Topology Of Manifolds Research Articles

A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

Quasiregular Mappings and Discrete Group Actions

We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices. We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7-9]).

Manifold topology, observables, and gauge group

The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.

Solutions of the KdV Equation through Analysis of Regular Symmetries

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

Partially Hyperbolic Dynamics and 3-Manifold Topology

This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow.

Local Extraction of 3D Time‐Dependent Vector Field Topology

AbstractWe present an approach to local extraction of 3D time‐dependent vector field topology. In this concept, Lagrangian coherent structures, which represent the separating manifolds in time‐dependent transport, correspond to generalized streak manifolds seeded along hyperbolic path surfaces (HPSs). Instead of expensive and numerically challenging direct computation of the HPSs by intersection of ridges in the forward and backward finite‐time Lyapunov exponent (FTLE) fields, our approach employs local extraction of respective candidates in the four‐dimensional space‐time domain. These candidates are subsequently refined toward the hyperbolic path surfaces, which provides unsteady equivalents of saddle‐type critical points, periodic orbits, and bifurcation lines from steady, traditional vector field topology. In contrast to FTLE‐based methods, we obtain an explicit geometric representation of the topological skeleton of the flow, which for steady flows coincides with the hyperbolic invariant manifolds of vector field topology. We evaluate our approach on analytical flows, as well as data from computational fluid dynamics, using the FTLE as a ground truth superset, i.e., we also show that FTLE ridges exhibit several types of false positives.

3D topological quantum computing

In this paper, we will present some ideas to use 3D topology for quantum computing extending ideas from a previous paper. Topological quantum computing used “knotted” quantum states of topological phases of matter, called anyons. But anyons are connected with surface topology. But surfaces have (usually) abelian fundamental groups and therefore one needs non-Abelian anyons to use it for quantum computing. But usual materials are 3D objects which can admit more complicated topologies. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere (see T. Asselmeyer-Maluga, Quantum Rep. 3 (2021) 153, arXiv:2102.04452 for previous work). The whole system is designed as knotted superconductor, where every crossing is a Josephson junction and the qubit is realized as flux qubit. We discuss the properties of this systems in particular the fluxion quantization by using the A-polynomial of the knot. Furthermore, we showed that 2-qubit operations can be realized by linked (knotted) superconductors again coupled via a Josephson junction.

Enlargeable length-structure and scalar curvatures

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings N↪M of one topological manifold into another. More specifically, we describe an algorithm for computing the homology Hn(X,A) and cohomology Hn(X,A) of a finite connected CW-complex X with local coefficients in a Zπ1X-module A when A is finitely generated over Z. It can be used, in particular, to compute the integral cohomology Hn(X˜H,Z) and induced homomorphism Hn(X,Z)→Hn(X˜H,Z) for the covering map p:X˜H→X associated to a finite index subgroup H<π1X, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree 2 homology group H2(X˜H,Z) distinguishes between the homotopy types of the complements X⊂R4 of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree 1 homology homomorphism H1(p−1(B),Z)→H1(X˜H,Z) distinguishes between the homeomorphism types of the complements X⊂R3 of the granny knot and the reef knot, where B⊂X is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology). Our open source implementation allows the user to experiment with further examples of knots, knotted surfaces, and other embeddings of spaces. We conclude the paper with an explanation of how the cohomology algorithm also provides an approach to computing the set [W,X]ϕ of based homotopy classes of maps f:W→X of finite CW-complexes over a fixed group homomorphism π1f=ϕ in the case where dim⁡W=n, π1X is finite and πiX=0 for 2≤i≤n−1.

Topological dynamics of volume-preserving maps without an equatorial heteroclinic curve

Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.

Classification of dark solitons via topological vector potentials.

Dark solitons are some of the most interesting nonlinear excitations and are considered to be the one-dimensional topological analogs of vortices. However, in contrast to their two-dimensional vortex counterparts, the topological characteristics of a dark soliton are far from fully understood because the topological charge defined according to the phase jump cannot reflect its essential property. Here, similar to the complex extension used in the exploration of the partition-function zeros to depict thermodynamic states, we extend the complex coordinate space to explore the density zeros of dark solitons. Surprisingly we find that these zeros constitute some pointlike magnetic fields, each of which has a quantized magnetic flux of elementary π. The corresponding vector potential fields demonstrate the topology of the Wess-Zumino term and can depict the essential characteristics of dark solitons. Then we classify the dark solitons according to the Euler characteristic of the topological manifold of the vector potential fields. Our study not only reveals the topological features of dark solitons but can also be applied to explore and identify new dark solitons with high topological complexity.

The Integral Manifolds of the N Body Problem

The “N-Body Problem” refers to the study of the dynamics of a system of N point masses, moving under their mutual gravitational attraction. For such a system, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets $${\mathfrak {M}}(c,h)$$ of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. In spite of their fundamental importance in the N-body problem, relatively little is known about the global structure of these spaces. In particular, for non-zero angular momentum, the manifolds exhibit a variety of discontinuities and complexities at collinear configurations. These discontinuities are removed by the introduction of a blow-up construction, which is then exploited to provide a description of the integral manifolds, in the sense of a reduction formula for the homology groups of the manifolds. That reduction formula is then applied to investigate the topology of the integral manifolds for energies just below zero. For four masses, the homology groups for this energy range are fully determined; for arbitrary N, the homology with real coefficients is determined. Comparison to the values for positive energies confirms that for all masses, the topology of the integral manifold changes at zero energy.

