Principal bundles on two-dimensional CW-complexes with disconnected structure group

2022 ◽  
pp. 1-16
Author(s):  
André G. Oliveira
Keyword(s):  
Lie Group ◽  
Classical Result ◽  
Topological Classification ◽  
Characteristic Classes ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Dimension 2 ◽  
Disconnected Structure

Abstract Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Cohomology of Homotopy 2-types

2019 ◽  
pp. 379-422
Author(s):  
Graham Ellis
Keyword(s):  
Perturbation Theory ◽  
Projective Plane ◽  
Integral Homology ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Crossed Modules ◽  
Homological Perturbation Theory ◽  
Homological Perturbation ◽  
Manual Classification

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.

Unstable homotopy classification of

1996 ◽  
Vol 119 (1) ◽  
pp. 119-137 ◽  
Author(s):  
John Martino ◽  
Stewart Priddy
Keyword(s):  
Lie Group ◽  
Compact Lie Group ◽  
Classifying Space ◽  
Homotopy Classification ◽  
Completion Process

For nilpotent spaces p-completion is well behaved and reasonably well understood. By p–completion we mean Bousfield–Kan completion with respect to the field Fp [BK]. For non-nilpotent spaces the completion process often has a chaotic effect, this is true even for small spaces. One knows, however, that the classifying space of a compact Lie group is Fp-good even though it is usually non-nilpotent.

TOPOLOGY AND LATTICES

1989 ◽  
Vol 04 (16) ◽  
pp. 1537-1547
Author(s):  
RADEL BEN-AV ◽  
SORIN SOLOMON
Keyword(s):  
Closed Form ◽  
Topological Charge ◽  
Spin Model ◽  
Topological Classification ◽  
Gauge Model ◽  
Homotopy Classes ◽  
Continuous Deformation ◽  
Definition Of ◽  
The Continuum

The concept of interpolation relates lattice configurations to continuum configurations. This relation induces from the continuum to the lattice the definitions of “continuous deformation”, topological classification and homotopy classes. The lattice homotopy classes obtained this way are separated by boundaries made out of “exceptional configurations” (EC). The EC boundaries allow the topological classification of the lattice configurations even in models in which it is impossible (or cumbersome) to give in a closed form, a definition of the topological charge in terms of lattice variables. We give the description of the EC boundaries for the 2-dimensional XY spin model and the 4-dimensional SU(2) gauge model.

TOPOLOGY AND LATTICES

1990 ◽  
Vol 05 (02) ◽  
pp. 427-437
Author(s):  
RADEL BEN-AV ◽  
SORIN SOLOMON
Keyword(s):  
Topological Charge ◽  
Spin Model ◽  
Topological Classification ◽  
Gauge Model ◽  
Homotopy Classes ◽  
Continuous Deformation ◽  
Definition Of ◽  
Close Form ◽  
The Continuum

The concept of interpolation relates lattice configurations to continuum configurations. This relation induces from the continuum to the lattice the definitions of “continuous deformation”, topological classification and homotopy classes. The lattice homotopy classes obtained this way are separated by boundaries made out of “exceptional configurations” (EC). The EC boundaries allow the topological classification of the lattice configurations even in models in which it is impossible (or cumbersome) to give in a close form, a definition of the topological charge in terms of lattice variables. We give the description of the EC boundaries for the 2 dimensional XY spin model and the 4 dimensional SU(2) gauge model.

scholarly journals Wavefront dislocations reveal the topology of quasi-1D photonic insulators

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Clément Dutreix ◽  
Matthieu Bellec ◽  
Pierre Delplace ◽  
Fabrice Mortessagne
Keyword(s):  
Local Density ◽  
Topological Defects ◽  
Wave Functions ◽  
Topological Classification ◽  
Local Density Of States ◽  
Topological Phase Transition ◽  
Interference Fringes ◽  
Classical Wave ◽  
Phase Singularities

AbstractPhase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation – a 2D phase singularity – in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

Topological classification of nodal-line semimetals in square-net materials

2021 ◽  
Vol 103 (16) ◽  
Author(s):  
Inho Lee ◽  
S. I. Hyun ◽  
J. H. Shim
Keyword(s):  
Nodal Line ◽  
Topological Classification

Lie group classification of the N-th-order nonlinear evolution equations

2011 ◽  
Vol 54 (12) ◽  
pp. 2553-2572 ◽  
Author(s):  
ShouFeng Shen ◽  
ChangZheng Qu ◽  
Qing Huang ◽  
YongYang Jin
Keyword(s):  
Lie Group ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Group Classification ◽  
Nonlinear Evolution Equations ◽  
Lie Group Classification

Invariants of weak equivalence in primitive matrices

2000 ◽  
Vol 20 (2) ◽  
pp. 611-626 ◽  
Author(s):  
RICHARD SWANSON ◽  
HANS VOLKMER
Keyword(s):  
Inverse Limit ◽  
Direct Application ◽  
Topological Classification ◽  
Weak Equivalence ◽  
Transition Matrices ◽  
Inverse Limit Spaces ◽  
Positive Dimension ◽  
One Dimensional ◽  
Dimension Group

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

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