Algebraic Topology

Je remercie DIEU tout puissant de m'avoir donné la foie, le courage pour réaliser ce modeste travail et qui a mis dans mon chemin les bonnes personnes et m'a con…é aux bonnes mains. Je tiens en premier lieu à exprimer mes plus vifs remerciements à Madame K.Khelou… ma promotrice pour l'intéressant sujet qu'elle m'a proposé. Ainsi que pour son aide sa patience ses conseils et ses encouragements sa grande disponibilité et son ouverture d'esprit qui m'ont aidé à mener à bien ce travail. Je lui suis également reconnaissante pour la connaissance qu'elle ma accordée. Il m'est impossible de lui exprimer toute ma gratitude en seulement quelques lignes. J'adresse ma profonde gratitude et mes remerciements les plus vifs à Monsieur A. Berboucha, pour m'avoir fait l'honneur de présider le jury de cette soutenance. Je tiens à remercier également Madame S.Allili-Zahar qui a accepté d'examiner cet humble travail. Mes derniers et profonds remerciements vont à mes chers parents, frère et soeur, pour leur soutien et leur con…ance en moi, sans oublier mes amis et tous ceux qui ont servis, de près ou de loin pour achever ce travail. . i Je dédie ce modeste travail à: Mes chers parents qui étaient toujours attentifs, a¤ectueux et compréhensifs qui m'ont soutenu durant les laborieuses années de mes études, à qui je témoigne toute ma gratitude. Mon cher frère: houssem et ma cher soeur: naila. Mes très chers grand-parents. Ma chere grande famille: mes oncles et mes tantes, mes cousins et mes cousines. Toutes mes amies. Toute ma famille chacun par son nom. Tous les étudiants de notre département, en particulier notre promotion. I. X C(I; J) : l'espace des fonctions continue sur I à valeurs dans J: [a; b]): l'espace des fonctions u mesurables sur [a; b] et véri…ant b R a ju (t)j p dt < 1: Notations et conventions Si la réponse est a¢ rmative, on passe à d'autres questions: comment la solution peut-elle être construite ou approchée? Combien de solutions y a-t-il ? Quelle est la structure de l'ensemble de toutes les solutions? Dans ce mémoire, on s'intéresse notamment à l'étude d'existence de points …xes positifs d'une contraction stricte d'ensembles en prolongeant le concept de l'indice du point …xe dé…ni pour les applications complètement continues à des classes d'applications plus larges (les contractions strictes d'ensembles et les applications condensantes). Une contraction stricte d'ensemble est une application telle que l'image de tout ensemble borné est en un certain sens plus compacte que l'ensemble lui-même. Le degré de non-compacité est mesuré à l'aide d'une fonction appelée mesure de non-compacité. Le premier qui a considéré la mesure de non-compacité d'un sous-ensemble A d'un espace métrique est K. Kuratowski en 1930. Dans les années cinquante, G. Darbo, A. Furi, M.A. Krasnosel'skii et autres ont appliqué di¤érentes mesures de non-compacité dans les théories du point …xe, des applications linéaires et des équations di¤érentielles et intégrales. Ce mémoire comprend trois chapitres: Le premier chapitre contient un ensemble de dé…nitions et de résultats qui nous seront indispensables à la compréhension de la suite de cette étude. Ces résultats concernent essentiellement les notions suivantes: la mesure de non-compacité de Kuratowski, les contractions strictes d'ensembles, les applications condensantes et les cônes. Dans le deuxième chapitre, nous présentons quelques théorèmes du point …xe sur les cônes pour les contractions strictes d'ensembles qui entraînent l'existence ou la multiplicité d'un point …xe positif. La première section de ce chapitre est consacrée à la dé…nition de l'indice du point …xe pour une contraction stricte d'ensembles. Dans la deuxième section, nous rassemblons quelques lemmes fondamentaux utiles pour la suite de ce travail, puis nous donnons quelques développements récents du célèbre théorème de Krasnosel'skii, établi en 1960 et développé par la suite par beaucoup d'auteurs, on peut citer les travaux de Guo ([14],1988), Avery et Anderson ([3], 2002), Avery, Anderson et Krueger ([4], 2006). Ces récents développements sont dû à Sun ([26], 1986), Meiqiang, Zhang et Weigao ([20], 2010) pour les contractions strictes d'ensembles. Dans la troisième section, nous nous intéressons à l'existence des points …xes multiples d'une contraction stricte d'ensembles dé…nie sur un cône dans un espace de Banach ordonné. Plus précisément, nous présentons

This article is concerned with the existence of positive solutions of a fourthorder p-Laplacian boundary value problem. Based on a priori estimates achieved by utilizing Jensen's integral inequalities for convex and concave functions, we use fixed point index theory to establish the existence of positive solutions for the above problem.

