Algebraic Topology

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.

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.

Pythagorean Theorem rr = xx+yy can be expressed as rr = (x+yi)(x-yi), thus rr = rr*, superposing two statements: one true under algebra with one false under epistemology; that way revealing the inner nature of complex numbers as the superposition property expressed in the topology of complex, conjugate and orthogonal states that are invariant with respect to their topological transformations.

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 "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,\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

Let A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra H. Suppose H 3 is a free abelian group of minimum possible rank, given the values the Möbius function μ : L 2 → Z takes on the rank 2 flats of A. Then the associated graded Lie algebra of G decomposes (in degrees ≥ 2) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by φ r (G) = X∈L 2 φ r (F μ(X) ), for r ≥ 2. We illustrate this new Lower Central Series formula with several families of examples.

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,

We provide all geometric polyhedral realizations of Mobius' torus with 7 vertices. There are no simplicia1 realizations having a higher geometric symmetry than Csaszar's. We confirm two conjectures about the realization space of Mobius' torus posed by J. Reay. Above all, the article shows oriented matroids to be a useful tool for investigating polyhedral structures. Nous presentons toutes les realisations geometriques polyedriques du tore de Mobius avec sept sommets. II n'existe pas de realisation simpliciale possedant une plus grande symetrie geometrique que celle de Cshszhr. Nous confirmons deux conjectures enoncees par J. Reay a propos de I'espace de realisation du tore de Mobius. Par dessus tout, cet article montre que les matro'ides orientees sont des outils efficaces pour la recherche de structures polyedriques. A.

The aim of this work is to develop a theory parallel to that of motivic complexes based on cycles and correspondences with coefficients in quadratic forms. This framework is closer to the point of view of A 1 -homotopy than the original one envisioned by Beilinson and set up by Voevodsky. Contents Introduction 1 Acknowledgments 2 Conventions 2 1. MW-transfers on sheaves 2 1.1. Reminder on MW-correspondences 2 1.2. MW-transfers 4 2. Framed correspondences 10 2.1. Definitions and basic properties 10 2.2. Presheaves 15 3. MW-motivic complexes 16 3.1. Derived category 16 3.2. The A 1 -derived category 20 3.3. The stable A 1 -derived category 26 4. MW-motivic cohomology 29 4.1. MW-motivic cohomology as ext-groups 29 4.2. Comparison with Chow-Witt groups 31 4.2.1. The naive Milnor-Witt presheaf 31 References 35 In all this work, we will fix a base field k assumed to be infinite perfect. All schemes considered will be assumed to be separated of finite type over k, unless explicitly stated. We will fix a ring of coefficients R. We will also consider a Grothendieck topology t on the site of smooth k-schemes, which in practice will be either the Nisnevich of the étale topology. In section 3 and 4 we will restrict to these two latter cases. 1.1. Reminder on MW-correspondences. 1.1.1. We will use the definitions and constructions of [CF14]. In particular, for any smooth schemes X and Y (with Y connected of dimension d), we consider the following group of finite MW-correspondences from X to Y : where T runs over the ordered set of reduced (but not necessarily irreducible) closed subschemes in X × Y whose projection to X is finite equidimensional and ω Y is the pull back of This definition is clearly functorial in S and it follows that we obtain a presheaf of Z-graded algebras on the category of smooth schemes via ). We denote by K MW * ,naive the associated Nisnevich sheaf of graded Z-algebras and observe that this definition naturally extends to essentially smooth k-schemes. Our next aim is to show that this naive definition in fact coincides with the definition of the unramified Milnor-Witt K-theory sheaf given in [Mor12, §3] (see also §1]). Indeed, let X be a smooth integral scheme. The ring homomorphism O(X) → k(X) induces a ring homomorphism ) and it is straightforward to check that elements in the image are unramified, i.e. that the previous homomorphism induces a ring homomorphism K MW * (O(X)) → K MW * (X). By the universal property of the associated sheaf, we obtain a morphism of sheaves K MW * ,naive → K MW * .

In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of a 2-truncated simplicial group. Therefore, the 2-crossed module is an algebraic homotopy model for the homotopy 3-types. Tricategories are a three-dimensional generalization of the bicategory concept. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product. Briefly, a Gray category is a semi-strict 3-category for homotopy 3-types. Naturally, the tricategory perspective is used in homotopy theory. The 2-crossed module is associated with the concept of the Gray category. The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups.

a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the lower central series functors Z k . The paper ends with an application to algebraic K-theory.

We formulate the concept of minimal fibration in the context of fibrations in the model category S C of C-diagrams of simplicial sets, for a small index category C. When C is an E I -category satisfying some mild finiteness restrictions, we show that every fibration of C-diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in S C over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations (Barratt et al. in Am J Math 81:639-657, 1959).

Đây là tập bài giảng cho các môn học về tôpô cho sinh viên đại học, gồm Tôpô Đại cương, Tôpô Đại số, và Tôpô Vi phân.

This is a set of lecture notes for courses in topology for undergraduate students, consisting of General Topology, Algebraic Topology, and Differential Topology.

Characterizations of paracompact finite C-spaces via continuous selections avoiding Z σ -sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.

Let f : X → Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space. We consider the following property of continuous maps g : It is shown that all maps g ∈ C(X, I n+1 ) with the above property form a dense G δ -set in the function space C(X, I n+1 ) equipped with the source limitation topology. Moreover, for every n + 1 ≤ m ≤ ω the space C(X, I m ) contains a dense G δ -set of maps having this property.

