## Abstract

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to differential graded operads corresponds to the cobar-duality of operads (which specializes to Koszul duality for Koszul operads). This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the space of metric ribbon graphs, known to be homotopy equivalent to the moduli space of Riemann surfaces, and the cohomology of a certain sheaf on the space of usual metric graphs.

Original language | English (US) |
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Pages (from-to) | 1878-1894 |

Number of pages | 17 |

Journal | Advances in Mathematics |

Volume | 218 |

Issue number | 6 |

DOIs | |

State | Published - Aug 20 2008 |

## Keywords

- Constructible sheaf
- Cyclic operad
- Graph homology
- Koszul duality
- Simplicial complex
- Verdier duality