In this paper, using Q *-closed sets, we introduce a new version of normality called Q *-normality, which is a weak form of normality. Further utilizing Q * g-closed sets, we obtain some characterizations of Q *-normal and normal spaces and also obtain some preservation theorems for Q *-normal spaces. Keywords-Q *-closed, g-closed, Q * g-closed sets, Q *-continuous and almost Q *-continuous functions, Normal, Q *-normal spaces.

In this paper we introduced the concepts of new separation axioms called SC*-separation axioms and H*-separation axioms by using SC* and H*- open sets in topological spaces. SC*- separation axioms i.e. SC*-C0, SC*-C1, weakly SC*-C0, and weakly SC*-C1 and H*-T½, H*-Tb, H*-Td -spaces. Also we obtained several properties of such spaces.

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of A.

An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.

Often described as the "Rosetta Stone of mathematics," the Langlands Program reveals profound structural correspondences across number theory, geometry, and representation theory. This paper proposes a metaphorical extension of its core principles to the analysis of poetic meaning and structure. Drawing on five key concepts-functorial mappings, cohomological bounds, spectral decomposition, self-duality, and contractibility-it introduces a framework for interpreting how poetic symbols move across interpretive layers, how coherence is formed or fractured, and how thematic unity emerges. Inspired by recent advances in the geometric Langlands Program, this approach models the logic of poetic meaning not as fixed interpretation, but as dynamic structural behaviour. Unlike methods focused on surface patterns such as metre, rhyme, or numerical sequences, it engages the deep architecture of symbolic expression. The framework is applied to three contrasting poetic texts-Paul Éluard's The World Is Blue as an Orange, Shakespeare's Sonnet 116, and Abol Froushan's For the People Who Came in From the Lost Century-to demonstrate its adaptability across surrealist, classical, and contemporary forms. The paper argues that mathematical thinking, when applied structurally rather than decoratively, can offer powerful new tools for literary analysis and illuminate the shared logic between symbolic mathematics and poetic expression.

Poetry presents unique challenges for natural language processing (NLP) due to its fragmented structure, intertextuality, and multimodal nature. Conventional NLP models struggle to capture its evolving semantic relationships, particularly across translations, historical contexts, and interpretative traditions. This paper introduces the Poetic Ontology Dataset (POD), a structured resource designed to embed poetic meaning as a dynamic, topological construct rather than a static textual entity. By applying sheaf theory, functorial mappings, and graph embeddings, we model poetic motifs and metaphors as interdependent structures within a meta-body-a network of poetic relations spanning time and cultures. Empirical validation compares AI-assisted meta-body analysis, derived through structured motif tracking and graph clustering, against traditional NLP embeddings. The results demonstrate that ontology-aware NLP models preserve semantic continuity and intertextual depth more effectively than conventional approaches. This work establishes a foundation for meaning-aware NLP architectures, bridging computational poetics, multimodal embeddings, and topological data analysis.

A characterization of simplicial objects in categories with finite products obtained by the reduced bar construction is given. The condition that characterizes such simplicial objects is a strictification of Segal's condition guaranteeing that the loop space of the geometric realization of a simplicial space X and the space X1 are of the same homotopy type. A generalization of Segal's result appropriate for bisimplicial spaces is given. This generalization gives conditions guaranteing that the double loop space of the geometric realization of a bisimplicial space X and the space X11 are of the same homotopy type.

Some sufficient conditions on a simplicial space X : ?op ? Top guaranteeing that X1 ? ?|X| were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar construction, providing an n-fold delooping of the classifying space of a category.

