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AbstractSuppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞ . Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q . [ABSTRACT FROM AUTHOR]

For any genuinely ramified morphism f : Y ⟶ X between irreducible smooth projective curves we prove that (Y × X Y) ∖ Δ ‾ is connected, where Δ ⊂ Y × X Y is the diagonal. Using this result the following are proved: (1) If f is further Morse then the Galois closure is the symmetric group S d , where d = degree (f). (2) The Galois group of the general projection, to a line, of any smooth curve X ⊂ P n of degree d , which is not contained in a hyperplane and contains a non-flex point, is S d. [ABSTRACT FROM AUTHOR]

AbstractIn a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

In this paper, our prime aim is to develop the concept of fundamental group structure in soft set theoretic approach. To execute this, first, we have defined soft homotopy of maps, soft path, soft loop, soft path homotopy, soft loop homotopy, product of soft loops, soft homotopy class, etc., at a soft element using generalized soft mappings and their important behaviors are studied. Finally, we have introduced the notion of fundamental group whose members are soft homotopy classes. Some examples are also discussed in different soft topological spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show that a complete shrinking general Ricci flow soliton system with condition is compact if and only if is bounded on , where is the 2-form with components . We also prove that a complete shrinking general Ricci flow system soliton has finite fundamental group. [ABSTRACT FROM AUTHOR]

Let G be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes c²1(S)/c2(S) of minimal nonsingular surfaces of general type S with S with π1(S)≃G are dense in the interval [1,3]. [ABSTRACT FROM AUTHOR]

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold Σ β 1 + β 2 in a unit sphere S d that provides trivial homology and fundamental groups. We also validate the instability of current flows in π 1 (Σ β 1 + β 2 ) . The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let X•=(X,DX)$X^{\bullet }=(X, D_{X})$ be a pointed stable curve over an algebraically closed field of characteristic p>0$p>0$ and ΠX•$\Pi _{X^{\bullet }}$ the admissible fundamental group of X•$X^{\bullet }$. We prove that there exists a group‐theoretical algorithm, whose input datum is the admissible fundamental group ΠX•$\Pi _{X^{\bullet }}$, and whose output data are the topological and the combinatorial structures associated with X•$X^{\bullet }$. This result can be regarded as a mono‐anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups. [ABSTRACT FROM AUTHOR]

A graph product G on a graph Γ is a group defined as follows: For each vertex v of Γ there is a corresponding non-trivial group Gv. The group G is the quotient of the free product of the Gv by the commutation relations [ G v , G w ] = 1 for all adjacent v and w in Γ. Semistability at ∞ is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at ∞ for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at ∞. While many classes of groups are known to contain only semistable at ∞ groups, there are only a few combination results for such groups. Our main theorem is such a result. It states that if G is a graph product on a finite graph Γ and each vertex group is finitely presented, then G has non-semistable fundamental group at ∞ if and only if there is a vertex v of Γ such that Gv is not semistable, and the subgroup of G generated by the vertex groups of vertices adjacent to v is finite (equivalently lk(v) is a complete graph and each vertex group of lk(v) is finite). Hence if one knows which vertex groups of G are not semistable and which are finite, then an elementary inspection of Γ determines whether or not G has semistable fundamental group at ∞. [ABSTRACT FROM AUTHOR]

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in CPn+1$\mathbb {CP}^{n+1}$. We prove that for n≥5$n\ge 5$, the Galois covers of any surfaces of minimal degree are simply‐connected surfaces of general type. [ABSTRACT FROM AUTHOR]

In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it "Core fundamental groupoid" and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., "f: M N is a homeomorphism if and only if f#: p1M 1N is a topological groupoid isomorphism". This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid. [ABSTRACT FROM AUTHOR]

