K}{\"{a}}hler
manifolds, finite simple graphs, right-angled {A}rtin groups, {B}estvina-{B}rady groups,
resonance varieties},
abstract = {A finite simple graph $\Gamma$ determines a right-angled {A}rtin group
$G_{\Gamma}$, with one generator for each vertex $v$, and with one commutator
relation $vw=wv$ for each pair of vertices joined by an edge. The {B}estvina-{B}rady
group $N_{\Gamma}$ is the kernel of the projection $G_{\Gamma} \to \mathbb{Z}$,
which sends each generator $v$ to $1$. We establish precisely which graphs $\Gamma$
give rise to quasi-{K}{\"{a}}ahler (respectively, {K}{\"{a}}ahler) groups $N_{\Gamma}$.
This yields examples of quasi-projective groups which are not commensurable (up to
finite kernels) to the fundamental group of any aspherical, quasi-projective variety.},
arxiv = {http://arxiv.org/abs/math.AG/0603446},
gsid = {16940023116361855160},
URL = {http://www.ams.org/distribution/jag/2008-17-01/S1056-3911-07-00463-8/home.html},
DOI = {10.1112/jlms/jdm045}
}
@article {Suciu:mathann08,
AUTHOR = {Kreck, Matthias and Suciu, Alexander I.},
TITLE = {Free abelian covers, short loops, stable length, and systolic
inequalities},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {340},
YEAR = {2008},
month = {},
NUMBER = {3},
PAGES = {709--729},
% note = {initial preprint together with Katz, Mikhail G.},
MRCLASS = {53C23 (57N65)},
MRNUMBER = {2358001 (2008k:53079)},
MRREVIEWER = {Florent Balacheff},
ZBLNUMBER = {1134.53019},
ZBREVIEWER = {Mircea Craioveanu},
keywords = {},
abstract = {We explore the geometry of the {A}bel-{J}acobi map $f$ from
a closed, orientable Riemannian manifold $X$ to its {J}acobi torus.
Applying {G}romov's filling inequality to the typical fiber of $f$,
we prove an interpolating inequality for two flavors of shortest
length invariants of loops. The procedure works, provided the lift
of the fiber is non-trivial in the homology of the maximal free
abelian cover, $\tilde{X}$, classified by $f$. We show that the
finite-dimensionality of the rational homology of $\tilde{X}$ is
a sufficient condition for the homological non-triviality of the fiber.
When applied to nilmanifolds, our \enquote{fiberwise} inequality typically
gives stronger information than the filling inequality for $X$ itself.
In dimension $3$, we present a sufficient non-vanishing condition in
terms of {M}assey products. This condition holds for certain manifolds
that do not fiber over their {J}acobi torus, such as $0$-framed surgeries
on suitable links. Our systolic inequality applies to surface bundles
over the circle (provided the algebraic monodromy has $1$-dimensional
coinvariants), even though the {M}assey product invariant vanishes for
some of these bundles.},
arxiv = {http://arXiv.org/abs/math.DG/0207143v1},
gsid = {501581416797590928},
URL = {http://www.springerlink.com/content/c2732614k8626687},
DOI = {10.1007/s00208-007-0182-3}
}
@article {Suciu:jlms07,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Algebraic invariants for {B}estvina-{B}rady groups},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society},
VOLUME = {76},
YEAR = {2007},
month = {},
NUMBER = {2},
PAGES = {273-292},
MRCLASS = {20F36 (20F14 57M07 57M27)},
MRNUMBER = {2363416},
MRREVIEWER = {},
ZBLNUMBER = {1176.20037},
ZBREVIEWER = {},
keywords = {Graph, flag complex, right-angled {A}rtin group,
{B}estvina--{B}rady group, lower central series, holonomy {L}ie
algebra, {C}hen {L}ie algebra, resonance variety, {A}lexander invariant},
abstract = {{B}estvina--{B}rady groups arise as kernels of length
homomorphisms from right-angled {A}rtin groups $G_\Gamma$ to the
integers. Under some connectivity assumptions on the flag complex
$\Delta_\Gamma$, we compute several algebraic invariants of such
a group $N_\Gamma$, directly from the underlying graph $\Gamma$.
As an application, we give examples of {B}estvina--{B}rady groups
which are not isomorphic to any {A}rtin group or arrangement group.},
arxiv = {http://arXiv.org/abs/math.GR/0603240},
gsid = {10009662441783203862},
URL = {http://jlms.oxfordjournals.org/cgi/content/abstract/jdm045},
DOI = {10.1112/jlms/jdm045}
}
@article {Suciu:pamq07,
author = {Denham, Graham and Suciu, Alexander I.},
title = {Moment angle complexes, monomial ideals, and {M}assey products},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {3},
YEAR = {2007},
month = {},
NUMBER = {1},
PAGES = {25-60},
MRCLASS = {55S30 (13F55, 16E05, 32Q55, 55P62, 57R19)},
MRNUMBER = {2330154 (2008g:55028)},
MRREVIEWER = {Marc Aubry},
ZBLNUMBER = {1169.13013},
ZBREVIEWER = {{M}artin D. Crossley},
KEYWORDS = {Moment-angle complex, cohomology ring, homotopy {L}ie algebra, {S}tanley-{R}eisner ring,
{E}ilenberg-{M}oore spectral sequence, cellular cochain algebra, formality, {M}assey product, triangulation,
{B}ier sphere, subspace arrangement, complex manifold},
abstract = {Associated to every finite simplicial complex $K$ there is a \enquote{moment-angle} finite {CW}-complex,
${\mathcal{Z}}_{K}$; if $K$ is a triangulation of a sphere, ${\mathcal{Z}}_{K}$ is a smooth, compact manifold.
Building on work of {B}uchstaber, {P}anov, and {B}askakov, we study the cohomology ring, the homotopy groups,
and the triple {M}assey 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.},
arxiv = {http://arxiv.org/abs/math.AT/0512497},
URL = {http://www.intlpress.com/site/pub/pages/journals/items/pamq/content/vols/0003/0001/a002/},
DOI = {10.4310/PAMQ.2007.v3.n1.a2},
gsid = {6679862638053175477}
}
@article {Suciu:adv06,
author = {Cohen, Daniel C. and Suciu, Alexander I.},
title = {Boundary manifolds of projective hypersurfaces},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {206},
YEAR = {2006},
month = {},
NUMBER = {2},
PAGES = {538-566},
MRCLASS = {14J70 (32S22 32S35)},
MRNUMBER = {2263714 (2007j:14064)},
MRREVIEWER = {Henry K. Schenck},
ZBLNUMBER = {1110.14036},
ZBREVIEWER = {Daniel Matei},
keywords = { Complex hypersurface, hyperplane arrangement, boundary manifold, cohomology ring, topological complexity,
resonance variety},
abstract = {We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface.
We show that, under certain {H}odge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially
determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is
completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the
{LS} category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.},
arxiv = {http://arXiv.org/abs/math.AT/0502506},
URL = {http://www.sciencedirect.com/science/article/pii/S0001870805003166},
DOI = {10.1016/j.aim.2005.10.003},
gsid = {10045347529432015336}
}
@article {Suciu:cmh06,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {When does the associated graded {L}ie algebra of an arrangement group decompose?},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {81},
YEAR = {2006},
NUMBER = {4},
PAGES = {859-875},
MRNUMBER = {2271225 (2007h:52028)},
MRREVIEWER = {Ivan V. Arzhantsev},
ZBLNUMBER = {1104.52009},
ZBREVIEWER = {},
keywords = {Hyperplane arrangement, lower central series, associated graded {L}ie algebra, holonomy {L}ie algebra, {C}hen {L}ie algebra},
abstract = {Let $\mathcal{A}$ be a complex hyperplane arrangement, with fundamental group $G$ and holonomy
{L}ie algebra $\mathfrak{H}$. Suppose $\mathfrak{H}_3$ is a free abelian group of minimum possible rank, given the
values the {M}{\"{o}}bius function $\mu\colon \mathcal{L}_2\to \Z$ takes on the rank $2$ flats of $\mathcal{A}$.
Then the associated graded {L}ie algebra of $G$ decomposes (in degrees $\ge 2$) as a direct product of free {L}ie algebras.
