In the present treatise progress in topological approach to Hall system physics is reported, including recent achievements in graphene. The homotopy methods of braid groups turn out to be of particular convenience in order to grasp peculiarity of 2D charged systems upon magnetic field resulting in Laughlin correlations. The real progress in understanding of structure and role of composite fermions in Hall system is provided. The crucial significance of carrier mobility apart from interaction in creation of the fractional quantum Hall effect (FQHE) is described and supported by recent graphene experiments. Recent progress in FQHE field including topological insulators and optical lattices was reviewed and commented in terms of braid group approach. The braid group methods are presented from more general point of view including proposition of pure braid group application.
Introduction; Elements of Hall System Physics in 2D Spaces; Topological Methods of Describing Systems of Many Particles at Various Manifolds; Cyclotron Braids for Multi-Particle Charged 2D Systems in a Strong Magnetic Field; Recent Progress in FQHE Field; Summary; Comments and Supplements.