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TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

Part of: Families, fibrations Representation theory of groups

Published online by Cambridge University Press:  11 October 2011

KAZUNORI NAKAMOTO  and
TAKESHI TORII
KAZUNORI NAKAMOTO*
Affiliation:
Center for Life Science Research, University of Yamanashi, Yamanashi 409-3898, Japan (email: nakamoto@yamanashi.ac.jp)
TAKESHI TORII
Affiliation:
Department of Mathematics, Okayama University, Okayama 700-8530, Japan (email: torii@math.okayama-u.ac.jp)
*
✉For correspondence; e-mail: nakamoto@yamanashi.ac.jp
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Abstract

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In this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−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 (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

Keywords

moduli of representationsrepresentation varietyBorel moldfree monoid

MSC classification

Secondary: 14D22: Fine and coarse moduli spaces 20C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 1 , August 2011 , pp. 55 - 87
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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