In particular, the ranks of the lower central series quotients of the group are given by
$\phi_r(G)=\sum _{X\in \mathcal{L}_2} \phi_r(F_{\mu(X)})$, for $r\ge 2$. We illustrate this new {L}ower {C}entral {S}eries
formula with several families of examples.},
arxiv = {http://arxiv.org/abs/math.CO/0309324},
URL = {http://www.ems-ph.org/journals/show_abstract.php?issn=0010-2571&vol=81&iss=4&rank=7},
DOI = {10.4171/CMH/77},
gsid = {7772275320129497097}
}
@article {Suciu:mich06,
author = {Denham, Graham and Suciu, Alexander I.},
title = {On the homotopy {L}ie algebra of an arrangement},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {Michigan Mathematical Journal},
VOLUME = {54},
YEAR = {2006},
NUMBER = {2},
PAGES = {319--340},
ISSN = {0026-2285},
MRCLASS = {17B70 (16S37 17B55)},
MRNUMBER = {2252762 (2007f:17039)},
MRREVIEWER = {Marc Aubry},
ZBLNUMBER = {1198.17012},
ZBREVIEWER = {},
keywords = {Holonomy and homotopy {L}ie algebras, hyperplane arrangement, {Y}oneda
algebra, {K}oszul algebra, {H}opf algebra, spectral sequence, homotopy groups},
abstract = {Let $A$ be a graded-commutative, connected $\mathbb{k}$-algebra
generated in degree $1$. The homotopy {L}ie algebra $\mathfrak{g}_A$
is defined to be the {L}ie algebra of primitives of the Yoneda algebra,
${\rm Ext}_{A}(\mathbb{k},\mathbb{k})$. Under certain homological
assumptions on $A$ and its quadratic closure, we express $\mathfrak{g}_A$
as a semi-direct product of the well-understood holonomy {L}ie algebra
$\mathfrak{h}_A$ with a certain $\mathfrak{h}_A$-module. This allows
us to compute the homotopy {L}ie algebra associated to the cohomology
ring of the complement of a complex hyperplane arrangement, provided some
combinatorial assumptions are satisfied. As an application, we give
examples of hyperplane arrangements whose complements have the same
Poincar{\'{e}} polynomial, the same fundamental group, and the same
holonomy {L}ie algebra, yet different homotopy {L}ie algebras.},
arxiv = {http://arXiv.org/abs/math.AT/0502417},
URL = {http://projecteuclid.org/euclid.mmj/1156345597},
DOI = {10.1307/mmj/1156345597},
gsid = {280832469447226187}
}
@article {Suciu:mathann06,
author = {Papadima, Stefan and Suciu, Alexander I.},
title = {Algebraic invariants for right-angled {A}rtin groups},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {334},
YEAR = {2006},
NUMBER = {3},
PAGES = {533--555},
ISSN = {0025-5831},
MRCLASS = {20F36 (13F55 55P62 57M07)},
MRNUMBER = {2207874 (2006k:20078)},
MRREVIEWER = {Eddy Godelle},
ZBLNUMBER = {1165.20032},
ZBREVIEWER = {},
Keywords = {Right-angled {A}rtin groups, lower central series, {C}hen groups, resonance varieties,
finite simplicial graphs, hyperplane arrangements},
abstract = {A finite simplicial graph $\Gamma$ determines a right-angled {A}rtin group $G_{\Gamma}$, with generators
corresponding to the vertices of $\Gamma$, and with a relation $vw=wv$ for each pair of adjacent vertices.
We compute the lower central series quotients, the {C}hen quotients, and the (first) resonance variety of
$G_{\Gamma}$, directly from the graph $\Gamma$. },
gsid = {38501648530934345},
arxiv = {http://arXiv.org/abs/math.GR/0412520},
URL = {http://www.springerlink.com/content/fx2681l300513430},
DOI = {10.1007/s00208-005-0704-9}
}
@article {Suciu:tams06,
author = {Schenck, Henry K. and Suciu, Alexander I.},
title = {Resonance, linear syzygies, {C}hen groups, and the
{B}ernstein-{G}elfand-{G}elfand correspondence},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {358},
YEAR = {2006},
NUMBER = {5},
PAGES = {2269--2289},
ISSN = {0002-9947},
MRCLASS = {52C35 (16E05)},
MRNUMBER = {2197444 (2007a:52026)},
MRREVIEWER = {Hiroaki Terao},
ZBLNUMBER = {1153.52008},
ZBREVIEWER = {},
keywords = {Hyperplane arrangement, {C}hen groups, resonance varieties, free resolutions, {BGG} correspondence},
abstract = {If $\mathcal{A}$ is a complex hyperplane arrangement, with complement $X$,
we show that the {C}hen ranks of $G=\pi_1(X)$ are equal to the graded {B}etti numbers of
the linear strand in a minimal, free resolution of the cohomology ring $A=H^*(X,\mathbb{k})$,
viewed as a module over the exterior algebra $E$ on $\mathcal{A}$:
\[
\theta_k(G) = \dim_{\k}{\rm Tor}^E_{k-1}(A,\k)_k, \quad \text{for $k\ge 2$},
\]
where $\mathbb{k}$ is a field of characteristic $0$. The {\it {C}hen ranks conjecture}\/ asserts that,
for $k$ sufficiently large, $\theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$,
where $h_r$ is the number of $r$-dimensional components of the projective resonance variety $\mathcal{R}^{1}(\mathcal{A})$.
Our earlier work on the resolution of $A$ over $E$ and the above equality yield a proof of the conjecture for graphic
arrangements. Using results on the geometry of $\mathcal{R}^{1}(\mathcal{A})$ and a
localization argument, we establish the inequality
\[
\theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, \quad \text{for $k\gg 0$},
\]
for arbitrary $\mathcal{A}$. Finally, we show that there is a polynomial $P(t)$ of degree equal to the dimension of
$\mathcal{R}^1(\mathcal{A})$, such that $\theta_k(G) = P(k)$, for all $k\gg 0$.},
gsid = {6690197364550213494},
arxiv = {http://arXiv.org/abs/math.AC/0502438},
URL = {http://www.ams.org/tran/2006-358-05/S0002-9947-05-03853-5},
DOI = {10.1090/S0002-9947-05-03853-5}
}
@unpublished {Suciu:dps05,
author = {Dimca, Alexandru and Papadima, Stefan and Suciu, Alexander I.},
title = {Formality, {A}lexander invariants, and a question of {S}erre},
note = {preprint, December 2005 (updated December 2007)},
gsid = {9270464303245803990},
arxiv = {http://arxiv.org/abs/math.AT/0512480},
URL = {https://www.researchgate.net/publication/2124633_Formality_Alexander_invariants_and_a_question_of_Serre}
}
@incollection {Suciu:emissary05,
AUTHOR = {Falk, Michael J. and Suciu, Alexander I.},
TITLE = {Complex hyperplane arrangements},
BOOKTITLE = {Emissary (MSRI Newsletter)},
VOLUME = {Spring},
PAGES = {4--6},
PUBLISHER = {Mathematical Sciences Research Institute},
ADDRESS = {Berkeley, CA},
YEAR = {2005},
keywords = {hyperplane arrangement, intersection lattice, fundamental group, resonance variety},
abstract = {In its simplest manifestation, an arrangement is merely a finite collection of lines in the real plane. The complement
of the lines consists of a finite number of polygonal regions. Determining the number of regions turns out to be a purely
combinatorial problem: one can easily find a recursion for the number of regions, whose solution is given by a formula
involving only the number of lines and the numbers of lines through each intersection point. This formula generalizes to
collections of hyperplanes in $\mathbb{R}^{\ell}$, where the recursive formula is satisfied by an evaluation of the
characteristic polynomial of the (reverse-ordered) poset of intersections. The study of characteristic polynomials forms
the backbone of the combinatorial, and much of the algebraic theory of arrangements, which were featured in the {MSRI}
workshop \enquote{Combinatorial Aspects of Hyperplane Arrangements} last November.
From the topological standpoint, a richer situation is presented by arrangements of complex hyperplanes,
that is, finite collections of hyperplanes in $\mathbb{C}^{\ell}$. In this case, the complement is connected,
and its topology, as reflected in the fundamental group or the cohomology ring for instance, is much more interesting. },
ARXIV = {http://arXiv.org/abs/math.AG/0505166},
URL = {http://www.msri.org/ext/Emissary/EmissarySpring05.pdf}
}
@article {Suciu:jalg05,
AUTHOR = {Matei, Daniel and Suciu, Alexander I.},
TITLE = {Counting homomorphisms onto finite solvable groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {286},
YEAR = {2005},
month = {apr},
NUMBER = {1},
PAGES = {161--186},
ISSN = {0021-8693},
MRCLASS = {20D10},
MRNUMBER = {2124813 (2006b:20034)},
MRREVIEWER = {Alexander Lubotsky},
ZBLNUMBER = {1116.20026},
ZBREVIEWER = {Andrea Lucchini},
keywords = {Solvable quotients, chief series, {G}asch{\"{u}}tz formula,
group cohomology, finite-index subgroups, {B}aumslag--{S}olitar groups,
parafree groups, braid groups},
abstract = {We present a method for computing the number of epimorphisms
from a finitely presented group $G$ to a finite solvable group $\G$,
which generalizes a formula of {G}asch{\"{u}}tz.
