Narrow Results

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree $d$ and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces ${{𝒫}}_d^{c\mathrm{\Theta }}$ of real monic univariate polynomials of degree $d$ whose real divisors avoid sequences of root multiplicities, taken from a given poset $\mathrm{\Theta }$ of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of ${{𝒫}}_d^{c\mathrm{\Theta }}$ and of some related topological spaces. We find explicit presentations for the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in $d$. We also show that ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ stabilizes for $d$ large. The mechanism that generates ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree $d$ polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder $ℝ\times S^1$, whose images avoid the tangency patterns from $\mathrm{\Theta }$ with respect to the generators of the cylinder.

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$-manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) ${\pi }_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2$, $3$ or $4$, or (b) $p=2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2$, $3$ or $4$, then $\mathrm{\text{dim}}{H}_1(M;F_p)<157.763\cdot \mathrm{\text{vol}}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that ${\pi }_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g=2,\dots ,8$. These results should be compared with those of our previous paper “The ratio of homology rank to hyperbolic volume, I,” in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by this paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of ${\pi }_1(M)$ in terms of $\mathrm{\text{vol}}M$, assuming that either ${\pi }_1(M)$ is $9$-free, or $M$ is closed and ${\pi }_1(M)$ is $5$-free.

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.

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 ${\mathbb{P}}^2$. 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 $X={\mathbb{P}}^2\setminus S$ [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 $\tilde{X}$ given by the complement of the Hesse configuration of 12 lines in ${\mathbb{P}}^2$. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of ${\pi }_1(\tilde{X})$, which we carry out in this note.

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.

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 $((M,g),(N,h),\varphi ,X,\rho ,\lambda )$ is compact if and only if $\left|\right|X\left|\right|$ is bounded on $M$. Also, we show that a complete shrinking Ricci–Bourguignon harmonic flow soliton has finite fundamental group.

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.

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.

In this paper, we study closed orientable Euclidean manifolds which are also known as flat three-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, Fox showed that the 3-torus is not a double branched covering of the 3-sphere. So, it is not a hyperelliptic manifold. In this paper, we show that all the remaining Euclidean 3-forms are hyperelliptic manifolds.

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

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 $p\in X$ generate a subgroup of the fundamental group of the unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, 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.

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 ${\mathrm{\Gamma }}_n^k$. This is the step towards “complexification” of the $G_n^k$ and ${\mathrm{\Gamma }}_n^k$ approach first developed in [V. O. Manturov, D. A. Fedoseev, S. Kim and I. M. Nikonov, On groups $G_n^k$ and ${\mathrm{\Gamma }}_n^k$: A study of manifolds, dynamics, and invariants, preprint (2019), arXiv:1905.08049].

Suppose $\mathrm{\Gamma }$ is a discrete group, and $\alpha \in Z^3(B\mathrm{\Gamma };A)$, with $A$ an abelian group. Given a representation $\rho :{\pi }_1(M)\to \mathrm{\Gamma }$, with $M$ a closed 3-manifold, put $F(M,\rho )=〈{(B\rho )}^{\ast }[\alpha ],[M]〉$, where $B\rho :M\to B\mathrm{\Gamma }$ is a continuous map inducing $\rho$ which is unique up to homotopy, and $〈-,-〉:H^3(M;A)\times H_3(M;\mathbb{Z})\to A$ is the pairing. We extend the definition of $F(M,\rho )$ to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing $F(M,\rho )$ when $M$ is given by a surgery along a link $L\subset S^3$. In particular, the Chern–Simons invariant can be computed this way.

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 $\pi$-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.

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.

In this paper, we consider polygroup $〈P,\circ ,1{,}^{-1}〉$ 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 $[{\mathbb{Z}}_2]$ and $[{\mathbb{Z}}_3]$ are 16 and 3, respectively, and the classes $[{\mathbb{Z}}_2\times {\mathbb{Z}}_2]$ and $[{\mathbb{Z}}_4]$ are singleton.

In this work, we prove that if a triangular algebra $A$ admits a strongly simply connected universal Galois covering for a given presentation, then the fundamental group associated to this presentation is free.

An analog of extended real line, $n$-sphere and $n$-disc in the setting of digital topology has been provided and the fundamental group of the digital circle has been computed which comes out to be the additive group $\mathbb{Z}$ of integers.

Let $Y(n,p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial–Meshulam model, and let $Y\sim Y(n,p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber and Kappeler, it is shown that for $p=o(1/n)$ with high probability ${\pi }_1(Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ which satisfy $3/n<p\ll n^{-46/47}$ with high probability, ${\pi }_1(Y)$ is not free. Here, we improve on both these results to show that there are explicit constants ${\gamma }_2<c_2<3$, so that for $p<{\gamma }_2/n$ with high probability $Y$ has free fundamental group and that for $p>c_2/n$ with high probability $Y$ has fundamental group which either is not free or is trivial.

Given a compact geodesic space $X$, we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.