Hopf Invariant Research Articles

Extrinsic and intrinsic nonlinear Hall effects across Berry-dipole transitions

Three-dimensional Hopf insulators are a class of topological phases beyond the tenfold-way classification. At the critical point of the transition between two distinct Hopf insulators with rotational symmetry, the band-touching points are point Berry dipoles with two opposite Berry charges overlapping in a mirror-symmetric way, and carry unique Berry curvature structures leading to a special quantization of Berry flux. Close to such Berry-dipole transitions, we find that the extrinsic and intrinsic nonlinear Hall conductivity tensors in the weakly doped regime are characterized by two universal functions of the ratio between doping level and bulk energy gap, and are directly proportional to the change in Hopf invariant across the transition. Our work suggests that the nonlinear Hall effects display a generalized-sense quantized behavior across Berry-dipole transitions, establishing a correspondence between nonlinear Hall effects and Hopf invariants.

ℰn–Hopf invariants

The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the $E_n$-bar construction on the cochains of $X$ and the homotopy groups of $X$. This pairing gives us a set of invariants of homotopy classes of maps from $S^m$ to a simplicial set $X$, this pairing can detect more homotopy classes of maps than the classical Hopf invariant. The second part of the paper is devoted to combining the $E_n$-Hopf invariants with the Koszul duality theory for $E_n$-operads to get a relation between the $E_n$-Hopf invariants of a space $X$ and the $E_{n+1}$-Hopf invariants of the suspension of $X$. This is done by studying the suspension morphism for the $E_\infty$-operad, which is a morphism from the $E_{\infty}$-operad to the desuspension of the $E_\infty$-operad. We show that it induces a functor from $E_\infty$-algebras to $E_\infty$-algebras, which has the property that it sends an $E_\infty$-model for a simplicial set $X$ to an $E_\infty$-model for the suspension of $X$. We use this result to give a relation between the $E_n$-Hopf invariants of maps from $S^m$ into $X$ and the $E_{n+1}$-Hopf invariants of maps from $S^{m+1}$ into the suspension of $X$. One of the main results we show here, is that this relation can be used to define invariants of stable homotopy classes of maps.

Dispersion of the Arnold’s Asymptotic Ergodic Hopf Invariant and a Formula for Its Calculation

The paper contains an example to describe magnetic fields in a conductive medium. The authors assume that new applications for turbulent magnetic fields in the case the magnetic field is not left- and right- polarized are possible.

Measuring Hopf links and Hopf invariants in a quenched topological Raman lattice

In a recent experimental work [Z. Wu et al., Science 354, 83 (2016)], the PKU-USTC group realized a two-dimensional two-band quantum anomalous Hall model on a square Raman lattice. By quenching the atom-laser detuning of such a Raman lattice, the time-dependent Bloch vectors for each quasimomentum points define a Hopf mapping from quasimomentum-time space $(k_x,k_y,t)\in T^3$ to the Bloch sphere $S^2$. The Hopf links between the preimages of any two Bloch vectors on $S^2$ can be measured experimentally through Bloch state tomography using spin-resolved time-of-flight measurement together with suitable radio-frequency manipulations. The dynamical Hopf invariants, which are quasimomentum-time integrations of Chern-Simons densities, can also be extracted experimentally through certain quench processes. As the Hopf invariant equals the Chern number of the post-quench Hamiltonian, the measurements of Hopf links and Hopf invariants provide an alternative way of understanding the topological phase diagram of the equilibrium system.

Majorana Zero Modes Protected by a Hopf Invariant in Topologically Trivial Superconductors.

Majorana zero modes are usually attributed to topological superconductors. We study a class of two-dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust Majorana zero modes in topological defects. The construction of this minimal single-band model is facilitated by the Hopf map and the Hopf invariant. This work will stimulate investigations of Majorana zero modes in superconductors in the topologically trivial regime.

Branch Processes of Vortex Filaments and Hopf Invariant Constraint on Scroll Wave

In this paper, by making use of Duan's topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function Z(x⃗, t). It is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes (splitting, merging, or encountering) during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the “exclusion principle" in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the “exclusion principle" is also protected by topology.

Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants

In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed. From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and 127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π n , there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem (in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1 manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle (by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required.

A New Route to the Interpretation of Hopf Invariant

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the ϕ-mapping topological current theory. The main purpose of this paper is to present a new theoretical framework, which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space R2n-1. For the sake of this purpose we introduce a topological tensor current, which can naturally deduce the (n-1)-dimensional topological defect in R2n-1 space. If these (n-1)-dimensional topological defects are closed oriented submanifolds of R2n-1, they are just the (n - 1)-dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher-dimensional knots can be constructed.

Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r. To each such link L we associate a geometrically natural characteristic map g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g_L. Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer nu, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r. We show that the pairwise linking numbers p, q and r of the components of L are equal to the degrees of its characteristic map g_L restricted to the 2-dimensional subtori, and that twice Milnor's mu-invariant for L is equal to Pontryagin's nu-invariant for g_L. When p, q and r are all zero, the mu- and nu-invariants are ordinary integers. In this case we use J. H. C. Whitehead's integral formula for the Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with a formula for the fundamental solution of the scalar Laplacian on the 3-torus as a Fourier series in three variables, to provide an explicit integral formula for nu, and hence for mu.