Key to this approach are the degree $1$ and $2$ cohomology
groups of $G$, with certain twisted coefficients. As an application,
we count low-index subgroups of $G$. We also investigate the finite
solvable quotients of the {B}aumslag--{S}olitar groups, the {B}aumslag
parafree groups, and the {A}rtin braid groups.},
gsid = {8187397391144572071},
ARXIV = {http://arXiv.org/abs/math.GR/0405122},
URL = {http://www.sciencedirect.com/science/article/pii/S0021869305000438},
DOI = {10.1016/j.jalgebra.2005.01.009}
}
@article {Suciu:gt04,
AUTHOR = {Papadima, Stefan and Suciu, Alexander I.},
TITLE = {Homotopy {L}ie algebras, lower central series and the {K}oszul property},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry and Topology},
VOLUME = {8},
YEAR = {2004},
PAGES = {1079--1125 (electronic)},
ISSN = {1465-3060},
MRCLASS = {55Q15 (16S37 20F14 57M25 57Q45)},
MRNUMBER = {2087079 (2005g:55022)},
MRREVIEWER = {Marc Aubry},
ZBLNUMBER = {1127.55004},
ZBREVIEWER = {},
keywords = {Homotopy groups, {W}hitehead product, rescaling, {K}oszul algebra, lower central series,
{Q}uillen functors, {M}ilnor--{M}oore group, {M}alcev completion, formal, coformal, subspace arrangement, spherical link},
abstract = {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 $H^*(X,\mathbb{Q})$ is a {K}oszul algebra. Then, the homotopy {L}ie algebra $\pi_*(\Omega Y)\otimes \mathbb{Q}$
equals, up to $k$-rescaling, the graded rational {L}ie algebra associated to the lower central series of $\pi_1(X)$.
If $Y$ is a formal space, this equality is actually equivalent to the {K}oszulness of $H^*(X,\mathbb{Q})$.
If $X$ is formal (and only then), the equality lifts to a filtered isomorphism between the {M}alcev completion of
$\pi_1(X)$ and the completion of $[\Omega S^{2k+1},\Omega Y]$. Among spaces that admit naturally defined homological
rescalings are complements of complex hyperplane arrangements, and complements of classical links. The
{R}escaling {F}ormula holds for supersolvable arrangements, as well as for links with connected linking graph.},
gsid = {1177731862088466514},
ARXIV = {http://arXiv.org/abs/math.AT/0110303},
URL = {http://www.msp.warwick.ac.uk/gt/2004/08/p030.xhtml},
DOI = {10.2140/gt.2004.8.1079}
}
@article {Suciu:imrn04,
AUTHOR = {Papadima, Stefan and Suciu, Alexander I.},
TITLE = {Chen {L}ie algebras},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2004},
NUMBER = {21},
PAGES = {1057--1086},
ISSN = {1073-7928},
MRCLASS = {17B70 (17D10 55P62)},
MRNUMBER = {2037049 (2004m:17043)},
MRREVIEWER = {Marc Aubry},
ZBLNUMBER = {1076.17007},
ZBREVIEWER = {Daniel Tanr\'e},
keywords = {Holonomy Lie algebra, Malcev completion, Chen Lie algebra, hyperplane arrangements, links in $S^3$},
abstract = {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 D. 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 W.S. 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 classical link
complements, and fundamental groups of complements of complex hyperplane arrangements. For link groups, we
sharpen {M}assey and {T}raldi's solution of the {M}urasugi conjecture. For arrangement groups, we prove that the
rational {C}hen {L}ie algebra is combinatorially determined.},
ARXIV = {http://arXiv.org/abs/math.GR/0307087},
URL = {http://imrn.oxfordjournals.org/cgi/content/abstract/2004/21/1057},
DOI = {10.1155/S1073792804132017},
gsid = {14025209372546073263}
}
@article {Suciu:agt03,
AUTHOR = {Cohen, Daniel C. and Denham, Graham and Suciu, Alexander I.},
TITLE = {Torsion in {M}ilnor fiber homology},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {3},
YEAR = {2003},
PAGES = {511--535 (electronic)},
ISSN = {1472-2747},
MRCLASS = {32S55 (32S22 55N25)},
MRNUMBER = {1997327 (2004d:32043)},
MRREVIEWER = {Daniel Matei},
ZBLNUMBER = {1030.32022},
ZBREVIEWER = {Theo de Jong},
KEYWORDS = {{M}ilnor fibration, characteristic variety, arrangement},
abstract = {In a recent paper, {D}imca and N{\'{e}}methi pose the problem of finding a homogeneous polynomial
$f$ such that the homology of the complement of the hypersurface defined by $f$ is torsion-free, but the homology
of the {M}ilnor fiber of $f$ has torsion. We prove that this is indeed possible, and show by construction that
for each prime $p$, there is a polynomial with $p$-torsion in the homology of the {M}ilnor fiber. The techniques
make use of properties of characteristic varieties of hyperplane arrangements.},
ARXIV = {http://arXiv.org/abs/math.GT/0302143},
URL = {http://www.msp.warwick.ac.uk/agt/2003/03/p016.xhtml},
DOI = {10.2140/agt.2003.3.511},
gsid = {18170373845516068134}
}
@article {Suciu:cras02,
AUTHOR = {Papadima, Stefan and Suciu, Alexander I.},
TITLE = {Rational homotopy groups and {K}oszul algebras},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris},
VOLUME = {335},
YEAR = {2002},
month = {},
NUMBER = {1},
PAGES = {53--58},
ISSN = {1631-073X},
MRCLASS = {55P62},
MRNUMBER = {1920995 (2003g:55015)},
MRREVIEWER = {Octavian Cornea},
ZBLNUMBER = {1006.55007},
ZBREVIEWER = {Daniel Tanr\'e},
keywords = {homotopy groups, cohomology ring, lower central series, rescaling, {K}oszul algebras},
abstract = {Let $X$ and $Y$ be finite-type CW-spaces ($X$ connected, $Y$
simply connected), such that the ring $H^*(Y,\mathbb{Q})$ is a $k$-rescaling of
$H^*(X,\mathbb{Q})$. If $H^*(X,\mathbb{Q})$ is a Koszul algebra, then the graded
{L}ie algebra $pi_*(\Omega Y) \otimes \mathbb{Q}$ is the $k$-rescaling of
$gr_*(pi_1 X) \otimes \mathbb{Q}$. If $Y$ is a formal space, then the converse
holds, and $Y$ is coformal. Furthermore, if $X$ is formal, with {K}oszul cohomology
algebra, there exist filtered group isomorphisms between the {M}alcev completion of
$pi_1 X$, the completion of $[\Omega S^{2k+1},\Omega Y]$, and the {M}ilnor--{M}oore group
of coalgebra maps from $H_*(\Omega S^{2k+1},\mathbb{Q})$ to $H_*(\Omega Y,\mathbb{Q})$.},
gsid = {14589120776570079053},
URL = {http://www.sciencedirect.com/science/article/pii/S1631073X02024202},
DOI = {10.1016/S1631-073X(02)02420-2}
}
@article {Suciu:tams02,
AUTHOR = {Schenck, Henry K. and Suciu, Alexander I.},
TITLE = {Lower central series and free resolutions of hyperplane arrangements},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {354},
YEAR = {2002},
NUMBER = {9},
PAGES = {3409--3433 (electronic)},
ISSN = {0002-9947},
MRCLASS = {16Exx (52C35)},
MRNUMBER = {1911506 (2003k:52022)},
MRREVIEWER = {Ruth Lawrence},
ZBLNUMBER = {1057.52015},
KEYWORDS = {lower central series; free resolution; hyperplane arrangement;
change of rings spectral sequence; Koszul algebra; linear strand;
graphic arrangement},
abstract = {If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,k)$ is the cohomology ring
of $M$ over a field of characteristic 0, then the ranks, $phi_k$, of the lower central series quotients of
$pi_1(M)$ can be computed from the {B}etti numbers, $b_{ii}=dim_{k} {\rm Tor}^A_i(k,k)_i$, of the linear strand in
a (minimal) free resolution of $k$ over $A$. We use the {C}artan--{E}ilenberg change of rings spectral sequence
to relate these numbers to the graded Betti numbers, $b'_{ij}=dim_{k} {\rm Tor}^E_i(A,k)_j$, of a (minimal) resolution
of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of {F}alk for $phi_3$, and obtain a
new formula for $phi_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a
question of {F}alk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is {K}oszul
iff the arrangement is supersolvable. We also give combinatorial lower bounds on the {B}etti numbers, $b'_{i,i+1}$,
of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it
is attained for all $i ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and
we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do
not attain the lower bound, unless they have no braid sub-arrangements), we show that $b'_{i,i+1}$ is determined
by the number of triangles and $K_4$ subgraphs in the graph.},
ARXIV = {http://arXiv.org/abs/math.AG/0109070},
URL = {http://www.ams.org/journal-getitem?pii=S0002-9947-02-03021-0},
DOI = {10.1090/S0002-9947-02-03021-0},
gsid = {2755621069366262032}
}
@article {Suciu:imrn02,
AUTHOR = {Matei, Daniel and Suciu, Alexander I.},
TITLE = {Hall invariants, homology of subgroups, and characteristic varieties},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2002},
NUMBER = {9},
PAGES = {465--503},
ISSN = {1073-7928},
MRCLASS = {20Fxx (20J05)},
MRNUMBER = {1884468 (2003d:20055)},
MRREVIEWER = {Alexander Lubotsky},
ZBLNUMBER = {1061.20040},
KEYWORDS = {Hall invariants, metabelian groups, characteristic varieties,
cohomology of groups, low-index subgroups, fundamental groups},
abstract = {Given a finitely-generated group $G$, and a finite group
$\Gamma$, Philip Hall defined $\delta_{\Gamma(G)}$ to be the
number of factor groups of $G$ that are isomorphic to $\Gamma$.
