Complete Topological Classification Research Articles

A Complete Topological Classification of the Space of Baire Functions on Ordinals

Considering the spaces Bp[1, α] of all Baire functions x: [1, α] → ℝ on the ordinal segments [1, α] that are endowed with the topology of pointwise convergence, we give a complete topological classification of these spaces.

G-degree for singular manifolds

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-degree 6 is obtained in dimension 3, where all 4-colored graphs represent singular manifolds.

Semi-local Liouville equivalence of complex Hamiltonian systems defined by rational Hamiltonian

Consider the holomorphic function f=az2+R(w), where z and w are complex variables and R is a rational function. Let Dξ0,ε be a small disc around ξ0∈C. Function f defines the foliation in the neighborhood f−1(Dξ0,ε) of the (singular) fiber f−1(ξ0). We give a complete topological classification of such foliations.

Classification of crystalline topological semimetals with an application to Na3Bi

Topological phases can not only be protected by internal symmetries (e.g., time-reversal symmetry), but also by crystalline symmetries, such as reflection or rotation symmetry. Recently a complete topological classification of reflection symmetric insulators, superconductors, nodal semimetals, and nodal superconductors has been established. In this article, after a brief review of the classification of reflection-symmetry-protected semimetals and nodal superconductors, we discuss an example of a three-dimensional topological Dirac semimetal, which exhibits time-reversal symmetry as well as reflection and rotation symmetries. We compute the surface state spectrum of this Dirac semimetal and identify the crystal lattice symmetries that lead to the protection of the surface states. We discuss the implications of our findings for the stability of the Fermi arc surface states of the Dirac material Na3Bi. Our analysis suggests that the Fermi arc of Na3Bi is gapped except at time-reversal invariant surface momenta, which is in agreement with recent photoemission measurements.

A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces

In a paper of Oshemkov and Sharko, three-colour graphs were used to make the topological equivalence of Morse-Smale flows on surfaces obtained by Peixoto more precise. In the present paper, in the language of three-colour graphs equipped with automorphisms, we obtain a complete (including realization) topological classification of gradient-like cascades on surfaces. Bibliography: 25 titles.

Existence of Majorana-Fermion Bound States on Disclinations and the Classification of Topological Crystalline Superconductors in Two Dimensions

We prove a topological criterion for the existence of a zero-energy Majorana bound state on a disclination, a rotation symmetry breaking point defect, in fourfold symmetric topological crystalline superconductors (TCS) in two dimensions. We first establish a complete topological classification of TCS using the Chern invariant and three integral rotation invariants. By analytically and numerically studying disclinations, we algebraically deduce a Z2 index that identifies the parity of the number of Majorana zero modes at a disclination. Surprisingly, we also find weakly protected Majorana fermions bound at the corners of superconductors with trivial Chern and weak invariants.

Topological equivalence of finitely determined real analytic plane-to-plane map germs

Generic smooth map germs $({\mathsf R}^2,0)\to ({\mathsf R}^2,0)$ are topologically equivalent to cones of mappings $S^1\to S^1$. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determined real analytic map germs $({\mathsf R}^2,0)\to ({\mathsf R}^2,0)$.

TOPOLOGY AND PERIODIC ORBITS OF RING-SHAPED POTENTIALS AS A GENERALIZED 4-D ISOTROPIC OSCILLATOR

In this work we study a generalized integrable biparametric family of 4-D isotropic oscillators. This family allows to treat, in a unified way, oscillators defined by the potentials given by Hartmann and Quesne and other ring-shaped systems. Using the Liouville–Arnold theorem and the analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. By this topological study and the calculation of the action-angle variables we obtain the full classification of periodic and quasiperiodic orbits for this system.

Topological classification of families of diffeomorphisms without small divisors

The author gives a complete topological classification for germs of one-parameter families of one-dimensional complex analytic diffeomorphisms without small divisors. In the non-trivial cases the topological invariants are given by some functions attached to the fixed points set plus the analytic class of the element of the family corresponding to the special parameter. The proof is based on the structure of the limits of orbits when we approach the special parameter.

