## The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points.(English)Zbl 1206.14051

The authors compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold $$M_n$$ of real points of the moduli space of stable genus zero curves with $$n$$ marked points. Further, it is shown that the rational homology operad of $$M_n$$ is the operad of Gerstenhaber 2-algebras, and analogies and differences between $$M_n$$ and the configuration space $$C_{n-1}$$ of $$n-1$$-tuples of distinct complex numbers are investigated. In particular it is shown that the spaces $$M_n$$ are not formal for $$n\geq 6$$.
More in detail, let $$\overline{\mathcal{M}}_{0,n}$$ be the moduli space of genus zero curves with $$n$$ marked points (this is a smooth projective variety over $$\mathbb{Q}$$), and let $$M_n=\overline{\mathcal{M}}_{0,n}(\mathbb{R})$$ the manifold of real points of $$\overline{\mathcal{M}}_{0,n}$$. For any ordered $$4$$-element subset $$\{i,j,k,l\}$$ of $$\{1,\dots,n\}$$ there is a natural map $$\phi_{ijkl}:M_n\to M_4\cong \mathbb{P}^1(\mathbb{R})$$ forgetting the points with labels outside $$\{i,j,k,l\}$$, so for any commutative ring $$R$$ there is a natural homomorphism of algebras $$\phi_{ijkl}^*: H^*(M_4;R)\to H^*(M_n;R)$$. Denote by $$\omega_{ijkl}(M_n)\in H^1(M_n;R)$$ the image via $$\phi_{ijkl}^*$$ of the standard generator of $$H^*(M_4;R)\cong\mathbb{Z}$$. The authors introduce, for every $$n\geq 3$$, a quadratic algebra $$\Lambda_n$$ as the skew-commutative algebra generated over $$\mathbb{Z}$$ by elements $$\omega_{ijkl}$$, $$1\leq i,j,k,l\leq n$$, which are antisymmetric in $$ijkl$$ and with defining relations $\omega_{ijkl}+\omega_{jklm}+\omega_{klmi}+\omega_{lmij}+\omega_{mijk}=0\tag{1}$
$\omega_{ijkl}\omega_{ijkm}\tag{2}$
$\omega_{ijkl}\omega_{lmpi}+\omega_{klmp}\omega_{pijk}+\omega_{mpij}\omega_{jklm}=0\tag{3}$ for distinct $$i,j,k,l,m,p$$. The ideal generated by the first two relations contains 2 times the third relation, so that this last relation becomes redundant in $$\Lambda_n\otimes R$$ as soon as $$2$$ is invertible in $$R$$. Moreover, if $$1/2\in R$$, the elements $$\omega_{ijkl}(M_n)$$ satisfy the relations $$(1)$$ and $$(2)$$ above. This is essentially a consequence of the isomophism $$H^2(M_5;\mathbb{Z})\cong \mathbb{Z}/2\mathbb{Z}$$. Therefore, for any commutative ring $$R$$ in which $$2$$ is invertible there is an homorphism of algebras $$f_n^R:\Lambda_n\otimes R\to H^*(M_n;R)$$ mapping $$\omega_{ijkl}$$ to $$\omega_{ijkl}(M_n)$$. The main result of the paper then states that $$f_n^\mathbb{Q}$$ is an isomorphism.
In particular this reduces the computation of the Poincaré polynomial of $$M_n$$ to the computation of the Poincaré polynomyal of the $$\mathbb{Z}$$-module $$\Lambda_n$$, which is shown to be $P_n(t)=\prod_{0\leq k<(n-3)/2}(1+(n-3-2k)^2t).$ Moreover, it is shown in the paper that $$H^*(M_n;\mathbb{Z})$$ does not have 4-torsion, and it is given a description of its 2-torsion. By this, the main result of the paper can be stengthened: it can be shown that $$H^*(M_n;\mathbb{Z})$$ does not have odd torsion [E. M. Rains, J. Topol. 3, No. 4, 786–818 (2010; Zbl 1213.14102)], and that $$f_n^\mathbb{Z}$$ is an isomorphism. The fundamental group of $$M_n$$ is the pure cactus group $$\Gamma_n$$, and it is shown in [M. Davis, T. Januszkiewicz, R. Scott, Adv. Math. 177, No. 1, 115–179 (2003; Zbl 1080.52512)] that $$M_n$$ is a $$K(\Gamma_n,1)$$. Hence the main result of the paper also gives a description of the cohomology of $$\Gamma_n$$.
