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 D 6.

Right-angled mock reflection and mock Artin groups

“…Because of this connection to stable real curves with marked points, the group J n is sometimes called the cactus group and π 1 (M ) the pure cactus group. The cohomology of M (and, hence, of the pure cactus group) has recently been computed in [6].…”

The Cohomology of Real De Concini–Procesi Models of Coxeter Type

“…One explanation for this connection is that M 0,n (R) can be regarded as the real De Concini-Procesi model of the hyperplane arrangement of type A n−2 ; in this context the relation with poset homology was generalized to an arbitrary subspace arrangement in [15]. It was then natural to try to extend the results of [6] and [14] to the other finite Coxeter types, and that is the goal of the present work.…”

“…Thus part (6) is proved, and the span of {ν ij } is at least contained in the (−1)-eigenspace of σ; however, (6) shows that σ acts trivially on the quotient by the span of {ν ij }, which proves (5).…”

“…This section is brief, since type A was done in [6], type B was effectively done in [10] (though the Poincaré polynomial formula is not as explicit as in type A), type D reduces to type B, and the other types can be calculated directly. (The Euler characteristics in the classical types were calculated in [5], [7], and [1]).…”

“…In [6], the second author and his collaborators studied the rational cohomology ring of M 0,n (R), the manifold of real points of the moduli space of stable genus 0 curves with n marked points. In [14], the second author described this ring as a representation of S n , giving a formula for the graded character.…”