We show how to compute the Hall invariants by cohomological and
combinatorial methods, when $G$ is finitely-presented, and $\Gamma$
belongs to a certain class of metabelian groups. Key to this approach
is the stratification of the character variety, ${\rm Hom}(G,\mathbb{K}^*)$,
by the jumping loci of the cohomology of $G$, with coefficients in
rank $1$ local systems over a suitably chosen field $\mathbb{K}$.
Counting relevant torsion points on these ``characteristic'' subvarieties
gives $\delta_{\Gamma(G)}$. In the process, we compute the
distribution of prime-index, normal subgroups $K\triangleleft G$
according to $\dim_{\mathbb{K}} H_1(K;\mathbb{K})$, provided
${\rm char}\, \mathbb{K}\ne |G:K|$. In turn, we use this distribution
to count low-index subgroups of $G$. We illustrate these
techniques in the case when $G$ is the fundamental group of
the complement of an arrangement of either affine lines in
$\mathbb{C}^{2}$, or transverse planes in $\mathbb{R}^4$.},
ARXIV = {http://arXiv.org/abs/math.GR/0010046},
URL = {http://imrn.oxfordjournals.org/cgi/content/abstract/2002/9/465},
DOI = {10.1155/S107379280210907X},
gsid = {11118518697914219836}
}
@article {Suciu:adv02,
AUTHOR = {Papadima, Stefan and Suciu, Alexander I.},
TITLE = {Higher homotopy groups of complements of complex hyperplane arrangements},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {165},
YEAR = {2002},
month = {jan},
NUMBER = {1},
PAGES = {71--100},
ISSN = {0001-8708},
MRCLASS = {55R80 (32S22 52C35)},
MRNUMBER = {1880322 (2003b:55019)},
MRREVIEWER = {Daniel C. Cohen},
ZBLNUMBER = {1019.52016},
ZBREVIEWER = {Michael J. Falk},
KEYWORDS = {hypersolvable arrangement, higher homotopy groups, minimal cell decomposition},
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 $\mathbb{Z}\pi_1$-module presentation of $\pi_p$,
the first non-vanishing higher homotopy group. We also
give a combinatorial formula for the $\pi_1$-coinvariants
of $\pi_p$.
For affine line arrangements whose cones are hypersolvable,
we provide a minimal resolution of $\pi_2$, and study some
of the properties of this module. For graphic arrangements
associated to graphs with no $3$-cycles, we obtain information
on $\pi_2$, directly from the graph. The $\pi_1$-coinvariants
of $\pi_2$ may distinguish the homotopy $2$-types of arrangement
complements with the same $\pi_1$, and the same Betti numbers
in low degrees.},
ARXIV = {http://arXiv.org/abs/math.AT/0002251},
URL = {http://www.sciencedirect.com/science/article/pii/S0001870801920237},
DOI = {10.1006/aima.2001.2023},
gsid = {7967937589913006815}
}
@incollection {Suciu:conm01,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Fundamental groups of line arrangements: Enumerative aspects},
BOOKTITLE = {Advances in algebraic geometry motivated by physics (Lowell,
MA, 2000)},
SERIES = {Contemp. Math.},
FSERIES = {Contemporary Mathematics},
VOLUME = {276},
PAGES = {43--79},
EDITOR = {Emma Previato},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2001},
MRCLASS = {14F35 (32S22 52C35 57M05)},
MRNUMBER = {1837109 (2002k:14029)},
MRREVIEWER = {Nguyen Viet Dung},
ZBLNUMBER = {0998.14012},
KEYWORDS = {complements of line arrangements; fundamental groups; characteristic varieties; resonance varieties; finite covers},
ABSTRACT = {This is a survey of some recent developments in the study of complements of line arrangements
in the complex plane. We investigate the fundamental groups and finite covers of those complements,
focusing on homological and enumerative aspects. The unifying framework for this study is the
stratification of the character variety of the fundamental group, $G$, by the jumping loci for
cohomology with coefficients in rank $1$ local systems. Counting certain torsion points on these
\enquote{characteristic} varieties yields information about the homology of branched and unbranched covers
of the complement, as well as on the number of low-index subgroups of its fundamental group.
We conclude with two conjectures, expressing the lower central series quotients of $G/G''$
(and, in some cases, $G$ itself) in terms of the closely related \enquote{resonance} varieties.
We illustrate the discussion with a number of detailed examples, some of which reveal new phenomena.},
ARXIV = {http://arXiv.org/abs/math.AG/0010105},
DOI = {10.1090/conm/276/04510},
gsid = {10588085294178530201}
}
@article {Suciu:topapp02,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Translated tori in the characteristic varieties of complex
hyperplane arrangements},
NOTE = {Arrangements in Boston: a Conference on Hyperplane
Arrangements (1999)},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {118},
YEAR = {2002},
NUMBER = {1-2},
PAGES = {209--223},
ISSN = {0166-8641},
MRCLASS = {32S22 (52C35)},
MRNUMBER = {1877726 (2002j:32027)},
MRREVIEWER = {Guangfeng Jiang},
ZBLNUMBER = {1021.32009},
ZBREVIEWER = {Margaret M. Bayer},
KEYWORDS = {hyperplane arrangement; characteristic variety; {O}rlik--{S}olomon algebra; translated tori},
abstract = {We give examples of complex hyperplane arrangements $\mathcal{A}$ for which the top
characteristic variety, $V_1(\mathcal{A})$, contains positive-dimensional irreducible components
that do not pass through the origin of the algebraic torus $(\mathbb{C}^*)^{|\mathcal{A}|}$.
