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Published March 2010 | Submitted
Journal Article Open

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

Abstract

We compute the Poincaré 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 = 6.

Additional Information

© 2010 Annals of Mathematics. Received July 25, 2005; Revised May 13, 2007. The authors are grateful to L. Avramov, C. De Concini, J. Morava, J. Morgan, and B. Sturmfels, for useful discussions and references. P.E. thanks the mathematics department of ETH (Zurich) for hospitality. The work of P.E. was partially supported by the NSF grant DMS-0504847 and the CRDF grant RM1-2545-MO-03. E.R. was supported in part by NSF Grant No. DMS-0401387. J.K. thanks the mathematics department of EPFL for hospitality. The work of J.K. was supported by NSERC and AIM. Finally, we would like to mention that at many stages of this work we made significant use of the Magma computer algebra system for algebraic computations.

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