A theory of PWD-structures

A general study is undertaken of product-wedge-diagonal (= PWD) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead's fat-wedge characterization of Lusternik–Schnirelmann category. From another viewpoint PWD-structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD-structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X → Y . A principal result of the paper uses Hopf invariants to formulate a Berstein–Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U * V that has PWD-structure defined canonically via the join structure in terms of diagonal maps on U and V.

Knotted Wave Dislocation with the Hopf Invariant

We study the wave dislocations with an induced gauge potential. The topological current characterized the wave dislocations is constructed with the dual of Abelian gauge field. And the topological charges and locations of the wave dislocations are determined by the φmapping topological current theory. Furthermore, it is shown that the knotted wave dislocations can be described with a Hopf invariant in the wave field. At last we discussed the evolution of the knotted wave dislocations.

Applications of combinatorial groups to Hopf invariant and the exponent problem

Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.

Homotopical dynamics II: Hopf invariants, smoothings and the Morse complex

The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse–Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincaré duality spaces as well as an extension of the Morse complex.

Hopf–Ganea invariants and weak LS category

In this work, we recall the definition of Hopf's invariants given by Ganea and we define a Hopf invariant adapted to a cone decomposition of a space. As Berstein and Hilton did, we apply it for the computation of the weak category of Lusternik and Schnirelmann of a space. An example provided by the cellular decomposition of Sp(3) shows the efficiency of this method.

Gauge fields associated with minimal surfaces and their Hopf invariants

We study gauge fields described by an (n - 1)-form in a -dimensional Riemannian manifold satisfying the equations , . We show that minimal surfaces may be associated with them that have a unique Hopf invariant. Some of these surfaces are stable but the issue is not settled completely.

Hopf invariants of the Berstein-Hilton-Ganea kind

We study Hopf invariants of a kind constructed classically by Berstein, Hilton and Ganea. Beyond basic properties of such Hopf invariants we also consider comparisons to other generalizations of the Hopf invariant as have been made by Nomura (1966), Walker (1979) and recently Marcum. Our principal example here of a Hopf invariant of the Berstein-Hilton-Ganea kind is the Ganea-Hopf invariant GHI: ΣΩ M(f,g) → ΩC f ∗ ΩC g . The main result we establish for GHI is a computational formula, one term of which involves the dual product of Arkowitz.

Homotopy Pullbacks and the Hopf Invariant

Journal of the London Mathematical SocietyVolume s2-19, Issue 1 p. 153-158 Notes and Papers Homotopy Pullbacks and the Hopf Invariant Marshall Walker, Marshall Walker York University, Downsview, OntarioSearch for more papers by this author Marshall Walker, Marshall Walker York University, Downsview, OntarioSearch for more papers by this author First published: February 1979 https://doi.org/10.1112/jlms/s2-19.1.153Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-19, Issue1February 1979Pages 153-158 RelatedInformation

Self-intersections and higher Hopf invariants

IN THIS paper we show how the well known models for loop spaces of Boardman and Vogt [3], James [5], May [9], and Segal[ lo], can be viewed in a natural way as “Thorn spaces for immersions”. Thus homotopy classes of maps into these models correspond to bordism classes of immersed manifolds with certain extra structures. By considering the multiple points of such immersions we obtain operations in homotopy theory. Special cases are the generalised and higher Hopf invariants of James[6], the Hopf ladder of Boardman and Steer[2], and the cohomotopy operations of Snaith [ 121, and Segal[ I I]. In 01 we establish the connection between models of loop spaces and structured immersions. In 02 we describe the process of “taking k-tuple points” in what might be termed an “external Hopf invariant”, and set out its properties in Theorem 2.2. We then get the Segal and Snaith operations by composing with a suitable “forgetful” function. A similar procedure is followed in 03 where the James operations are described. We are grateful to the referee for many helpful comments.

Hopf Invariants for Reduced Products of Spheres

Let $S_m^n$ be the $m$th reduced product complex of the even dimensional sphere ${S^n}$. Using ’cup’-products, James defined a Hopf invariant homomorphism \[ H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf {Z}}\] such that $H_2^n$ is the classical Hopf invariant. Extending the result of Adams on $H_2^n$ we determine the image of $H_m^n$. Partial calculations were made by Hardie and Shar.

Hopf invariants for reduced products of spheres

Let S m n S_m^n be the m m th reduced product complex of the even dimensional sphere S n {S^n} . Using ’cup’-products, James defined a Hopf invariant homomorphism \[ H m n : π m n − 1 ( S m − 1 n ) → Z H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf {Z}} \] such that H 2 n H_2^n is the classical Hopf invariant. Extending the result of Adams on H 2 n H_2^n we determine the image of H m n H_m^n . Partial calculations were made by Hardie and Shar.