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 {O}rlik--{S}olomon 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.},
ARXIV = {http://arXiv.org/abs/math.AG/9912227},
gsid = {2541340059672056701},
URL = {http://www.sciencedirect.com/science/article/pii/S0166864101000529},
DOI = {10.1016/S0166-8641(01)00052-9}
}
@book {Suciu:arrbos02,
TITLE = {Arrangements in {B}oston: a {C}onference on {H}yperplane
{A}rrangements},
EDITOR = {Cohen, Daniel C. and Suciu, Alexander I.},
NOTE = {Papers from the conference held at Northeastern University,
Boston, MA, June 12--15, 1999,
Topology Appl. {\bf 118} (2002), no. 1-2},
PUBLISHER = {North-Holland Publishing Co.},
ADDRESS = {Amsterdam},
VOLUME = {118},
YEAR = {2002},
NUMBER = {1-2},
PAGES = {v--viii and 1--274},
ISSN = {0166-8641},
MRCLASS = {00B25 (20-06 52-06)},
MRNUMBER = {1877711 (2002h:00013)},
ZBLNUMBER = {0983.00038},
abstract = {This Special Issue is devoted to the activities surrounding \enquote{Arrangements in Boston:
A Conference on Hyperplane Arrangements,} which was held at Northeastern University, 12--15 June 1999.},
URL = {http://www.sciencedirect.com/science/journal/01668641/118/1-2},
DOI = {10.1016/S0166-8641(01)00055-4},
gsid = {2628550103319635517}
}
@article {Suciu:gafa01,
AUTHOR = {Katz, Mikhail G. and Suciu, Alexander I.},
TITLE = {Systolic freedom of loop space},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {11},
YEAR = {2001},
NUMBER = {1},
PAGES = {60--73},
ISSN = {1016-443X},
MRCLASS = {53C23 (55M99)},
MRNUMBER = {1829642 (2002c:53067)},
MRREVIEWER = {John F. Oprea},
ZBLNUMBER = {1048.53030},
ZBREVIEWER = {H. Gollek},
KEYWORDS = {systolic freedom; systole; total volume; submanifold; loop space; rational homotopy},
abstract = {Given a pair of integers $m$ and $n$ such that $1 < m < n$, we show that every $n$-dimensional
manifold admits metrics of arbitrarily small total volume, and possessing the following property: every
$m$-dimensional submanifold of less than unit $m$-volume is necessarily torsion in homology.
This result is different from the case of a pair of complementary dimensions, for which a direct geometric
construction works and gives the analogous theorem in complete generality. In the present paper, we use
{S}ullivan's telescope model for the rationalisation of a space to observe systolic freedom.},
ARXIV = {http://arXiv.org/abs/math.DG/0106153},
gsid = {2031395487854113991},
URL = {http://www.springerlink.com/content/8xf1kr4b43ratj08/},
DOI = {10.1007/PL00001672}
}
@incollection {Suciu:aspm00,
AUTHOR = {Matei, Daniel and Suciu, Alexander I.},
TITLE = {Cohomology rings and nilpotent quotients of real and complex
arrangements},
BOOKTITLE = {Arrangements---Tokyo 1998},
SERIES = {Adv. Stud. Pure Math.},
FSERIES = {Advanced Studies in Pure Mathematics},
VOLUME = {27},
PAGES = {185--215},
PUBLISHER = {Kinokuniya},
ADDRESS = {Tokyo},
YEAR = {2000},
ISBN = {978-4314101400},
EDITOR = {Michael Falk and Hiroaki Terao},
MRCLASS = {32S22 (20F34 55R80)},
MRNUMBER = {1796900 (2002b:32045)},
MRREVIEWER = {Nguyen Viet Dung},
ZBLNUMBER = {0974.32020},
KEYWORDS = {cohomology rings; nilpotent quotients; real and complex
hyperplane arrangements; {O}rlik-{S}olomon algebra},
abstract = {For an arrangement with complement $X$ and fundamental
group $G$, we relate the truncated cohomology ring, $H^{\le 2}(X)$,
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 {O}rlik-{S}olomon
algebra mod $p$. As an application, we establish the cohomology
classification of $2$-arrangements of $n\le 6$ planes in $\mathbb{R}^4$.},
DOI = {10.2969/aspm/02710185},
URL = {https://projecteuclid.org/ euclid.aspm/1534788972},
ARXIV = {http://arXiv.org/abs/math.GT/9812087},
gsid = {12810245655501976816}
}
@article {Suciu:camb99,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {Characteristic varieties of arrangements},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {127},
YEAR = {1999},
NUMBER = {1},
PAGES = {33--53},
ISSN = {0305-0041},
MRCLASS = {32S22 (52C35)},
MRNUMBER = {1692519 (2000m:32036)},
MRREVIEWER = {Hiroaki Terao},
ZBLNUMBER = {0963.32018},
KEYWORDS = {arrangement of complex hyperplanes; characteristic subvariety;
Alexander invariants; reflection arrangements; monomial groups},
abstract = {The $k^{\text{th}}$ {F}itting ideal of the {A}lexander invariant $B$
of an arrangement $\mathcal{A}$ of $n$ complex hyperplanes defines a
characteristic subvariety, $V_k(\mathcal{A})$, of the algebraic
torus $\mathbb{T}$. In the combinatorially determined case where $B$
decomposes as a direct sum of local {A}lexander invariants, we
obtain a complete description of $V_k(\mathcal{A})$. For any arrangement
$\mathcal{A}$, we show that the tangent cone at the identity of this variety
coincides with $\RR^{1}_{k}(A)$, one of the cohomology support
loci of the {O}rlik--{S}olomon algebra. Using work of {A}rapura,
we conclude that all irreducible components of $V_{k}(\A)$ which
pass through the identity element of $\mathbb{T}$ are combinatorially
determined, and that $\RR^{1}_{k}(A)$ is the union of a subspace
arrangement in $\mathbb{C}^n$, thereby resolving a conjecture of
{F}alk. We use these results to study the reflection
arrangements associated to monomial groups.},
ARXIV = {http://arXiv.org/abs/math.AG/9801048},
GSID = {7302019678268589445},
URL = {http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=37609},
DOI = {10.1017/S0305004199003576}
}
@incollection {Suciu:conm99,
AUTHOR = {Katz, Mikhail G. and Suciu, Alexander I.},
TITLE = {Volume of {R}iemannian manifolds, geometric inequalities, and
homotopy theory},
BOOKTITLE = {Tel Aviv Topology Conference: Rothenberg Festschrift (1998)},
SERIES = {Contemp. Math.},
VOLUME = {231},
PAGES = {113--136},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
EDITOR = {Michael Farber, Wolfgang L{\"u}ck, Shmuel Weinberger},
MRCLASS = {53C23 (55Q15)},
MRNUMBER = {1705579 (2000i:53063)},
MRREVIEWER = {Andrea Sambusetti},
ZBLNUMBER = {0967.53024},
ZBREVIEWER = {Vladimir Yu. Rovenskij},
KEYWORDS = {volume; stable systole; systolic freedom; coarea inequality;
isoperimetric inequality; surgery; Whitehead product; loop space;
{E}ilenberg-{M}ac{L}ane space; ordinary systole},
abstract = {We outline the current state of knowledge regarding geometric inequalities of systolic type,
and prove new results, including systolic freedom in dimension $4$. Namely, every compact, orientable,
smooth $4$-manifold $X$ admits metrics of arbitrarily small volume such that every orientable, immersed
surface of smaller than unit area is necessarily null-homologous in $X$.},
gsid = {3813152774748486218},
ARXIV = {http://arXiv.org/abs/math.DG/9810172},
URL = {http://www.ams.org/books/conm/231/3357},
DOI= {10.1090/conm/231/03357},
}
@article {Suciu:top00,
AUTHOR = {Matei, Daniel and Suciu, Alexander I.},
TITLE = {Homotopy types of complements of {$2$}-arrangements in
{${\mathbf{R}}\sp 4$}},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {39},
YEAR = {2000},
NUMBER = {1},
PAGES = {61--88},
ISSN = {0040-9383},
MRCLASS = {55R80 (52C35)},
MRNUMBER = {1710992 (2000h:55028)},
MRREVIEWER = {Peter Orlik},
Zblnumber = {0940.55010},
Zbreviewer = {Vincent Moulton},
keywords = {arrangement; line configuration; link; braid; characteristic variety},
abstract = {We study the homotopy types of complements of arrangements of $n$ transverse
planes in $\mathbb{R}^4$, obtaining a complete classification for $n \le 6$, and lower bounds
for the number of homotopy types in general. Furthermore, we show that the homotopy type of
a $2$-arrangement in $\mathbb{R}^4$ is not determined by the cohomology ring, thereby
answering a question of {Z}iegler. 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.},
gsid = {3220690496186230146},
ARXIV = {http://arXiv.org/abs/math.GT/9712251},
URL = {http://www.sciencedirect.com/science/article/pii/S0040938398000585},
DOI = {10.1016/S0040-9383(98)00058-5}
}
@article {Suciu:mrl98,
AUTHOR = {Babenko, Ivan K. and Katz, Mikhail G. and Suciu, Alexander I.},
TITLE = {Volumes, middle-dimensional systoles, and {W}hitehead products},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {5},
YEAR = {1998},
NUMBER = {4},
PAGES = {461--471},
ISSN = {1073-2780},
MRCLASS = {53C23 (53C20 55Q15)},
MRNUMBER = {1653310 (99m:53084)},
MRREVIEWER = {Athanase Papadopoulos},
ZBLNUMBER = {0933.53022},
ZBREVIEWER = {A. P. Stone},
KEYWORDS = {Whitehead product; $k$-systole; systolic freedom},
abstract = {Let $X$ be a closed, orientable, smooth manifold of dimension $2m \ge 6$
with torsion-free middle-dimensional homology. We construct metrics on $X$ of arbitrarily
small volume, such that every orientable, middle-dimensional submanifold of less
than unit volume necessarily bounds. Thus, {L}oewner’s theorem has no higher-dimensional analogue.},
gsid = {4657159027319540013},
ARXIV = {http://arXiv.org/abs/math.DG/9707116},
URL = {http://intlpress.com/site/pub/files/_fulltext/journals/mrl/1998/0005/0004/MRL-1998-0005-0004-a004.pdf},
DOI = {10.4310/MRL.1998.v5.n4.a4}
}
