Let G be the fundamental group of a closed manifold X and \alpha: G —> U(n) a finite dimensional unitary representation, i.e. a flat unitary vector bundle over X.
To these data, one can associate a class [\alpha] in the (topological) K-theory group with R/Z-coefficients.
In this talk, after introducing the needed tools, we describe the construction of the class and discuss its role of secondary invariant.
By means of an operator algebraic point of view, the class [\alpha] can be generalised it to the noncommutative setting of a discrete group G suitably acting on a C^*-algebra A.
Based on joint work with Paolo Antonini and Georges Skandalis.
