##
**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.

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.

Reviewer: Domenico Fiorenza (Roma)

### 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 |

### Keywords:

moduli spaces; Lie algebras; Gerstenhaber algebras; operads; Poincaré polynomials; formal spaces
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\textit{P. Etingof} et al., Ann. Math. (2) 171, No. 2, 731--777 (2010; Zbl 1206.14051)

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