@article {Suciu:tams99,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {Alexander invariants of complex hyperplane arrangements},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {351},
YEAR = {1999},
NUMBER = {10},
PAGES = {4043--4067},
ISSN = {0002-9947},
MRCLASS = {52B30 (20F34 57M05)},
MRNUMBER = {1475679 (99m:52019)},
MRREVIEWER = {Richard Randell},
ZBLNUMBER = {0945.20024},
ZBREVIEWER = {V. V. Chueshev},
KEYWORDS = {Alexander invariants; {C}hen groups; {G}assner representation; fundamental groups;
braid monodromy homomorphisms; pure braid groups; presentations},
abstract = {Let $\mathcal{A}$ be an arrangement of complex hyperplanes. The fundamental group of
the complement of $\mathcal{A}$ is determined by a braid monodromy homomorphism from a finitely
generated free group to the pure braid group. Using the {G}assner representation of the pure braid group,
we find an explicit presentation for the {A}lexander invariant of $\mathcal{A}$. From this presentation,
we obtain combinatorial lower bounds for the ranks of the {C}hen groups of $\mathcal{A}$. We also provide
a combinatorial criterion for when these lower bounds are attained.},
ARXIV = {http://arXiv.org/abs/math.AG/9703030},
gsid = {1966826052746853923},
URL = {http://www.ams.org/tran/1999-351-10/S0002-9947-99-02206-0/},
DOI = {10.1090/S0002-9947-99-02206-0}
}
@article {Suciu:jpaa98,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {Homology of iterated semidirect products of free groups},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {126},
YEAR = {1998},
NUMBER = {1-3},
PAGES = {87--120},
ISSN = {0022-4049},
MRCLASS = {20J05 (20F36 57M05)},
MRNUMBER = {1475679 (99e:20064)},
MRREVIEWER = {Paul Igodt},
ZBLNUMBER = {0908.20033},
ZBREVIEWER = {Michael J. Falk},
KEYWORDS = {Group cohomology; braid groups; fundamental groups; fiber-type
arrangements; {M}ilnor fibers; cohomology vanishing theorems;
{C}oxeter arrangements; {B}urau and {G}assner representations},
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 {M}agnus, {B}urau, and {G}assner. Our construction
also yields algorithms for computing the homology of the {M}ilnor 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.},
ARXIV = {http://arXiv.org/abs/math.AG/9503002},
gsid = {6042016949369869548},
URL = {http://www.sciencedirect.com/science/article/pii/S0022404996001533},
DOI = {10.1016/S0022-4049(96)00153-3}
}
@article {Suciu:cmh97,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {The braid monodromy of plane algebraic curves and
hyperplane arrangements},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {72},
YEAR = {1997},
NUMBER = {2},
PAGES = {285--315},
ISSN = {0010-2571},
MRCLASS = {52B30 (14H30 20F36 57M05)},
MRNUMBER = {1470093 (98f:52012)},
MRREVIEWER = {Lee Rudolph},
ZBLNUMBER = {0959.52018},
ZBREVIEWER = {A. Lipovski},
KEYWORDS = {Braid monodromy; plane algebraic curve; hyperplane arrangement;
fundamental group of the complement of an algebraic curve; polynomial
cover; braid group; wiring diagram; intersection lattice},
abstract = {To a plane algebraic curve of degree $n$, {M}oishezon associated
a braid monodromy homomorphism from a finitely generated free group
to {A}rtin's braid group $B_n$. Using {H}ansen's polynomial covering
space theory, we give a new interpretation of this construction.
Next, we provide an explicit description of the braid monodromy of an
arrangement of complex affine hyperplanes, by means of an associated
\enquote{braided wiring diagram.} The ensuing presentation of the fundamental
group of the complement is shown to be {T}ietze-I equivalent to the
{R}andell-{A}rvola presentation. Work of {L}ibgober then implies
that the complement of a line arrangement is homotopy equivalent to
the $2$-complex modeled on either of these presentations.
Finally, we prove that the braid monodromy of a line arrangement
determines the intersection lattice. Examples of {F}alk then show
that the braid monodromy carries more information than the
group of the complement, thereby answering a question of {L}ibgober.},
ARXIV = {http://arXiv.org/abs/math.AG/9608001},
GSID = {8881203904084384409},
URL = {http://www.springerlink.com/content/h9aamqbq1704rn6d/},
DOI = {10.1007/s000140050017}
}
@article {Suciu:jpaa95,
AUTHOR = {Farjoun, Emmanuel Dror and Jekel, Solomon M. and Suciu,
Alexander I.},
TITLE = {Homology of jet groups},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {102},
YEAR = {1995},
month = {jul},
NUMBER = {1},
PAGES = {17--24},
ISSN = {0022-4049},
MRCLASS = {20J05},
MRNUMBER = {1350206 (97g:20060)},
MRREVIEWER = {},
ZBLNUMBER = {0848.57036},
ZBREVIEWER = {John McCleary},
KEYWORDS = {Jet groups; spectral sequence; group homology},
abstract = {In this paper we compute the second homology of the discrete
jet groups. The $n$-th jet group, $J_n$, is the group, under composition
followed by truncation, of invertible, orientation-preserving real $n$-jets
at $0$. Consider the homomorphism $D\colon J_n \to \mathbb{R}^+$ obtained
by projecting onto the first coefficient. The main result of this paper is:
The map $D_*\colon H_2(J_n) \to H_2(\mathbb{R}^+)$ is an isomorphism.},
gsid = {6259253554684852086},
URL = {http://www.sciencedirect.com/science/article/pii/0022404995000552},
DOI = {10.1016/0022-4049(95)00055-2}
}
@article {Suciu:jlms95,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {On {M}ilnor fibrations of arrangements},
JOURNAL = {J. London Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {51},
YEAR = {1995},
NUMBER = {1},
PAGES = {105--119},
ISSN = {0024-6107},
MRCLASS = {32S55 (52B30)},
MRNUMBER = {1310725 (96e:32034)},
MRREVIEWER = {Richard Randell},
ZBLNUMBER = {0814.32007},
ZBREVIEWER = {Daniel C.Cohen},
KEYWORDS = {{M}ilnor fiber; homogeneous polynomial; hyperplane arrangements;
Betti numbers; algebraic monodromy},
abstract = {We use covering space theory and homology with local coefficients
to study the {M}ilnor fiber of a homogeneous polynomial. These techniques are applied
in the context of hyperplane arrangements, yielding an explicit algorithm for
computing the {B}etti numbers of the {M}ilnor fiber of an arbitrary real central
arrangement in $\mathbb{C}^3$, as well as the dimensions of the eigenspaces of the
algebraic monodromy. We also obtain combinatorial formulas for these invariants of the
{M}ilnor fiber of a generic arrangement of arbitrary dimension using these methods.},
gsid = {7826294628801091701},
URL = {http://jlms.oxfordjournals.org/cgi/content/abstract/51/1/105},
DOI = {10.1112/jlms/51.1.105}
}
@incollection {Suciu:conm95,
AUTHOR = {Cohen, Daniel C. and Suciu, Alexander I.},
TITLE = {The {C}hen groups of the pure braid group},
BOOKTITLE = {The {\v{C}}ech centennial (Boston, MA, 1993)},
SERIES = {Contemp. Math.},
VOLUME = {181},
PAGES = {45--64},
EDITOR = {Mila Cenkl and Haynes Miller},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1995},
MRCLASS = {20F14 (20F36 57M25)},
MRNUMBER = {1320987 (96c:20055)},
MRREVIEWER = {Colin Maclachlan},
ZBLNUMBER = {0833.20047},
ZBREVIEWER = {A. M. Akimenkov},
KEYWORDS = {lower central series, {C}hen groups, pure braid groups,
Hilbert series, graded modules, Gr{\"{o}}bner bases},
abstract = {The {C}hen groups of a group are the lower central series quotients
of its maximal metabelian quotient. We show that the {C}hen groups of the pure braid
group $P_n$ are free abelian, and we compute their ranks. The computation of these
{C}hen groups reduces to the computation of the {H}ilbert series of a certain graded
module over a polynomial ring, and the latter is carried out by means of a
{G}r{\"{o}}bner basis algorithm. This result shows that, for $n \ge 4$,
the group $P_n$ is not a direct product of free groups.},
gsid = {2903466370899826001},
URL = {http://www.ams.org/books/conm/181/2029},
DOI = {10.1090/conm/181/02029}
}
@article{Suciu:bams93,
AUTHOR = {Dwyer, William G. and Jekel, Solomon M. and Suciu,
Alexander I.},
TITLE = {Homology isomorphisms between algebraic groups made discrete},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {25},
YEAR = {1993},
NUMBER = {2},
PAGES = {145--149},
ISSN = {0024-6093},
MRCLASS = {20J05 (57T99)},
MRNUMBER = {1204066 (94f:20097)},
MRREVIEWER = {Lekh Raj Vermani},
ZBLNUMBER = {0801.20017},
ZBREVIEWER = {Li Fuan},
KEYWORDS = {homology groups; split exact sequence; discrete groups;
diagonalizable endomorphism; connected affine algebraic group;
unipotent radical; descending central series},
abstract = {Consider a split exact sequence of discrete groups
$$\{1\}\to G \to \Gamma \overset\pi\to
{\underset\sigma\to\rightleftarrows} \Gamma/G \to \{1\}.$$
Suppose there exists a normal series
$G=G_0 \triangleright G_1 \triangleright \cdots \triangleright G_n
\triangleright G_{n+1}={1}$, such that (1) $G_i/G_{i+1}$ is a rational
vector space for $i=0, cdots, n$; (2) $G_i/G_{i+1}$ is contained in the
center of $G/G_{i+1}$ for $i=0, cdots, n$; (3) there exists an element
in the center of $\Gamma/G$ that induces a diagonalizable endomorphism of
each $G_i/G_{i+1}$ with all eigenvalues rational and greater than $1$.
