The authors compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold of real points of the moduli space of stable genus zero curves with marked points. Further, it is shown that the rational homology operad of is the operad of Gerstenhaber 2-algebras, and analogies and differences between and the configuration space of -tuples of distinct complex numbers are investigated. In particular it is shown that the spaces are not formal for .
More in detail, let be the moduli space of genus zero curves with marked points (this is a smooth projective variety over ), and let the manifold of real points of . For any ordered 4-element subset of there is a natural map forgetting the points with labels outside , so for any commutative ring there is a natural homomorphism of algebras . Denote by the image via of the standard generator of . The authors introduce, for every , a quadratic algebra as the skew-commutative algebra generated over by elements , , which are antisymmetric in and with defining relations
for distinct . The ideal generated by the first two relations contains 2 times the third relation, so that this last relation becomes redundant in as soon as 2 is invertible in . Moreover, if , the elements satisfy the relations and above. This is essentially a consequence of the isomophism . Therefore, for any commutative ring in which 2 is invertible there is an homorphism of algebras mapping to . The main result of the paper then states that is an isomorphism.
In particular this reduces the computation of the Poincaré polynomial of to the computation of the Poincaré polynomyal of the -module , which is shown to be
Moreover, it is shown in the paper that 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 does not have odd torsion [E. M. Rains, J. Topol. 3, No. 4, 786–818 (2010; Zbl 1213.14102)], and that is an isomorphism. The fundamental group of is the pure cactus group , and it is shown in [M. Davis, T. Januszkiewicz, R. Scott, Adv. Math. 177, No. 1, 115–179 (2003; Zbl 1080.52512)] that is a . Hence the main result of the paper also gives a description of the cohomology of .
Next, the authors investigate the operadic properties of the spaces . The operation of attaching genus zero curves at marked points endows the collection of spaces with the structure of topological operad, and therefore the collection of their homologies 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 has very different topological properties from those of its complex counterpart . Indeed, is a , its Poincaré polynomial has a simple factorization, its Betti numbers grow polynomially in and its homology is a finitely generated operad, where is simply connected, its Poincaré polynomial does not have a simple factorization, its Betti numbers grow exponentially in and its homology operad is not finitely generated. On the other hand, the topological properties of just mentioned are enjoyed by the configuration spaces of -tuples of distinct points in (in particular, the homology operad of the topological operad is the operad govening Gerstenhaber algebras). The analogy between and 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 is a quadratic algebra and so one can consider its quadratic dual algebra , which is the universal enveloping algebra of a quadratic Lie algebra . On the other hand, one can construct a Lie algebra directly from the group , 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 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 over a field of characteristic zero, there are natural representations , which are shown to factor through . The proof of this fact is based on Drinfel’d quantization of the representations [V. G. Drinfel’d, Algebra Anal. 1, No. 6, 114–148 (1989; Zbl 0718.16033)]. Also it is conjectured that the algebra is Koszul.
On the other hand, it is shown in the paper that for the Malcev Lie algebra of is not isomorphic to the degree completion of , and so in particular the spaces are not formal for . This fact reflects an essential difference between the pure cactus group and the pure braid group.