Qualitative analysis of the phase flow of a Manev system in a rotating reference frame

The rotating two-body Manev problem is defined by means of the Hamiltonian function with (α, β)∈ℝ+×ℝ being two structural parameters. Using the Liouville–Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. This analysis, in some aspects very computational, is made with the help of a standard commercial mathematical package.

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

Abstract In this paper, we consider an integrable approximation of the planar motion of a gyrostat in Newtonian interaction with a spherical rigid body. We then describe the Hamiltonian dynamics, in the fibers of constant total angular momentum vector of an invariant manifold of motion. Finally, using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. The results can be applied to study two-body roto-translatory problems where the rotation of one of them has a strong influence on the orbital motion of the system.

The topological classification of minimal surfaces in ℝ3

We give a complete topological classification of properly embedded minimal surfaces in Euclidian three-space

The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the motion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold theorem and some specifics techniques, we obtained a complete topological classification of the phase flow associated to this system. The action-angle variables regions are obtained. These variables allow us to calculate the modified Keplerian elements of this problem useful to elaborate a perturbation theory. The results of this work have a direct application to the study of two body roto-translatory pro-blems where the rotation of one of them influences strongly in the orbital motion of the system. In particular, we can apply these results to binary asteroids.

Classification of homotopy Wall's manifolds

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a Wall's manifold (defined as a mapping torus of a Dold manifold), introduced by Wall in his 1960 Annals paper on cobordism, have been done by determining: (1) the normal invariants of Wall's manifolds, (2) the surgery obstruction of a normal invariant and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy Wall's manifolds (to be made precise in the body of the paper). Consequently classification results of automorphisms (self homeomorphisms, and self PL-homeomorphisms) of Dold manifolds follow.

Topological classification of multiple saddle connections

We give a complete topological classification of three dimensional dynamical systems with a non-return configuration of multiple saddle-connections along the skeleton of a normal crossings divisor. Our classifying space has a geometrical description, is finite and depends only on the distribution of the eigenvalues.

Classification of homotopy real Milnor manifolds

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a real Milnor manifold (hypersurface of type (1,1) of a product of two real projective spaces) have been done by determining: (1) the normal invariants of the real Milnor manifolds, (2) the surgery obstruction of a normal invariant, and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy real Milnor manifolds (to be made precise in the body of the paper).

Topological equivalence for chains of saddle-connections

In this paper we give a complete topological classification for multiple saddle-connections of a real analytic vector field along an axis Γ in an ambient space of dimension three, under the assumption that Γ is the intersection of two invariant surfaces D 1 and D 2.

Orbits of Braid Groups on Cacti

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to S_n or A_n, where n is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on ``cacti'' (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.

Magnetic skyrmions and their lattices in triplet superconductors

Complete topological classification of solutions in SO(3) symmetric Ginzburg-Landau free energy has been performed and a new class of solutions in weak external magnetic field carrying two units of magnetic flux has been identified. These solutions, magnetic skyrmions, do not have singular core like Abrikosov vortices and at low magnetic field become lighter for strongly type II superconductors. As a consequence, the lower critical magnetic field Hc1 is reduced by a factor of log(kappa). Magnetic skyrmions repel each other as 1/r at distances much larger then magnetic penetration depth forming relatively robust triangular lattice. Magnetic induction near Hc1 increases gradually as (H-Hc1)^2. This agrees very well with experiments on heavy fermion superconductor UPt3. Newly discovered Ru based compounds Sr2RuO4 and Sr2YRu(1-x)Cu(x)O6 are other possible candidates to possess skyrmion lattices. Deviations from exact SO(3) symmetry are also studied.

Topological classification of (n−1)-convex sets

We investigate the properties of (n−1)-convex sets associated with the properties of conjugate sets. We give a complete topological classification of (n−1)-convex sets.