| Abstract: | 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  . Although the characteristic classes of principal bundles are trivial when  , their classical Chern–Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps  to the cohomology group  , where S is null‐cobordant  ‐manifold, once a G‐invariant polynomial p of degree r on  is fixed. For  , this gives a homomorphism  . The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections  , modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set  of connections with vanishing (0, 2)‐part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented. |
