Topological components of surface group representations and signature

We study the topological components of the surface group representations into SL(2,R) and PSL(2,R). Some components correspond to geometric structures on surface, especially discrete and faithful ones to hyperbolic structures. Utilizing the signature formula established by Kim-Pansu-Wan, we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies. There are modified Milnor-Wood inequality for boundary elliptic and parabolic representations. This is a joint work with X.Wan.