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.
- Borel mold
- free monoid
- phrases moduli of representations
- representation variety
ASJC Scopus subject areas