Let $E_n$ be Morava $E$-theory and let $G \subset G_n$ be a finite subgroup of $G_n$, the extended Morava stabilizer group. Let $E_{n}^{tG}$ be the Tate spectrum, defined as the cofiber of the norm map $N:(E_n)_{hG} \to E_n^{hG}$. We use the Tate spectral sequence to calculate $\pi_*E_{p-1}^{tG}$ for $G$ a maximal finite $p$-subgroup, and $p$ an odd prime. We show that $E_{p-1}^{tG} \simeq \ast$, so that the norm map gives equivalence between homotopy fixed point and homotopy orbit spectra. Our methods also give a calculation of $\pi_*E_{p-1}^{hG}$, which is a folklore calculation of Hopkins and Miller.

The orbital chromatic polynomial introduced by Cameron and Kayibi counts the number of proper $\lambda$-colorings of a graph modulo a group of symmetries. In this paper, we establish expansions for the orbital chromatic polynomial of the $n$-cycle for the group of rotations and its full automorphism group. Besides, we obtain a new proof of Fermat&#39;s Little Theorem.

We propose a mathematical framework for modeling identity recursion and symbolic encoding in topologically nontrivial spacetimes. Drawing from linear algebra (as classically formulated in Hoffman and Kunze), differential topology, and closed timelike loop geometries, we define the Recursive Encryptment Manifold (REM) as a structure where temporal coiling enables phase adjacency and cyclic semantic embedding. This formalizes aspects of the 'Encryptment Thesis' in algebraic and topological terms.

This paper proposes a new theoretical framework—Dimensional Invariance Theory—which redefines the structure of physical reality by assigning a single unchanging “father” constant to each dimension. Drawing from general relativity, quantum mechanics, algebraic topology, and higher-dimensional geometry, the theory posits that in any given dimension, only one element may remain invariant while all others must change to preserve its constancy. Beginning with light as the governing force of the third dimension, we derive time as its dependent “child,” and extend the logic into four dimensions using the tesseract geometry to define a new governing constant: agency. This superdimensional agency acts not through spontaneous creation or collapse but through traversal across a pre-encoded state-space, aligning with topological algebra where all possible outcomes must already exist for meaningful choice to be made. The theory critiques mainstream quantum interpretations of randomness and superposition, offering instead a deterministic structure governed by vertical hierarchies of invariance and horizontal constraints of dimensional exclusivity. Building on Kip Thorne’s depiction of the tesseract and relativistic time manipulation, we derive new geometric-metric formulations that introduce agency as a fourth-dimensional constant orthogonal to time. This work aims to unify the logical architecture of physical laws with the philosophical conditions required for causality, agency, and consciousness, laying groundwork for a new class of theoretical physics rooted in structure rather than uncertainty.

A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F ) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The main result of this paper is the determination of those arc complexes Arc(F ) that are also spherical. This classification has consequences for Riemann's moduli space of curves via its known identification with an analogous arc complex in the punctured case with no boundary. Namely, the essential singularities of the natural cellular compactification of Riemann's moduli space can be described.

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

A clutter on a ground set E is a collection of incomparable (inclusionwise) subsets of E. A map on clutters is a function from the class of all clutters on E to itself. Examples of maps on clutters are the blocking and the complementary maps. Let M be a matroid; the complementary and blocker maps give some relations between the clutters of M; however, b(H), the blocker of the hyperplanes of M, has not been studied. An axiomatic characterization of b(H) seems difficult to find, so in this paper we give some structural properties of such a clutter. In particular, necessary conditions about the dual rank of the members of b(H) are given. We show how b(H) can be decomposed in the blockers of the hyperplanes of submatroids of M. A characterization of the clutter of hyperplanes of a matroid and its blocker through forbidden minors is also given.

A clutter on a ground set E is a collection of incomparable (inclusionwise) subsets of E. A map on clutters is a function from the class of all clutters on E to itself. Examples of maps on clutters are the blocking and the complementary maps. Let M be a matroid; the complementary and blocker maps give some relations between the clutters of M ; however, b(H), the blocker of the hyperplanes of M , has not been studied. An axiomatic characterization of b(H) seems difficult to find, so in this paper we give some structural properties of such a clutter. In particular, necessary conditions about the dual rank of the members of b(H) are given. We show how b(H) can be decomposed in the blockers of the hyperplanes of submatroids of M . A characterization of the clutter of hyperplanes of a matroid and its blocker through forbidden minors is also given.

