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

@article{Alm2015BrownsDM,
  title={Brown's dihedral moduli space and freedom of the gravity operad},
  author={J. Alm and Dan Erik Petersen},
  journal={arXiv: Algebraic Geometry},
  year={2015}
}
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… 

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