  In this paper we tackle the problem of constructing explicit examples of
topological cocycles of Roberts' net cohomology, as defined abstractly by
Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum
field theory on the two dimensional Einstein cylinder. After deriving some
crucial results of the algebraic framework of quantization, we address the
problem of the construction of the topological cocycles. All constructed
cocycles lead to unitarily equivalent representations of the fundamental group
of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces).
The construction is carried out using only Cauchy data and related net of local
algebras on the circle.