Next, the authors investigate the operadic properties of the spaces $$M_n$$. The operation of attaching genus zero curves at marked points endows the collection of spaces $$M_n$$ with the structure of topological operad, and therefore the collection of their homologies $$H_*(M_n;\mathbb{Q})$$ is an operad in the symmetric monoidal category of $$\mathbb{Z}$$-graded $$\mathbb{Q}$$-vector spaces. It is shown that this operad is the operad governing Gerstenhaber 2-algebras: it is generated by a graded commutative associative product of degree 0 with unit and by a skew-graded commutative ternary 2-bracket of degree -1, such that the 2-bracket is a derivation in each variable and satisfies a quadratic Jacobi identity in the space of 5-ary operations.
Finally, the analogy with braid groups is discussed. One sees that the space $$M_n=\overline{\mathcal{M}}_{0,n}(\mathbb{R})$$ has very different topological properties from those of its complex counterpart $$\overline{\mathcal{M}}_{0,n}(\mathbb{C})$$. Indeed, $$M_n$$ is a $$K(\pi,1)$$, its Poincaré polynomial has a simple factorization, its Betti numbers grow polynomially in $$n$$ and its homology is a finitely generated operad, where $$\overline{\mathcal{M}}_{0,n}(\mathbb{C})$$ is simply connected, its Poincaré polynomial does not have a simple factorization, its Betti numbers grow exponentially in $$n$$ and its homology operad is not finitely generated. On the other hand, the topological properties of $$M_n$$ just mentioned are enjoyed by the configuration spaces $$C_{n-1}$$ of $$n-1$$-tuples of distinct points in $$\mathbb{C}$$ (in particular, the homology operad of the topological operad $$C_{n-1}$$ is the operad govening Gerstenhaber algebras). The analogy between $$M_n$$ and $$C_{n-1}$$ and between their fundamental groups (the pure cactus group and the pure braid group, respectively) had already been remarked and investigated in [S. L. Devadoss, Contemp. Math. 239, 91–114 (1999; Zbl 0968.32009); J. Morava, arXiv:math/0109086; A. Henriques, J. Kamnitzer, Duke Math. J. 132, No. 2, 191–216 (2006; Zbl 1123.22007)]. This analogy suggests the following construction. The cohomology algebra $$H^*(M_n;\mathbb{Q})$$ is a quadratic algebra and so one can consider its quadratic dual algebra $$U_n$$, which is the universal enveloping algebra of a quadratic Lie algebra $$L_n$$. On the other hand, one can construct a Lie algebra $$\mathcal{L}_n$$ directly from the group $$\Gamma_n$$, by taking the associated graded of the lower central series filtration and then quotienting by the 2-torsion. The authors construct a surjective homomorphism of graded Lie algebras $$\psi_n:L_n\to \mathcal{L}_n$$ and conjecture it is actually an isomorphism as in the braid group case. More precisely, given a Lie algebra with a coboundary Lie quasibialgebra structure $$(\mathfrak{g},\varphi)$$ over a field of characteristic zero, there are natural representations $$\beta_{n,\mathfrak{g},\varphi}:L_n\to U(\mathfrak{g})^{\otimes n-1}$$, which are shown to factor through $$\psi_n:L_n\to \mathcal{L}_n$$. The proof of this fact is based on Drinfel’d quantization of the representations $$\beta_{n,\mathfrak{g},\varphi}$$ [V. G. Drinfel’d, Algebra Anal. 1, No. 6, 114–148 (1989; Zbl 0718.16033)]. Also it is conjectured that the algebra $$U_n$$ is Koszul.
On the other hand, it is shown in the paper that for $$n\geq 6$$ the Malcev Lie algebra of $$\Gamma_n$$ is not isomorphic to the degree completion of $$L_n\otimes \mathbb{Q}$$, and so in particular the spaces $$M_n$$ are not formal for $$n\geq 6$$. This fact reflects an essential difference between the pure cactus group and the pure braid group.

### MSC:

 14H10 Families, moduli of curves (algebraic) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 18D50 Operads (MSC2010) 17B99 Lie algebras and Lie superalgebras
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