English

We present a method for the generation of a pure quad mesh approximating a discrete manifold of arbitrary topology that preserves the patch layout characterizing the intrinsic object structure. A three-step procedure constitutes the core of our approach which first extracts the patch layout of the object by a topological partitioning of the digital shape, then computes the minimal surface given by the boundaries of the patch layout (basic quad layout) and then evolves it towards the object boundaries. The Lagrangian evolution of the initial surface (basic quad layout) in the direction of the gradient of the signed distance function is smoothed by a mean curvature term. The direct control over the global quality of the generated quad mesh is provided by two types of tangential redistributions: area-based, to equally distribute the size of the quads, and angle-based, to preserve quad corner angles. Experimental results showed that the proposed method generates pure quad meshes of arbitrary topology objects, composed of well-shaped evenly distributed elements with few extraordinary vertices.

On reduced complexity of closed piecewise linear 5-manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of $$\mathbb {S}^2 \times \mathbb {S}^3$$ among closed connected spin PL 5-manifolds.

Topological Quantum Computing and 3-Manifolds

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

Homotopy of gauge groups over high-dimensional manifolds

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n− 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold M_{c^2} with double simplex {widetilde{{mathcal {D}}}}_1, seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of M_{c^2}={widetilde{{mathcal {D}}}}_1, leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to {widetilde{{mathcal {D}}}}_1. Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on M_{c^2} as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

A Refined Lines/Regions and Lines/Lines Topological Relations Model Based on Whole-Whole Objects Intersection Components

Refined topological relations play an important role in spatial database quality control. Currently, there is no unified and reasonable method to represent refined line/region and line/line topological relations in two-dimensional (2D) space. In addition, the existing independent line/region and line/line models have some drawbacks such as incomplete type discrimination and too many topological invariants. In this paper, a refined line/region and line/line topological relations are represented uniformly by the sequence, dimension, and topological type of the intersection components. To make the relevant definitions conform to the traditional cognitions in 2D Euclidean space, the (simple) spatial object is defined based on manifold topology, and the spatial intersection components are defined based on the whole-whole object intersection set. Then the topological invariant of node degree is introduced, and the adjacent point kinds (e.g., “Null”, “On”, “In”, and “Out”) are defined to distinguish the intersection component types. Excluding impossible and symmetrical types, 29 types of intersection-lines (including 21 between lines/regions and 8 between lines/lines), and 6 types of intersection-points (including 2 between lines/regions and 4 between lines/lines) are classified. On this basis, a node degree-based whole-whole object intersection sets (N-WWIS) model for refined line/region and line/line topological relations is presented, and it can be combined with the Euler number-based whole object intersection and difference (E-WID) model (coarse level) to form a hierarchical representation method of topological relations. Furthermore, a prototype system based on the N-WWIS model for automatic topological integrity checking is developed and some evaluation experiments are conducted with OpenStreetMap (OSM) data is presented based on the classification of intersection components. The experimental results show that the N-WWIS model will enable the geographic information systems (GIS) community to develop automated topological conflict checking and dealing tools for spatial data updates and quality control.

Properties of the Null Distance and Spacetime Convergence

AbstractThe null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov–Hausdorff, and Sormani–Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that nonuniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

О максимальной размерности алгебр Ли инфинитезимальных аффинных преобразований касательных расслоений с синектической связностью А.П. Широкова

The theory of tangent bundles over a differentiable manifold M be­longs to the geometry and topology of manifolds and is an intensively developing area of the theory of fiber spaces. The foundations of the theo­ry of fibered spaces were laid in the works of S. Eresman, A. Weil, A. Mo­ri­moto, S. Sasaki, K. Yano, S. Ishihara. Among Russian scientists, tangent bund­les were investigated by A. P. Shirokov, V. V. Vishnevsky, V. V. Shu­rygin, B. N. Shapukov and their students. In the study of automorphisms of generalized spaces, the question of infinitesimal transformations of connections in these spaces is of great importance. K. Yano, G. Vrancianu, P. A. Shirokov, I. P. Egorov, A. Z. Pet­rov, A. V. Aminova and others have studied movements in different spac­es. The works of K. Sato and S. Tanno are devoted to the motions and au­tomorphisms of tangent bundles. Infinitesimal affine collineations in tan­gent bundles with a synectic connection were considered by H. Shadyev. At present, the question of the motions of fibered spaces is considered in the works of A. Ya. Sultanov, in which infinitesimal transformations of a bundle of linear frames with a complete lift connection, the Lie algebra of holomorphic affine vector fields in arbitrary Weyl bundles are investi­gated. In this paper we obtain exact upper bounds for the dimensions of Lie algebras of infinitesimal affine transformations in tangent bundles with a synectic connection A. P. Shyrokov.