This paper is built on the following observation: the purity of the mixed Hodge
structure on the cohomology of Brown’s moduli spaces is essentially equivalent to the
freeness of the dihedral operad underlying the gravity operad. We prove these two
facts by relying on both the geometric and the algebraic aspects of the problem:
the complete geometric description of the cohomology of Brown’s moduli
spaces and the coradical filtration of cofree cooperads. This gives a conceptual
proof of an identity of Bergström and Brown which expresses the Betti
numbers of Brown’s moduli spaces via the inversion of a generating series. This
also generalizes the Salvatore–Tauraso theorem on the nonsymmetric Lie
operad.
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