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


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

Massachusetts Institute of Technology
Department of Mathematics
77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States

André Henriques

Universiteit Utrecht
Mathematisch Instituut
Postbus 80 010
3508 TA Utrecht
The Netherlands

Joel Kamnitzer

Department of Mathematics
University of Toronto
Room 6290, 40 St. George Street
Toronto, Ontario M5S 2E4

Eric M. Rains

Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
United States