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let S be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane ℙ 2 . Such S has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of X = ℙ 2 ∖ S [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. X has a 9-fold Galois covering space X ̃ given by the complement of the Hesse configuration of 12 lines in ℙ 2 . Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of π 1 (X ̃) , which we carry out in this note. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give an intrinsic fundamental group and coverings for categories enriched over an abelian symmetric monoidal category (V , ⊗ , I). We relate this to various categorical constructions such as the universal covering, smash products and connected gradings of a small V -category B. Finally, we apply these methods to obtain an extension of Cohen-Montgomery duality theorem for coactions to the symmetric monoidal category (V , ⊗ , I). [ABSTRACT FROM AUTHOR]

The present paper focuses on the study of fundamental group of Minkowski space with the fine topology which has been found to contain uncountably many subgroups isomorphic to the additive group of integers. [ABSTRACT FROM AUTHOR]

Our main purpose in this paper is to give a counterexample to some interesting conjectures on a rational fibred surface with multiple fibers proposed by R. V. Gurjar and D.-Q. Zhang. [ABSTRACT FROM AUTHOR]

It is widely known that the fundamental group of a Lie group, and in general a symmetric space, is abelian. In the current paper it is demonstrated that any finitely generated abelian group is the fundamental group of a compact Lie group. In addition, it is proved that for any arbitrary group there is a differentiable manifold of dimension greater than 3 whose fundamental group is that arbitrary group. [ABSTRACT FROM AUTHOR]

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotropy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotropy quotient of the Hopf algebroid corresponding to flat connections over the digraph. [ABSTRACT FROM AUTHOR]

Generalizing MacPherson-Vilonen's method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on C² stratified with respect to a singular plane curve and the category of n-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case. [ABSTRACT FROM AUTHOR]

In this paper we study the cubical stuctures and fundamental groups of real toric spaces. We give an explicit presentation of the fundamental group of the real toric space over a simple polytope. Then using this presentation, we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view. [ABSTRACT FROM AUTHOR]

It has been recently observed that a topological decomposition of the Pauli group, as central product of the quaternion group of order eight and the cyclic group of order four, influences some significant dynamical systems in mathematical physics. The connection between groups of symmetries and dynamical systems is in fact well known, but looking specifically at the algebraic and topological decompositions of the Pauli group, we find conditions for the existence of a Riemannian 3-manifold whose fundamental group is epimorphically mapped onto a central product. [ABSTRACT FROM AUTHOR]

In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3‐space R3. An embedding of a graph into R3 is said to be linear, if it sends every edge to be a line segment. And we say that an embedding f of a graph G into R3 is free, if π1(R3−f(G)) is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In the 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: How much can the complete graph Kn be enlarged so that the linear‐freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least 3 is linearly free, if it has less than eight vertices. The conditional inequality is strict, because we found a graph with eight vertices which are not linearly free. It is also proved that for n,m≤6 the complete bipartite graph Kn,m is linearly free. [ABSTRACT FROM AUTHOR]

For a formal scheme $\mathfrak {X}$ of finite type over a complete rank-one valuation ring, we construct a specialization morphism \[ \pi^{\mathrm{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}_1(\mathfrak{X}_k) \] from the de Jong fundamental group of the rigid generic fiber to the Bhatt–Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak {X}$ and normalizations of the irreducible components of $\mathfrak {X}_k$ , and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent. [ABSTRACT FROM AUTHOR]

This study aims to investigate L B -valued GFA from algebraic and topological perspectives, where L stands for residuated lattice and B is a set of propositions about the general fuzzy automata, in which its underlying structure is a complete infinitely distributive lattice. Further, the concepts of L B -valued general fuzzy automata (or simply L B -valued GFA) contractible spaces, L B -valued GFA path homotopy, L B -valued GFA retraction, L B -valued GFA deformation retraction, L B -valued GFA path connected space and L B -valued GFA homotopy equivalent space are introduced and explicated. In addition, L B -valued GFA fundamental groups are proposed and studied. Regarding these issues, some properties are also established and explained. [ABSTRACT FROM AUTHOR]

Let Y be an integral projective complex curve. To representations of the topological fundamental group of Y in the general linear group, we associate generalised parabolic vector bundles and Hitchin pairs. We use this correspondence to study the vector bundles and Hitchin pairs on Y associated to representations of the fundamental group in case the curve has singularities of type A 2 s , A 2 s - 1 or ordinary s-points. [ABSTRACT FROM AUTHOR]

Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$. Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject. [ABSTRACT FROM AUTHOR]

Studying self-similar solutions of geometric flows on manifolds plays an important role in understanding geometrical and topological properties of underlying manifolds. In this paper, we prove that a complete shrinking Ricci–Bourguignon harmonic flow soliton ((M , g) , (N , h) , ϕ , X , ρ , λ) is compact if and only if | | X | | is bounded on M. Also, we show that a complete shrinking Ricci–Bourguignon harmonic flow soliton has finite fundamental group. [ABSTRACT FROM AUTHOR]

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov n -space X with curvature bounded below, i.e. small loops at p ∈ X generate a subgroup of the fundamental group of the unit ball B 1 (p) that contains a nilpotent subgroup of index ≤ w (n) , where w (n) is a constant depending only on the dimension n. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence. (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets. [ABSTRACT FROM AUTHOR]

Mehta and Seshadri proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with the set of equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this paper, we discuss the Mehta–Seshadri correspondence over an irreducible projective curve with at most nodes as singularities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of fundamental group for soft topological spaces. To do so, we define soft paths, soft loops and the notion of ξ -soft path homotopy, and study some of their basic properties. We also show that the fundamental group of an ε -soft topological group is commutative. Finally, we define the category soft topological of spaces and prove that π 1 soft is a functor from this category to the category of groups. [ABSTRACT FROM AUTHOR]

This paper reviews some aspects of simplicial complexes. The aspects reviewed in this paper are very key in building undoubtedly strong concept as reviewed on simplicial complexes. The aspects reviewed take into a consideration: how simplices are used to construct simplicial complexes. Key areas covered at the preliminary part of this paper are simplices, face of a simplex, more on facet of a simplex, orientation of simplices, boundary of simplices, interior of a simplex etc. Geometric objects such as points, edges, triangles and tetrahedrons were the materials used in constructing simplicial complexes. We also looked at the geometric realization theorem, skeleton of simplicial complexes, maximal elements and free face, theorems under simplicial complexes and creation of simplicial complexes. At the results and discussion part, Betti numbers and Euler characteristic of simplicial complexes were computed and chimed in with real life examples. The paper was finalized by chipping in some applications of simplicial complexes. In all, the paper has opened up more and interesting study into simplicial complexes and algebraic topology at large. [ABSTRACT FROM AUTHOR]

Let X be an algebraic surface of degree 5, which is considered a branch cover of CP² with respect to a generic projection. The surface has a natural Galois cover with Galois group S5 . In this paper, we deal with the fundamental groups of Galois covers of degree 5 surfaces that degenerate to nice plane arrangements; each of them is a union of five planes such that no three planes meet in a line. [ABSTRACT FROM AUTHOR]

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued smooth functions from some subspace of the space of smooth functions with isolated singularities on 2-torus T² with respect to the action of the group of diffeomorphisms of T². [ABSTRACT FROM AUTHOR]

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space. [ABSTRACT FROM AUTHOR]

Graph manifolds are compact orientable 3–manifolds obtained by gluing several copies of D 2 × S 1 and N 2 × S 1 together by homeomorphisms of some components of their boundaries (D 2 is the 2–disc and N 2 denotes the 2–disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots. [ABSTRACT FROM AUTHOR]

In this paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups G n k and Γ n k . This is the step towards "complexification" of the G n k and Γ n k approach first developed in [V. O. Manturov, D. A. Fedoseev, S. Kim and I. M. Nikonov, On groups G n k and Γ n k : A study of manifolds, dynamics, and invariants, preprint (2019), arXiv:1905.08049]. [ABSTRACT FROM AUTHOR]