Let F be a disjoint iteration semigroup of C n di eomorphisms mapping a real open interval I 6 = ? onto I . It is proved that if F has a dense orbit possesing a subset of the second category with the Baire property, then F = fft : ft (x) = f 1 (f(x) + t) for every x 2 I ; t 2 Rg for some C n di eomorphism f of I onto the set of all reals R. The paper generalizes some results of J.A.Baker and G.Blanton 3]. Throughout this paper I 6 = ? denotes an open interval. Rand N are the sets of all reals and positive integers, respectively. In connection with a problem raised by O.Bor vka and F.Neumann (cf. e.g. 8]), J.A.Baker and J.Blanton 3](Theorem 1) (cf. also 2], 4] and 5]) have proved that every complete and disjoint group F of C n bijections from I to I has the form F = F f] := ff t : f t (x) = f 1 (f(x)+t) for x 2 I; t 2 Rg for some C n di eomorphism f of I onto R. For n = 0 this follows also from some earlier results of J.Acz el 1]. Let us remind (cf. 2]{ 5]) that a family of functions F I I is said to be disjoint provided the graphs of any two distinct members of F are disjoint (i.e. if f; g 2 F and f(a) = g(a) for some a 2 I, then f = g) and F is complete if S F = I I, where S F := f(a; f(a)) : a 2 I; f 2 Fg. Further, we say that F has a dense orbit provided there is b 2 I such that the set F(b) := ff(b) : f 2 Fg is dense in I. Clearly, if f : I ! R is a C n di eomorphism (i.e. f is a C n bijection and f 0 (x) 6 = 0 for all x 2 I), then F f] is a complete disjoint group of C n functions (cf. 2]{ 5]). We generalize the outcome from 3]. Namely we will prove the given below theorem. Theorem 1. Let n be a positive integer and F be a semigroup of C n bijection from I onto I. Suppose that F has a dense orbit F(b) possesing a subset D F(b) of the second category with the Baire property (cf. e.g. 7], p.599) and F fig is 1991 Mathematics Subject Classi cation: 26A18, 39B12.

Casual structure can take the form of cone bundles on a manifold, more general local preorders on a topological space, or simplicial orientations implicit in a simplicial set. This note takes a triangulation of a conal manifold M to mean an isomorphism between M and the locally preordered geometric realization of a simplicial set. Such triangulations completely encode causal and topological structure combinatorially. This note characterizes the triangulability of closed conal manifolds as the fibrewise freeness and generativity of the cone bundle.

In this paper, we question the structure of number sets, particularly the real numbers R. By analyzing the construction of the integers Z and rationals Q, we show that certain fundamental properties of classical mathematics, such as the multiplication of negative numbers and the addition of fractions, could be reinterpreted under a different logical framework. We also discuss the idea that there is no "universal base" allowing all rational numbers to be expressed in a single form. Our work follows the reflections initiated by Fibonacci in the Liber Abaci, Peano in his axioms, and other research in set theory and the philosophy of mathematics.

In this paper, we question the structure of number sets, particularly the real numbers R. By analyzing the construction of the integers Z and rationals Q, we show that certain fundamental properties of classical mathematics, such as the multiplication of negative numbers and the addition of fractions, could be reinterpreted under a different logical framework. We also discuss the idea that there is no "universal base" allowing all rational numbers to be expressed in a single form. Our work follows the reflections initiated by Fibonacci in the Liber Abaci, Peano in his axioms, and other research in set theory and the philosophy of mathematics.

Исследовательские программы, присутствующие в развитии теорий
гомологии и когомологии, следуют тенденции инфинитного матезиса. Постулирование симметризующих структур и повторное распространение случаев, требующих симметризации, является motum infinitum и представляет собой диалектику математизации между платонической (обобщающей) и аристотелевской (контекстуализирующей) тенденциями.

