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Zbl 1206.14051
Etingof, Pavel; Henriques, André; Kamnitzer, Joel; Rains, Eric M.
The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. (English)
[J] Ann. Math. (2) 171, No. 2, 731-777 (2010). ISSN 0003-486X; ISSN 1939-0980/e

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 $n\geq 6$. \par 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. \par 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 [{\it 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 [{\it 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 $\Gamma_n$. \par 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. \par 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 [{\it S. L. Devadoss}, Contemp. Math. 239, 91--114 (1999; Zbl 0968.32009); {\it J. Morava}, arXiv:math/0109086; {\it 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 $\beta_{n,\mathfrak{g},\varphi}$ [{\it 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. \par 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.
[Domenico Fiorenza (Roma)]

MSC 2000:: *14H10 Families, algebraic moduli (curves)
16E40 Homology and cohomology theories for assoc. rings
18D50 Operads
17B99 Lie algebras

Keywords: moduli spaces; Lie algebras; Gerstenhaber algebras; operads; Poincaré polynomials; formal spaces

Citations: Zbl 1080.52512; Zbl 0968.32009; Zbl 1123.22007; Zbl 0718.16033; Zbl 1213.14102

Cited in: Zbl 1223.14030

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