### Abstract

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.

Original language | English |
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Pages (from-to) | 55-87 |

Number of pages | 33 |

Journal | Journal of the Australian Mathematical Society |

Volume | 91 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1 2011 |

### Keywords

- Borel mold
- free monoid
- phrases moduli of representations
- representation variety

### ASJC Scopus subject areas

- Mathematics(all)