Then the map $pi$ induces an isomorphism $pi_* \colon H_*(B \Gamma,Z)
\to H_*(B (\Gamma/G), \mathbb{Z})$.},
URL = {http://blms.oxfordjournals.org/cgi/reprint/25/2/145},
DOI = {10.1112/blms/25.2.145}
}
@article {Suciu:cmh92,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Inequivalent frame-spun knots with the same complement},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {67},
YEAR = {1992},
month = {dec},
NUMBER = {1},
PAGES = {47--63},
ISSN = {0010-2571},
MRCLASS = {57Q45},
MRNUMBER = {1144613 (93a:57026)},
MRREVIEWER = {Jerome P. Levine},
ZBLNUMBER = {789.57014},
ZBREVIEWER = {Cherry Kearton},
KEYWORDS = {High-dimensional knots, frame-spinning construction,
generalized {P}ontryagin--{T}hom construction, homotopy groups of
spheres, {G}luck twist, inequivalent knots with the same complement},
abstract = {One of the basic questions of knot theory is: Is every $n$-knot determined by its complement?
For $n=1$, have recently given an affirmative answer to this question. For $n\ge 2$, there are at most two
$n$-knots with the same complement. A knot which is determined by its complement is called reflexive.
Knots that are spun, superspun, $2$-twist-spun, simple, stable, and some others are known to be reflexive.
{C}appell and {S}haneson gave the first examples of knots which are not determined by their complements. Their
method works for each $n\ge 2$, as long as certain square matrices of size $n+1$ exist, but such matrices
have been found only for $n=2, 3, 4, 5$. We prove here that there exist non-reflexive $n$-knots for every
$n$ congruent to $3$ or $4$ modulo $8$. We construct these $n$-knots by frame-spinning odd-twist-spun $2$-knots
with aspherical closed fiber, showing in the process that such $2$-knots are not reflexive. The basic idea
is to translate the question of reflexivity for the frame-spun $n$-knots into a question about homotopy
groups of spheres, via a generalized {P}ontrjagin--{T}hom construction. In turn, this question can be answered
using deep work of Mahowald on the image of the $J$-homomorphism in the EHP sequence.},
URL = {http://www.springerlink.com/content/k2054n6263477407/},
URL = {http://retro.seals.ch/digbib/view?rid=comahe-003:1992:67::9},
DOI = {10.1007/BF02566488},
gsid = {683480399342721237}
}
@article {Suciu:mathann91,
AUTHOR = {Klein, John R. and Suciu, Alexander I.},
TITLE = {Inequivalent fibred knots whose homotopy {S}eifert pairings are isometric},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {289},
YEAR = {1991},
month = {mar},
NUMBER = {4},
PAGES = {683--701},
ISSN = {0025-5831},
MRCLASS = {57Q45},
MRNUMBER = {1103043 (92d:57015)},
MRREVIEWER = {Cherry Kearton},
ZBLNUMBER = {711.57015},
KEYWORDS = {},
abstract = {The problem of classifying knots in terms of a finite list of invariants is a fundamental
question of knot theory. Suppose $V$ is a {S}eifert surface for an $n$-knot in $S^{n+2}$.
{F}arber showed that the isometry class of the homotopy {S}eifert pairing on $V$
determines the isotopy class of $V$, provided $n\ge 5$ and $V$ is $r$-connected, with $3r\ge n+1$.
Subsequently, {R}ichter improved the connectivity range in {F}arber's theorem to $3r\ge n$.
