The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

• 成果类型: Article
• 署名作者: Etingof, Pavel; Henriques, Andre; Kamnitzer, Joel; Rains, Eric M.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.171.731
• 发表日期: 2010
• 页码: 731-777
• 关键词: quantum cohomology

摘要

We compute the Poincare polynomial and the cohomology algebra with rational coefficients of the manifold M-n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M-n. As was shown by the fourth author, the cohomology of M-n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M-n is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld's theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra L-n of H* (M-n, Q) (associated to such quasibialgebras) factors through the the natural projection of L-n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M-n. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces M-n are not formal starting from n D = 6.