Characterizations of paracompact finite C-spaces via continuous selections avoiding Z σ -sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using Haken-Matveev technique, for recognizing virtual links. This paper may be read as both an introduction and as a research paper. For more about Haken-Matveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL].

We introduce a new regular relation δ on a given group G and show that δ is a congruence relation on G, concerning module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ. We conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

The aim of this work is to introduce representations of BiHom-left-symmetric algebras. and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a BiHom-left-symmetric algebra is introduced. We will prove that it can generate trivial linear deformations of a BiHom-left-symmetric algebra.

A well known problem of B. Grünbaum [Grü60] asks whether for every continuous mass distribution (measure) dµ = f dm on R n there exist n hyperplanes dividing R n into 2 n parts of equal measure. It is known that the answer is positive in dimension n = 3 [Had66] and negative for n ≥ 5, [Avis84] [Ram96]. We give a partial solution to Grünbaum's problem in the critical dimension n = 4 by proving that each measure µ in R 4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional, affine subspace L of R 4 . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on the Koschorke's exact singularity sequence [Kosch81] and the remarkable properties of the essentially unique, balanced, binary Gray code in dimension 4, [Toot56] [Knu01].

We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. We have no continuous dependence, but manage to utilize the "homological continuous dependence" of the heavily covered point. This allows us to infer some corollaries in a usual way. We also give an elementary argument to prove the simplest of these corollaries.

In this paper, we examine knots froma different angles, we provide a beginner's perspective, along with pro-viding related terminology, which may aid the researcher, in addition to the Monodromy, Euler character-istic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, finally we conclude with future recommendations.

In this paper, we examine knots froma different angles, we provide a beginner's perspective, along with pro-viding related terminology, which may aid the researcher, in addition to the Monodromy, Euler character-istic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, finally we conclude with future recommendations.

Without the following, this paper would have never seen the light, hence it is my greatest pleasure to mention the following helpers, not tutors, in ascending Alphabetic order: 1. Google: main research buddy, mainly with 'books' and 'scholar'.
2.Numdam: for the help with LateX's syntax understanding and comprehension by the share of tex papers, besides offering academic work in English and French. 3.Simon Jubert: for recommending "Overleaf.com", a thanks might not be enough, as without him, all effort, progress, and work would have gone in vein, with the wind "Une grande merci 'a vous".
4. SLMath: for offering resources, both physical and online, in addition to offering a set of references and links, and their welcoming hospitality.
In this paper, we examine knots froma different angles, we provide a beginner’s perspective, along with providing related terminology, which may aid the researcher, in addition to the Monodromy, Euler characteristic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, finally we conclude with future recommendations.

This paper explores the directed forest complex of food web graphs, which model the flow of energy in ecosystems. By applying discrete Morse theory, we construct near-perfect discrete Morse vector fields on the directed forest complex and provide necessary conditions for these fields to be perfect. Our results reveal deep connections between the combinatorial structure of food web graphs and their topological properties.

Abstract:
In this paper, we examine knots froma different angles, we provide a beginner’s perspective, along with pro- viding related terminology, which may aid the researcher, in addition to the Monodromy, Euler character- istic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, finally we conclude with future recommendations.
Thanks:
Without the following, this paper would have never seen the light, hence it is my greatest pleasure to mention the following helpers, not tutors, in ascending Alphabetic order: 1. Google: main research buddy, mainly with 'books' and 'scholar'. 2.Numdam: for the help with LateX's syntax understanding and comprehension by the share of tex papers, besides offering academic work in English and French.

3.Simon Jubert: for recommending "Overleaf.com", a thanks might not be enough, as without him, all effort, progress, and work would have gone in vein, with the wind "Une grande merci 'a vous".
4. SLMath: for offering resources, both physical and online, in addition to offering a set of references and links, and their welcoming hospitality.

In this paper, we give an overview on investigations into the Sperner property of posets and particularly the posets induced by applying natural orders to Coxeter groups. We first explore an elementary proof of Sperner’s original theorem using Hall’s matching condition and discuss the limitations of this technique. We will then discuss the development of the stronger normalized matching condition and normalized flow property and their applications to the problem of Sperner, and finally review recent applications of these techniques to the natural orders of Coxeter groups.

In this paper, we examine knots from different angles, we provide a beginner’s perspective, along with providing related terminology, which may aid the researcher, in addition to the f, Euler characteristic function, Fundamental groups which are touched upon, focusing on knots their types and homologies, and polynomials. besides that a short appendix, explaining various topics relating to knots and topology in general; Finally we conclude with future recommendations.

We study polynomials with complex coefficients which are nondegenerate with respect to their Newton polyhedron through data on contact loci, motivic nearby cycles and motivic Milnor fiber. Introducing an explicit description of these quantities we can answer in part to questions concerning the motivic Milnor fiber of the restriction of a singularity to a hyperplane and the integral identity conjecture in the context of nondegenerate singularities. Moreover, in the same context, we give calculations on the Hodge spectrum, the cohomology groups of the contact loci as well as their compatibility with natural automorphisms.

A re-upload of  the latest paper. In there, we explore the intriguing realm of low dimensional manifolds, particularly knots, and their historical journey from antiquity,  Renaissance, up until today. A Calculi section has been added, and a personal opinion subsubsection to enrich the academic discourse, in describing the curvature, how linear approximations does not work as their substitute, as we pay for the consequence of choosing our tools. mistakes —Most importantly curious to hear your thoughts on these constructive concepts, and finally Ricci flow is mentioned as a working example for comprehension and pondering upon  Enjoy the read!​