  Homotopy braid group description including cyclotron motion of charged
interacting 2D particles at strong magnetic field presence is developed in
order to explain, in algebraic topology terms, Laughlin correlations in
fractional quantum Hall systems. There are introduced special cyclotron braid
subgroups of a full braid group with one dimensional unitary representations
suitable to satisfy Laughlin correlation requirements. In this way an
implementation of composite fermions (fermions with auxiliary flux quanta
attached in order to reproduce Laughlin correlations) is formulated within
uniform for all 2D particles braid group approach. The fictitious
fluxes-vortices attached to the composite fermions in a traditional formulation
are replaced with additional cyclotron trajectory loops unavoidably occurring
when ordinary cyclotron radius is too short in comparison to particle
separation and does not allow for particle interchanges along single-loop
cyclotron braids. Additional loops enhance the effective cyclotron radius and
restore particle interchanges. A new type of 2D particles--composite anyons is
also defined via unitary representations of cyclotron braid subgroups. It is
demonstrated that composite fermions and composite anyons are rightful 2D
particles, not auxiliary compositions with fictitious fluxes and are associated
with cyclotron braid subgroups instead of the full braid group, which may open
also a new opportunity for non-Abelian composite anyons for topological quantum
information processing applications, due to richer representations of subgroup
than of a group.
