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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 n6.

More in detail, let ¯ 0,n be the moduli space of genus zero curves with n marked points (this is a smooth projective variety over ), and let M n = ¯ 0,n () the manifold of real points of ¯ 0,n . For any ordered 4-element subset {i,j,k,l} of {1,,n} there is a natural map ϕ ijkl :M n M 4 1 () forgetting the points with labels outside {i,j,k,l}, so for any commutative ring R there is a natural homomorphism of algebras ϕ ijkl * :H * (M 4 ;R)H * (M n ;R). Denote by ω ijkl (M n )H 1 (M n ;R) the image via ϕ ijkl * of the standard generator of H * (M 4 ;R). The authors introduce, for every n3, a quadratic algebra Λ n as the skew-commutative algebra generated over by elements ω ijkl , 1i,j,k,ln, which are antisymmetric in ijkl and with defining relations

ω ijkl +ω jklm +ω klmi +ω lmij +ω mijk =0(1)
ω ijkl ω ijkm (2)
ω ijkl ω lmpi +ω klmp ω pijk +ω mpij ω jklm =0(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 Λ n R as soon as 2 is invertible in R. Moreover, if 1/2R, the elements ω ijkl (M n ) satisfy the relations (1) and (2) above. This is essentially a consequence of the isomophism H 2 (M 5 ;)/2. Therefore, for any commutative ring R in which 2 is invertible there is an homorphism of algebras f n R :Λ n RH * (M n ;R) mapping ω ijkl to ω ijkl (M n ). The main result of the paper then states that f n 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 -module Λ n , which is shown to be

P n (t)= 0k<(n-3)/2 (1+(n-3-2k) 2 t)·

Moreover, it is shown in the paper that H * (M n ;) 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 ;) does not have odd torsion [E. M. Rains, J. Topol. 3, No. 4, 786–818 (2010; Zbl 1213.14102)], and that f n is an isomorphism. The fundamental group of M n is the pure cactus group Γ 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(Γ n ,1). Hence the main result of the paper also gives a description of the cohomology of Γ 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 ;) is an operad in the symmetric monoidal category of -graded -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 = ¯ 0,n () has very different topological properties from those of its complex counterpart ¯ 0,n (). Indeed, M n is a K(π,1), its Poincaré polynomial has a simple factorization, its Betti numbers grow polynomially in n and its homology is a finitely generated operad, where ¯ 0,n () 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 (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 ;) 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 n directly from the group Γ 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 ψ n :L n 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 (𝔤,φ) over a field of characteristic zero, there are natural representations β n,𝔤,φ :L n U(𝔤) n-1 , which are shown to factor through ψ n :L n n . The proof of this fact is based on Drinfel’d quantization of the representations β n,𝔤,φ [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 n6 the Malcev Lie algebra of Γ n is not isomorphic to the degree completion of L n , and so in particular the spaces M n are not formal for n6. This fact reflects an essential difference between the pure cactus group and the pure braid group.


MSC:
14H10Families, algebraic moduli (curves)
16E40Homology and cohomology theories for associative rings
18D50Operads
17B99Lie algebras