Suppose Γ is a discrete group, and α ∈ Z 3 (B Γ ; A) , with A an abelian group. Given a representation ρ : π 1 (M) → Γ , with M a closed 3-manifold, put F (M , ρ) = 〈 (B ρ) ∗ [ α ] , [ M ] 〉 , where B ρ : M → B Γ is a continuous map inducing ρ which is unique up to homotopy, and 〈 − , − 〉 : H 3 (M ; A) × H 3 (M ; ℤ) → A is the pairing. We extend the definition of F (M , ρ) to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing F (M , ρ) when M is given by a surgery along a link L ⊂ S 3 . In particular, the Chern–Simons invariant can be computed this way. [ABSTRACT FROM AUTHOR]

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π -orbifold group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank 2 are 2 -bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered. [ABSTRACT FROM AUTHOR]

The paper is a survey of my research related to applications of abelian groups to non-commutative groups some of which are direct ones, but the emphasis lies on indirect ones. [ABSTRACT FROM AUTHOR]

Here, it is shown a compact generalized quasi-Einstein or briefly a GQE manifold has the finite fundamental group. Next, under certain assumptions, a rigidity result shows that a compact GQE manifold is diffeomorphic to the standard n-sphere. As a corollary, if the scalar curvature is constant, then it is isometric to the standard n-sphere. Meanwhile, a trivialization criterion and existence of harmonic vector fields on a GQE are studied. [ABSTRACT FROM AUTHOR]

In this paper, we show that each soft topological group is a strong small soft loop transfer space at the identity element. This indicates that the soft quasitopological fundamental group of a soft connected and locally soft path connected space, is a soft topological group. [ABSTRACT FROM AUTHOR]

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid Π 1 (X) has well-defined dense products if and only if X admits a generalized universal covering space. [ABSTRACT FROM AUTHOR]

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the first result, we give an upper bound for the stable commutator length of generalized torsion elements. [ABSTRACT FROM AUTHOR]

Topological surgery in dimension 3 is intrinsically connected with the classification of 3 -manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3 -dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3 -dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3 -dimensional surgery as rotations of the decompactified 2 -sphere. Each rotation produces a different decomposition of the 3 -sphere which corresponds to a different visualization of the 4 -dimensional process of 3 -dimensional surgery. [ABSTRACT FROM AUTHOR]

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of modified flag curvature for Finsler metrics which naturally come from Hessian comparison theorem. We then establish a Toponogov-type comparison theorem for Finsler manifolds with nonnegative-modified flag curvature. As its applications, we prove that the fundamental group of any forward complete Finsler manifold (M, F) with nonnegative-modified flag curvature is finitely generated; furthermore, if additional M has bounded Cartan tensor, then M is of finite topological type. [ABSTRACT FROM AUTHOR]

Here it is shown that any Finslerian compact Yamabe soliton with bounded above scalar curvature is of constant scalar curvature. Furthermore, this extension of Yamabe solitons is developed for inequalities and among the others, it is proved that a forward complete non-compact shrinking Yamabe soliton has finite fundamental group and its first cohomology group vanishes, providing the scalar curvature is strictly bounded above. [ABSTRACT FROM AUTHOR]

For Riemannian manifolds with a smooth measure (M , g , e − f d v g) , we prove a generalized Myers compactness theorem when Bakry–Emery Ricci tensor is bounded from below and f is bounded. [ABSTRACT FROM AUTHOR]

Let K be a nontrivial knot in S 3 with the exterior E (K) , and denote π 1 (E (K)) by G (K). We prove that for any hyperbolic knot K and any nontrivial element g ∈ G (K) , there are only finitely many Dehn fillings of E (K) which trivialize g. We also demonstrate that there are infinitely many nontrivial elements in G (K) which cannot be trivialized by nontrivial Dehn fillings. [ABSTRACT FROM AUTHOR]

In this paper, we consider polygroup 〈 P , ∘ , 1 , − 1 〉 and prove necessary and sufficient conditions such that P is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say G , classifies in the class [ G ]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [ ℤ 2 ] and [ ℤ 3 ] are 16 and 3, respectively, and the classes [ ℤ 2 × ℤ 2 ] and [ ℤ 4 ] are singleton. [ABSTRACT FROM AUTHOR]