  We study stability of multiple conducting edge states in a topological
insulator against all multi-particle perturbations allowed by the time-reversal
symmetry. We model a system as a multi-channel Luttinger liquid, where the
number of channels equals the number of Kramers doublets at the edge. We show
that in the clean system with N Kramers doublets there always exist relevant
perturbations (either of superconducting or charge density wave character)
which always open (N-1) gaps. In the charge density wave regime, (N-1) edge
states get localised. The single remaining gapless mode describes sliding of
'Wigner crystal' like structure. Disorder introduces multi-particle
backscattering processes. While the single-particle backscattering turns out to
be irrelevant, the two-particle process may localise this gapless, in
translation invariant system, mode. Our main result is that an interacting
system with N Kramers doublets at the edge may be either a trivial insulator or
a topological insulator for N=1 or 2, depending on density-density repulsion
parameters whereas any higher number N>2 of doublets gets fully localised by
the disorder pinning irrespective of the parity issue.
