Topology of the representation varieties with borel mold for unstable cases

In this paper we show that, in the stable case, when m≥2 _n-1 the cohomology ring H ^*(Rep _n(m) _B) of the representation variety with Borel mold Rep _n(m) _B andH ^* ((F _n(C ^m(C ^m)); H δ (C _n)) Λ (s _n-1.., s _n-1) are isomorphic as algebras. Here the degree of si is 2m-'3 when 1i<n. In the unstable cases, when m2n-'2, we also calculate the cohomology group H ^*(Repn(m) B) when n=3,4. In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn ( ²)-PGL n (C) , where Fn ( ²) is the configuration space of n distinct points in ². We regard Rep n (2)B as a scheme over, and show that the Picard group Pic (Rep _n (2)B) of Rep _n (2)B is isomorphic to Z/n. We give an explicit generator of the Picard group.