We show here that {F}arber's theorem is false outside this range. Using a \enquote{diff-spinning}
construction, we show that, for each integer $n\ge 9$, there exist infinitely many pairs of
fibered $n$-knots whose homotopy {S}eifert pairings are isometric, yet whose complements
are not homotopy equivalent.},
URL = {http://www.springerlink.com/content/k2k262hp36m74124/},
DOI = {10.1007/BF01446596},
gsid = {9846312469091373847}
}
@article {Suciu:tams90,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Iterated spinning and homology spheres},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {321},
YEAR = {1990},
month = {sep},
NUMBER = {1},
PAGES = {145--157},
ISSN = {0002-9947},
MRCLASS = {57N65 (55Q52 57Q45 57R19)},
MRNUMBER = {987169 (90m:57014)},
MRREVIEWER = {Laurence R. Taylor},
ZBLNUMBER = {725.57010},
ZBREVIEWER = {Jerome P. Levine},
KEYWORDS = {Spinning manifolds; homology spheres; homotopy type},
abstract = {Given a closed $n$-manifold $M^n$ and a tuple of positive integers $P$,
let $\sigma_P M$ be the $P$-spin of $M$. If $M \not\simeq S^n$ and $P\ne Q$ (as
unordered tuples), it is shown that $\sigma_P M \not\simeq \sigma_Q M$ if either
(1) $H_*(M) \not\cong H_*(S^n)$, (2) $\pi_1(M)$ is finite, (3) $M$ is aspherical, or
(4) $n=3$. Applications to the homotopy classification of homology spheres and knot
exteriors are given.},
URL = {http://www.ams.org/journals/tran/1990-321-01/S0002-9947-1990-0987169-3/},
DOI = {10.1090/S0002-9947-1990-0987169-3},
gsid = {8010097610648139376}
}
@article {Suciu:pac88,
AUTHOR = {Suciu, Alexander I.},
TITLE = {The oriented homotopy type of spun {$3$}-manifolds},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {131},
YEAR = {1988},
NUMBER = {2},
PAGES = {393--399},
ISSN = {0030-8730},
MRCLASS = {57N13 (55P15 57M99)},
MRNUMBER = {922225 (89d:57020)},
MRREVIEWER = {Cameron McA. Gordon},
ZBLNUMBER = {0594.57008},
ZBREVIEWER = {},
KEYWORDS = {Spinning $3$-manifolds, framing, homotopy type},
abstract = {We show that the oriented homotopy type of a spun $3$-manifold
is determined by the fundamental group and the choice of framing.},
URL = {http://projecteuclid.org/euclid.pjm/1102689936},
DOI = {10.2140/pjm.1988.131.393},
gsid = {4989033156251303419}
}
@article {Suciu:mz87,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Immersed spheres in {$\mathbf{CP}\sp 2$} and
{$S\sp 2\times S\sp 2$}},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {196},
YEAR = {1987},
month = {mar},
NUMBER = {1},
PAGES = {51--57},
ISSN = {0025-5874},
MRCLASS = {57R95 (57N13 57R42)},
MRNUMBER = {907407 (88j:57038)},
MRREVIEWER = {Don{\v{c}}o Dimovski},
ZBLNUMBER = {608.57025},
ZBREVIEWER = {},
KEYWORDS = {Four-manifolds, immersed spheres, intersection form},
abstract = {If $M$ is a compact, connected, simply-connected,
smooth $4$-manifold, and gamma is a class in $H_2(M; \mathbb{\Z})$,
define $d_{\gamma}$ to be the minimum number of double points of
immersed spheres representing $\gamma$. We use a theorem of
S. K. Donaldson to provide lower bounds for $d_{\gamma}$, for
$\gamma$ certain homology classes in rational surfaces.},
URL = {http://www.springerlink.com/content/gg5677l137p214h3/},
DOI = {10.1007/BF01179266},
gsid = {7992054833126082985}
}
@article {Suciu:jlms87,
AUTHOR = {Plotnick, Steven P. and Suciu, Alexander I.},
TITLE = {Fibered knots and spherical space forms},
JOURNAL = {J. London Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {35},
YEAR = {1987},
NUMBER = {3},
PAGES = {514--526},
ISSN = {0024-6107},
MRCLASS = {57Q45},
MRNUMBER = {889373 (88f:57038)},
MRREVIEWER = {Jonathan A. Hillman},
ZBLNUMBER = {587.57009},
ZBREVIEWER = {},
KEYWORDS = {Knots in the $4$-sphere, fibrations, spherical space form, monodromy, {G}luck reconstruction},
abstract = {Let $K$ be a knotted sphere in $S^{n+1}$, with closed fiber a spherical
space form, $S^n/\pi$. If $n=3$, and the monodromy has odd order, then $K$ and its {G}luck
reconstruction, $K^*$, are inequivalent. On the other hand, if $n>3$, then $\pi$ is cyclic,
and $K$ is equivalent to $K^*$.},
URL = {http://jlms.oxfordjournals.org/cgi/reprint/s2-35/3/514.pdf},
DOI = {10.1112/jlms/s2-35.3.514},
gsid = {7127795096010552269}
}
@article {Suciu:topapp87,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Homology $4$-spheres with distinct $k$-invariants},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {25},
YEAR = {1987},
month = {feb},
NUMBER = {1},
PAGES = {103--110},
ISSN = {0166-8641},
MRCLASS = {57N13 (55S45 57Q45 57R65)},
MRNUMBER = {874982 (88f:57021)},
MRREVIEWER = {Don{\v{c}}o Dimovski},
ZBLNUMBER = {617.57008},
ZBREVIEWER = {Dusan Repov\v{s}},
KEYWORDS = {Homology $4$-spheres, $k$-invariants},
abstract = {We exhibit integral-homology $4$-spheres with isomorphic
$pi_1$ and $pi_2$ (as $pi_1$-modules), but with distinct $k$-invariants.},
URL = {http://www.sciencedirect.com/science/article/pii/0166864187900794},
DOI = {10.1016/0166-8641(87)90079-4},
gsid = {13255181481930725197}
}
@article {Suciu:camb85,
AUTHOR = {Suciu, Alexander I.},
TITLE = {Infinitely many ribbon knots with the same fundamental group},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {98},
YEAR = {1985},
NUMBER = {3},
PAGES = {481--492},
ISSN = {0305-0041},
MRCLASS = {57Q45 (57M05)},
MRNUMBER = {803607 (87a:57025)},
MRREVIEWER = {Don{\v{c}}o Dimovski},
ZBLNUMBER = {596.57013},
ZBREVIEWER = {Yasutaka Nakanishi},
KEYWORDS = {Knots in the 4-sphere, homotopy groups},
abstract = {A knot $K = (S^{n+2}, S^{n})$ is a ribbon knot if $S^{n}$ bounds
an immersed disc $D^{n+1}$ in $S^{n+2}$ with no triple points and such that
the components of the singular set are $n$-discs whose boundary $(n-1)$-spheres
either lie on $S^{n}$ or are disjoint from $S^{n}$. Pushing $D^{n+1}$ into $D^{n+3}$
produces a ribbon disc pair $D = (D^{n+3}, D^{n+1})$, with the ribbon knot
$(S^{n+2}, S^{n})$ on its boundary. The double of a ribbon $(n+1)$-disc pair is an
$(n+1)$-ribbon knot. Every $(n+1)$-ribbon knot is obtained in this manner.},
URL = {http://journals.cambridge.org/action/displayAbstract?aid=2095324},
DOI = {10.1017/S0305004100063684},
gsid = {6126303753692943427}
}
@article {Suciu:cmh85,
AUTHOR = {Plotnick, Steven P. and Suciu, Alexander I.},
TITLE = {{$k$}-invariants of knotted {$2$}-spheres},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {60},
YEAR = {1985},
month = {dec},
NUMBER = {1},
PAGES = {54--84},
ISSN = {0010-2571},
MRCLASS = {57Q45 (55P15)},
MRNUMBER = {787662 (86i:57026)},
MRREVIEWER = {John G. Ratcliffe},
ZBLNUMBER = {568.57017},
ZBREVIEWER = {Cherry Kearton},
KEYWORDS = {Knots in the 4-sphere, homotopy type},
abstract = {This paper studies some questions concerning homotopy
type invariants of smooth four-dimensional knot complements.
Higher-dimensional knot theory diverges sharply from classical knot theory
in this respect. A knot complement $S^4\setminus S^2$ has the homotopy
type of a $3$-complex, so a natural question is whether the homotopy
theory of knot complements in $S^4$ can be as complicated as that of
arbitrary $3$-complexes. The main result of this paper indicates
that the answer is yes.},
gsid = {5789357417350690039},
URL = {http://www.springerlink.com/content/q7440uh56552136t},
DOI = {10.1007/BF02567400}
}
@phdthesis{Suciu:thesis,
author = {Suciu, Alexander I.},
title = {Homotopy Type Invariants of Four-Dimensional Knot Complements},
school = {Columbia University},
address = {New York, NY},
year = {1984},
month = {May},
note = {Ph.D. thesis. Photocopy: UMI-8427479, Ann Arbor, MI},
MRNUMBER = {2633816},
keywords = {Knots in the 4-sphere, homotopy type},
abstract = {This thesis studies the homotopy type of smooth four
dimensional knot complements. In contrast with the classical case,
high-dimensional knot complements with fundamental group different
from are never aspherical. The second homotopy group already provides
examples of the way in which a knot in $S^4$ can fail to be determined
by its fundamental group ({C}. {M}c{A}. {G}ordon, {S}. {P}. {P}lotnick).
A natural class of knots to investigate is ribbon knots. They bound
immersed disks with \enquote{ribbon singularities.} A method is given for
computing $\pi_2$ of such knot complements. I show that there are
infinitely many ribbon knots in $S^4$ with isomorphic $\pi_1$ but
distinct $\pi_2$ (viewed as $\pi_1$-modules). They appear as boundaries
of distinct ribbon disk pairs with the same exterior. These knots have
the fundamental group of the spun trefoil, but none in a spun knot.
To a four-dimensional knot complement, one can associate a certain
cohomology class, the first $k$-invariant of {E}ilenberg, {M}ac{L}ane
and {W}hitehead. In a joint paper, {P}lotnick and I showed that there are
arbitrarily many knots in $S^4$ whose complements have isomorphic $\pi_1$
and $\pi_2$ (as $\pi_1$-modules), but distinct $k$-invariants. Here I prove
this result using examples which are somewhat more natural and easier
to produce. They are constructed from a fibered knot with fiber a
punctured lens space and a ribbon knot by surgery.
The proofs involve writing down explicit cell complexes, computing
twisted cohomology groups, combinatorial group theory and calculations
in group rings.},
gsid = {13409863007741117343},
URL = {https://web.northeastern.edu/suciu/papers/thesis.pdf},
URL = {http://proquest.umi.com/pqdlink?did=751356951&Fmt=7&clientId=79356&RQT=309&VName=PQD}
}