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 $n=6$.
Pavel Etingof 1 ; André Henriques 2 ; Joel Kamnitzer 3 ; Eric M. Rains 4
@article{10_4007_annals_2010_171_731,
author = {Pavel Etingof and Andr\'e Henriques and Joel Kamnitzer and Eric M. Rains},
title = {The cohomology ring of the real locus of the moduli space of stable curves of genus $0$ with marked points},
journal = {Annals of mathematics},
pages = {731--777},
year = {2010},
volume = {171},
number = {2},
doi = {10.4007/annals.2010.171.731},
mrnumber = {2630055},
zbl = {1206.14051},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.731/}
}
TY - JOUR AU - Pavel Etingof AU - André Henriques AU - Joel Kamnitzer AU - Eric M. Rains TI - The cohomology ring of the real locus of the moduli space of stable curves of genus $0$ with marked points JO - Annals of mathematics PY - 2010 SP - 731 EP - 777 VL - 171 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.731/ DO - 10.4007/annals.2010.171.731 LA - en ID - 10_4007_annals_2010_171_731 ER -
%0 Journal Article %A Pavel Etingof %A André Henriques %A Joel Kamnitzer %A Eric M. Rains %T The cohomology ring of the real locus of the moduli space of stable curves of genus $0$ with marked points %J Annals of mathematics %D 2010 %P 731-777 %V 171 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.731/ %R 10.4007/annals.2010.171.731 %G en %F 10_4007_annals_2010_171_731
Pavel Etingof; André Henriques; Joel Kamnitzer; Eric M. Rains. The cohomology ring of the real locus of the moduli space of stable curves of genus $0$ with marked points. Annals of mathematics, Tome 171 (2010) no. 2, pp. 731-777. doi: 10.4007/annals.2010.171.731
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