  We describe in detail a counterexample to the topological classification of
free fermion systems. We deal with a one dimensional chain of Majorana fermions
with an unusual T symmetry. The topological invariant for the free fermion
classification is an integer, but with the introduction of interactions it
becomes well defined only modulo 8. We illustrate this in the microscopic model
of the Majorana chain by constructing an explicit path between two distinct
free phases whose topological invariants are equal modulo 8, along which the
system remains gapped. The path goes through a strongly interacting region. We
also find the field theory interpretation of this phenomenon. There is a second
order phase transition between the two phases in the free theory which can be
avoided by going through the strongly interacting region. We show that this
transition is in the 2D Ising universality class, where a first order phase
transition line, terminating at a second order transition, can be avoided by
going through the analogue of a high temperature paramagnetic phase. In fact,
we construct the full phase diagram of the system as a function of the thermal
operator (i.e. the mass term that tunes between the two phases in the free
theory) and two quartic operators, obtaining a first order Peierls transition
region, a second order transition region, and a region with no transition.
