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

By Pavel Etingof, A. Henriques, Joel Kamnitzer, Eric M. Rains. 2005

116 citations recorded by the source corpus.

We compute the Poincare 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a50719446c3c3d5c30f6ed28aece6e0bcd13398d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Gröbner–Shirshov bases and their calculation (2013, background): “Corollary 16 ([100]) The Drinfeld–Kohno Lie algebra Ln is an iterated semidirect product of free Lie algebras Ai generated by Ti = {ti j | i < j ≤ n − 1}, for i = 1, .”
Groups and Lie algebras corresponding to the Yang–Baxter equations (2005, result): “tional holonomy Lie algebras). The same holds for the groups QTrn, n≥ 4. Summarizing, we may say that the properties of the groups Trn, QTr nare similar to those of the pure cactus group Γ studied in [EHKR]. Remark. 1. We warn the reader that the published version of this paper, [BEER], contains a s”
Open Gromov-Witten disk invariants in the presence of an anti-symplectic involution (2013, background): “The real moduli space RMk,0 is not orientable if k > 4 as shown in [2] and [5].”