Flat connections and cohomology invariants

The main goal of this article is to construct some geometric invariants for the topology of the set. of flat connections on a principal. G-bundle P -> M Although the characteristic classes of principal bundles are trivial when F not equal empty set, their classical Chern-Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps S -> F to the cohomology group H2(r-k) (M, R), where.. is null-cobordant (k-1)-manifold, once a.. -invariant polynomial p of degree r on Lie(G) is fixed. For S = S = Sk-1, this gives a homomorphism pi(k-1)(F) -> H2r-k (M, R). The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections. /Gau.., modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set F-0,F-2 of connections with vanishing (0, 2)-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.