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

• DOI: 10.24033/asens.2340
• Publication Date: 2018-05-08T17:05:17Z
• Authors: Johan Alm; Dan Petersen
• Keywords: Mathematics - Algebraic Geometry; Mathematics - Quantum Algebra; FOS: Mathematics; Algebraic Topology (math.AT); Quantum Algebra (math.QA); Mathematics - Algebraic Topology; 0101 mathematics; Algebraic Geometry (math.AG); 01 natural sciences

Francis Brown introduced a partial compactification $M_{0,n}^��$ 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 $M_{0,n}^��$. This says in particular that $H^\bullet(M_{0,n}^��)$ 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 $H^\bullet(M_{0,n}^��)$. We show that our results are equivalent to the claim that $H^k(M_{0,n}^��)$ 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 $M_{0,n}^��$ 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.<br/>39 pages. v2: significant revision, to appear in the Annales Scientifiques de l'ENS<br/>