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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 $n\ge 6$.

More in detail, let ${\overline{ℳ}}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points (this is a smooth projective variety over $ℚ$), and let ${M}_{n}={\overline{ℳ}}_{0,n}\left(ℝ\right)$ the manifold of real points of ${\overline{ℳ}}_{0,n}$. For any ordered 4-element subset $\left\{i,j,k,l\right\}$ of $\left\{1,\cdots ,n\right\}$ there is a natural map ${\phi }_{ijkl}:{M}_{n}\to {M}_{4}\cong {ℙ}^{1}\left(ℝ\right)$ forgetting the points with labels outside $\left\{i,j,k,l\right\}$, so for any commutative ring $R$ there is a natural homomorphism of algebras ${\phi }_{ijkl}^{*}:{H}^{*}\left({M}_{4};R\right)\to {H}^{*}\left({M}_{n};R\right)$. Denote by ${\omega }_{ijkl}\left({M}_{n}\right)\in {H}^{1}\left({M}_{n};R\right)$ the image via ${\phi }_{ijkl}^{*}$ of the standard generator of ${H}^{*}\left({M}_{4};R\right)\cong ℤ$. The authors introduce, for every $n\ge 3$, a quadratic algebra ${{\Lambda }}_{n}$ as the skew-commutative algebra generated over $ℤ$ by elements ${\omega }_{ijkl}$, $1\le i,j,k,l\le n$, which are antisymmetric in $ijkl$ and with defining relations

${\omega }_{ijkl}+{\omega }_{jklm}+{\omega }_{klmi}+{\omega }_{lmij}+{\omega }_{mijk}=0\phantom{\rule{2.em}{0ex}}\left(1\right)$
${\omega }_{ijkl}{\omega }_{ijkm}\phantom{\rule{2.em}{0ex}}\left(2\right)$
${\omega }_{ijkl}{\omega }_{lmpi}+{\omega }_{klmp}{\omega }_{pijk}+{\omega }_{mpij}{\omega }_{jklm}=0\phantom{\rule{2.em}{0ex}}\left(3\right)$

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 ${\omega }_{ijkl}\left({M}_{n}\right)$ satisfy the relations $\left(1\right)$ and $\left(2\right)$ above. This is essentially a consequence of the isomophism ${H}^{2}\left({M}_{5};ℤ\right)\cong ℤ/2ℤ$. Therefore, for any commutative ring $R$ in which 2 is invertible there is an homorphism of algebras ${f}_{n}^{R}:{{\Lambda }}_{n}\otimes R\to {H}^{*}\left({M}_{n};R\right)$ mapping ${\omega }_{ijkl}$ to ${\omega }_{ijkl}\left({M}_{n}\right)$. The main result of the paper then states that ${f}_{n}^{ℚ}$ 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 $ℤ$-module ${{\Lambda }}_{n}$, which is shown to be

${P}_{n}\left(t\right)=\prod _{0\le k<\left(n-3\right)/2}\left(1+{\left(n-3-2k\right)}^{2}t\right)·$

Moreover, it is shown in the paper that ${H}^{*}\left({M}_{n};ℤ\right)$ 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 ${H}^{*}\left({M}_{n};ℤ\right)$ does not have odd torsion [E. M. Rains, J. Topol. 3, No. 4, 786–818 (2010; Zbl 1213.14102)], and that ${f}_{n}^{ℤ}$ 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 $K\left({{\Gamma }}_{n},1\right)$. 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 ${H}_{*}\left({M}_{n};ℚ\right)$ is an operad in the symmetric monoidal category of $ℤ$-graded $ℚ$-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 ${M}_{n}={\overline{ℳ}}_{0,n}\left(ℝ\right)$ has very different topological properties from those of its complex counterpart ${\overline{ℳ}}_{0,n}\left(ℂ\right)$. Indeed, ${M}_{n}$ is a $K\left(\pi ,1\right)$, its Poincaré polynomial has a simple factorization, its Betti numbers grow polynomially in $n$ and its homology is a finitely generated operad, where ${\overline{ℳ}}_{0,n}\left(ℂ\right)$ 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 $ℂ$ (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 ${H}^{*}\left({M}_{n};ℚ\right)$ 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 ${ℒ}_{n}$ 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 ${\psi }_{n}:{L}_{n}\to {ℒ}_{n}$ 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 $\left(𝔤,\varphi \right)$ over a field of characteristic zero, there are natural representations ${\beta }_{n,𝔤,\varphi }:{L}_{n}\to U{\left(𝔤\right)}^{\otimes n-1}$, which are shown to factor through ${\psi }_{n}:{L}_{n}\to {ℒ}_{n}$. The proof of this fact is based on Drinfel’d quantization of the representations ${\beta }_{n,𝔤,\varphi }$ [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 $n\ge 6$ the Malcev Lie algebra of ${{\Gamma }}_{n}$ is not isomorphic to the degree completion of ${L}_{n}\otimes ℚ$, and so in particular the spaces ${M}_{n}$ are not formal for $n\ge 6$. This fact reflects an essential difference between the pure cactus group and the pure braid group.

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