246 research outputs found
The principal fibration sequence and the second cohomotopy set
Let be a principal fibration with classifying map . It
is well-known that the group acts on with orbit space
the image of p_#, where p_#: [X,E] -> [X,B]. The isotropy subgroup of the
map of to the base point of is also well-known to be the image of . The isotropy subgroups for other maps can definitely
change as does.
The set of homotopy classes of lifts of to the free loop space on is
a group. If has a lift to , the set p_#^{-1}(f) is identified with the
cokernel of a natural homomorphism from this group of lifts to .
As an example, is enumerated for 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
variables, as binary neural networks and cellular automata. The main difficulty
is the choice of a suitable topology to study the limit . 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
Topological Defects and Gapless Modes in Insulators and Superconductors
We develop a unified framework to classify topological defects in insulators
and superconductors described by spatially modulated Bloch and Bogoliubov de
Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with
adiabatic parameters r surrounding the defect and belong to any of the ten
symmetry classes defined by time reversal symmetry and particle-hole symmetry.
The topological classes for such defects are identified, and explicit formulas
for the topological invariants are presented. We introduce a generalization of
the bulk-boundary correspondence that relates the topological classes to defect
Hamiltonians to the presence of protected gapless modes at the defect. Many
examples of line and point defects in three dimensional systems will be
discussed. These can host one dimensional chiral Dirac fermions, helical Dirac
fermions, chiral Majorana fermions and helical Majorana fermions, as well as
zero dimensional chiral and Majorana zero modes. This approach can also be used
to classify temporal pumping cycles, such as the Thouless charge pump, as well
as a fermion parity pump, which is related to the Ising non-Abelian statistics
of defects that support Majorana zero modes.Comment: 27 pages, 15 figures, Published versio
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