Brown's dihedral moduli space and freedom of the gravity operad

Abstract

Francis Brown introduced a partial compactification $M_{0,n}^\delta$ of the moduli space $M_{0,n}$. We prove that the gravity cooperad, given by the degree-shifted cohomologies of the spaces $M_{0,n}$, is cofree as a nonsymmetric anticyclic cooperad; moreover, the cogenerators are given by the cohomology groups of $M_{0,n}^\delta$. This says in particular that $H^\bullet(M_{0,n}^\delta)$ injects into $H^\bullet(M_{0,n})$. As part of the proof we construct an explicit diagrammatically defined basis of $H^\bullet(M_{0,n})$ which is compatible with cooperadic cocomposition, and such that a subset forms a basis of $H^\bullet(M_{0,n}^\delta)$. We show that our results are equivalent to the claim that $H^k(M_{0,n}^\delta)$ has a pure Hodge structure of weight $2k$ for all $k$, and we conclude our paper by giving an independent and completely different proof of this fact. The latter proof uses a new and explicit iterative construction of $M_{0,n}^\delta$ from $\mathbb{A}^{n-3}$ by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of $\overline M_{0,n}$ from $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$, respectively.