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.


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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