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

TL;DR

This paper computes the cohomology algebra of the real moduli space of genus 0 curves with n points, revealing its quadratic structure, torsion properties, and operadic relations, with implications for fundamental group structure.

Contribution

It provides the first detailed description of the cohomology ring, operad structure, and torsion phenomena of the real moduli space M_n, including conjectures on Koszulity and fundamental group isomorphisms.

Findings

Cohomology algebra is quadratic and conjecturally Koszul.

Determined 2-local torsion in cohomology.

Identified the rational homology operad as 2-Gerstenhaber.

Abstract

We compute the Poincare polynomial and the cohomology algebra with rational coefficeints 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 E. Rains in arXiv:math/0610743 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.