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Graph homology: Koszul and Verdier duality. (English) Zbl 1158.55006

Recall that a cyclic operad is a structure that models a category of algebras in vector spaces (or chain complexes) for wich the notion of an invariant inner product makes sense. To each cyclic operad \(\mathcal O\) is associated a combinatorial complex \(C^{\Gamma\mathcal O}_*\) defined by graphs whose vertices are labelled by elements of \(\mathcal O\). The graph homology of \(\mathcal O\) is the homology of the graph complex \(C^{\Gamma\mathcal O}_*\) associated to \(\mathcal O\).
The authors associate to any cyclic operad \(\mathcal O\) a sheaf \(\mathcal F^{\mathcal O}\) on a simplicial complex formed by graphs whose edges are equipped with a metric. The graph homology of \(\mathcal O\) is isomorphic to the cohomology of the sheaf \(\mathcal F^{\text{Comm}}\) associated to the commutative operad \(\text{Comm}\) on the space of metric graphs.
The authors prove that Verdier’s dual of \(\mathcal F^{\mathcal O}\) is (up to an orientation twist) isomorphic in the derived category of sheaves to a sheaf \(\mathcal F^{D\mathcal O}\), where \(D\mathcal O\) refers to the cobar dual operad of \(\mathcal O\). In the case where \(\mathcal O\) is a Koszul operad, this sheaf \(\mathcal F^{D\mathcal O}\) is also quasi-isomorphic to the sheaf \(\mathcal F^{\mathcal O^!}\) associated to the Koszul dual operad of \(\mathcal O\). They apply this result to the commutative and Lie operads in order to give an interpretation of dual representations, in terms of graph homology, of the homology of outer automorphism groups.
They give a variant of their results for ribbon graphs. In this contex, the construction gives an interpretation of duality results for the homology of moduli spaces of Riemann surfaces.

MSC:

55N30 Sheaf cohomology in algebraic topology
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
18D50 Operads (MSC2010)
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References:

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