The principal fibration sequence and the second cohomotopy set

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of p_#, where p_#: [X,E] -> [X,B]. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, \Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set p_#^{-1}(f) is identified with the cokernel of a natural homomorphism from this group of lifts to $[X, \Omega C]$. As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.Comment: 12 page

Cohomotopy sets of 4-manifolds

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.Comment: 20 pages, 6 figures; this version to appear in the FreedmanFest (G&T Monographs, Volume 18

Majorana Fermions and Non-Abelian Statistics in Three Dimensions

We show that three dimensional superconductors, described within a Bogoliubov de Gennes framework can have zero energy bound states associated with pointlike topological defects. The Majorana fermions associated with these modes have non-Abelian exchange statistics, despite the fact that the braid group is trivial in three dimensions. This can occur because the defects are associated with an orientation that can undergo topologically nontrivial rotations. A new feature of three dimensional systems is that there are "braidless" operations in which it is possible to manipulate the groundstate associated with a set of defects without moving or measuring them. To illustrate these effects we analyze specific architectures involving topological insulators and superconductors.Comment: 4 pages, 2 figures, published versio

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.Comment: Proc. 44th ISSAC, July 15--18, 2019, Beijing, Chin

Supercurrent coupling in the Faddeev-Skyrme model

Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Riemannian manifold and the target space is a Kaehler manifold. It is shown that supercurrent coupling destroys the topological stability enjoyed by the usual Faddeev-Skyrme model, so that there can be no globally stable knot solitons in this model. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit-charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. A family of stable solutions is identified, though these may exist only in spaces of even dimension. Finally, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R is unstable for all R. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the uncoupled Faddeev-Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low-energy regime of two-component superconductors.Comment: 17 page

Lagrangian matching invariants for fibred four-manifolds: I

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The `Lagrangian matching invariants' are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold. They fit into a field theory which assigns Floer homology groups to fibred 3-manifolds. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I is devoted to the symplectic geometry of these Lagrangians.Comment: 72 pages, 4 figures. v.2 - numerous small corrections and clarification

Decomposing locally compact groups into simple pieces

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product.Comment: Index added; minor change

Rohlin's invariant and gauge theory III. Homology 4--tori

This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional cohomology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper47.abs.htm

Discrete Dynamical Systems Embedded in Cantor Sets

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $N$ variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit $N\to\infty$. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running top to bottom in figures, to appear in J. Math. Phy

Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.Comment: 22 pages, 8 figures. arXiv admin note: text overlap with arXiv:1101.337