Дата выпуска: 12/29/2022 Модульные пространствапрекрасный пример структуры, "заранее выбранной" математикой для математической и физической пользы. Учитывая окружающее пространство S, можно связать объект O(S) через функцию присваивания, которая кодирует элементы соответствующей геометрии и алгебры. Полученная в результате коллекция структур является неуникальной параметризацией точек S. Этот процесс соответствует построению гомологических групп, но может быть применен в строго геометрических контекстах. O(S) содержит симметрии некоторого события на S, при этом эквивалентные события отображаются на эквивалентные O(S)-структуры (Katz & Mazur 108). Несколько канонических примеров дают мотивацию для этого процесса. Проективное пространство в n-мерном пространстве является пространством модуля для (n+1)-мерных вещественных (или комплексных) векторов, которые проходят через начало координат. Следуя этой интуиции, структуры Грассмана действуют на вектор (m, V, F), где m>0-целое число, а Vвекторное пространство (абстрактная коллекция векторов) над заданным полем F. Стеки модулей могут быть адаптированы для обеспечения пространств параметров для проективных поверхностей заданных родов, а также для общих алгебраических многообразий (Harris & Morrison 187). Высокоразмерные алгебраические многообразия, особенно для эллиптических кривых и многообразий, часто являются объектами изучения в современной математике, и предоставляют методы решения таких, казалось бы, несвязанных проблем, как Последняя теорема Ферма, используя соответствующие теории многообразий (в частности, теорию Таниямы-Шимуры). Кроме того, пространства модулей моделируют различные явления в теоретической физике, наиболее известным из которых является топологическая квантовая теория поля. Классическим контекстом для данного исследованиякак и для многих вопросов топологической или гомологической теорииявляется гладкое многообразие M, в частности, с ассоциированной римановой или келеровой структурой. То, что пространства модулей моделируют многие события в теоретической физике и чистой математике, объяснимо с помощью конструкции. Пространство модулейэто структура ante-rem, постулированная для формализации различных математических контекстов, и может быть названа аристотелевским артефактом. Теории многообразий, орбифолдов, полей и Грассмана, которые мотивируют определение пространства модулей, обеспечивают основную причину его эффективности в моделировании различных контекстов.

In this paper, we consider an enriched orthogonality for classes of spaces, with respect to groupoids, simplicial sets and spaces themselves. This point of view allows one to characterize homotopy equivalences, shape and strong shape equivalences. We show that there exists a class of spaces, properly containing ANR-spaces, for which shape and strong shape equivalences coincide. For such a class of spaces homotopy orthogonality implies enriched orthogonality.

We employ the perspective of the functional equation satised by the classical Fourier transform to derive the Helgason Fourier transform map Ω l (G/K, W)-→ Ω k (G/K × G/P, V [χ]) : f-→ f : G/K × G/P → V [χ] : (x, b)-→ f (x, b) (for W-valued differential forms f ∈ Ω l (G/K, W)) as the G-invariant vector bundle-valued differential form f on the product space G/K×G/P whose image under the vector bundle-valued Poisson transform is the fibre convolution-integral ϕ U σ,ν τ,l,k * f on G/K, where ϕ U σ,ν τ,l,k is the W-valued τ-spherical l-form on G/K. Explicitly, we prove that f l,k,λ (x, b) = (C o (λ)-1 ∧ β V (λ)) • (G/K

The symmetric hit problem was introduced for the flrst time by the author in his thesis ((5)). The aim of this paper is to solve an important open problem posed in ((7)), in an special case, which is one of the fundamental results in the studies of the symmetric hit problem.

In this paper we describe and continue the study begun in of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E * , then a Floer E * -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E-module spectra, generalizing the stable homotopy realization criteria given in . Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners.

The aim of the present paper is to study nonlinear system of partial differential equations (PDEs) involving both complexand real-valued unknown functions. We shall extend the use of the first integral method "based on the theory of commutative algebra" to construct new solutions to the coupled Higgs field equations, the Davey-Sterwatson (DS) equations and the coupled Klein-Gordon-Zakharov equations. All the algebraic computations in this work are performed using Mathematica software.

We study the topological structure of the symmetry group of the standard model, G SM = U (1)×SU ( )×SU (3). Locally, G SM ∼ = S 1 ×(S 3 ) 2 ×S 5 . For SU (3), which is an S 3 -bundle over S 5 (and therefore a local product of these spheres) we give a canonical gauge i.e. a canonical set of local trivializations. These formulae give the matrices of SU (3) in terms of points of spheres. Globally, we prove that the characteristic function of SU (3) is the suspension of the Hopf map S 3 h -→S 2 . We also study the case of SU (n) for arbitrary n, in particular the cases of SU (4), a flavour group, and of SU (5), a candidate group for grand unification. We show that the 2-sphere is also related to the fundamental symmetries of nature due to its relation to SO 0 (3, 1), the identity component of the Lorentz group, a subgroup of the symmetry group of several gauge theories of gravity.

This paper introduces the Unified Process Relational Framework (UPRF), a minimalist axiomatic system designed to demonstrate how fundamental mathematical constants-particularly π-can necessarily emerge from primitive relational structures, without being presupposed a priori. Framed within the anti-substantialist philosophical perspective, which rejects mathematical objects as primitive entities, we construct a formal system based on five elementary operators with precisely defined algebraic properties. The framework exhibits a fundamental bipartition between pivot-free expressions and expressions with exactly one pivot, which we characterize through the Non-Duplication Theorem. We develop the categorical structure of this framework, establishing universal properties of key objects and proving the Absolute Uniqueness of Pivot theorem. This article constitutes the foundational first part of a series that will rigorously demonstrate how π emerges as both a universal limit and a fundamental lower bound from these relational structures. Our approach offers a novel, rigorous formalization of the anti-substantialist position, suggesting that mathematical constants are not mysterious pre-existing entities, but necessary structural properties emerging from more fundamental relational concepts.

It is known that there is a unique concordance class in the free homotopy class of S^1× pt ⊂ S^1 × S^2. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are invertible. The knots K⊂ S^1× S^2 with hyperbolic complements and trivial symmetry group are special interest here, because they can be used to generate absolutely exotic compact 4-manifolds by the recipe given by Akbulut and Ruberman. Here we built absolutely exotic manifold pairs by this construction, and show that this construction keeps the Stein property of the 4-manifolds we start out with. By using this we establish the existence of an absolutely exotic contractible Stein manifold pair, and absolutely exotic homotopy S^1× B^3 Stein manifold pair.

This paper presents a rigorous, purely mathematical framework utilizing noncommutative geometry to prove the Riemann Hypothesis, asserting that all non-trivial zeros of the Riemann ζ function lie on the critical line Re(s) = 1/2. We introduce a "dual-rule" algebraic system comprising two C*-algebras: A math , encoding prime distribution, and A dual , reflecting zeros duality. By constructing a spectral triple (A, H, D) and analyzing its K-theory and cyclic homology, we demonstrate that any non-trivial zero off the critical line induces topological obstructions in the K-theory of the associated 7-dimensional noncommutative manifold, contradicting fundamental algebraic properties. This approach avoids physical analogies, offering a novel algebraic-topological path to the Riemann Hypothesis.

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.

The Dwyer-Fried invariants of a finite cell complex X are the subsets \Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,\Q) which parametrize the regular \Z^r-covers of X having finite Betti numbers up to degree i. In previous work, we showed that each \Omega-invariant is contained in the complement of a union of Schubert varieties associated to a certain subspace arrangement in H^1(X,\Q). Here, we identify a class of spaces for which this inclusion holds as equality. For such &quot;straight&quot; 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,\Q). In general, though, translated components in the characteristic varieties affect the answer.

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

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

We use augmented commutative differential graded algebra (acdga) models to study G-representation varieties of fundamental groups π " π 1 pMq 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 Hompπ, Gq and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when M is either a compact Kähler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group G is a non-abelian subalgebra of sl 2 pCq.

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra of a group, from finitely presented, commutator-relators groups to arbitrary finitely presented groups. Using the notion of "echelon presentation," we give an explicit formula for the cupproduct in the cohomology of a finite 2-complex. This yields an algorithm for computing the corresponding holonomy Lie algebra, based on a Magnus expansion method. As an application, we discuss issues of graded-formality, filtered-formality, 1-formality, and mildness. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.

We study the topology of the boundary manifold of a line arrangement in CP 2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial .G/, and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri-Neumann-Strebel invariants of G . From the Alexander 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. For Fred Cohen on the occasion of his sixtieth birthday

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

We survey the cohomology jumping loci and the Alexander-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 π 1 (X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.

We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried 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 Riemann surfaces and complements of hyperplane arrangements. Contents 1. Introduction 1 2. Characteristic and resonance varieties 5 3. The Dwyer-Fried invariants 9 4. The Bieri-Neumann-Strebel-Renz invariants 11 5. Relating the Ω-invariants and the Σ-invariants 15 6. Toric complexes 18 7. Quasi-projective varieties 20 8. Configuration spaces 23 9. Hyperplane arrangements 25 10. Acknowledgements 29 References 29