Graph homology: Koszul and Verdier duality

A Lazarev,A.A Voronov

doi:10.1016/j.aim.2008.03.022

A Lazarev, A.A Voronov

Open Access

https://doi.org/10.1016/j.aim.2008.03.022

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Journal: Advances in Mathematics	Publication Date: Apr 21, 2008
Citations: 33	License type: publisher-specific-oa

Affiliation: University of Leicester, University of Minnesota

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.

Highlights

The popularity of graph homology owes largely to the fact that the cohomology of two important spaces in mathematics, the classifying space Yn of the outer automorphism group of the free group on n generators and the moduli space Mg,n of Riemann surfaces of genus g with n punctures, even though generally intractable, may be computed via a deceptively simple combinatorial construction, called graph homology, see M
The appearance of Koszul-dual operads in the right-hand side as corresponding to the homology vs. cohomology with compact supports in the left-hand side is quite suggestive: it hints on a relationship between some kind of Poincare duality for spaces and Koszul duality for operads
In this paper we show that this relationship takes place and prove more general results, Theorems 3.9 and 4.3, which show that up to an orientation twist, Verdier duality on the moduli space of graphs transfers a certain constructible sheaf corresponding to an operad O to the sheaf corresponding to the dg-dual operad DO, which is quasi-isomorphic to the Koszul-dual operad O!, if O happens to be Koszul

Summary

Introduction

The popularity of graph homology owes largely to the fact that the cohomology of two important spaces in mathematics, the classifying space Yn of the outer automorphism group of the free group on n generators and the (decorated) moduli space Mg,n of Riemann surfaces of genus g with n punctures, even though generally intractable, may be computed via a deceptively simple combinatorial construction, called graph homology, see M. Cyclic operad, Koszul duality, constructible sheaf, Verdier duality, simplicial complex. This work was completed during the first author’s stay at IHES and he wishes to express his gratitude to this institution for excellent working conditions. Of higher cohomology for corresponding sheaves, while in our paper it translates into a duality statement between highly nontrivial cohomology groups of spaces of metric graphs. As pointed out by the referee of this paper, stronger results must hold for certain compactifications of our moduli spaces; cyclic operads need to be replaced with modular operads in this more general setting. We mention that the relationship between Koszul and Verdier dualities (in a different context) was observed in the paper [17]

Verdier duality for simplicial complexes

Equivariant Verdier duality

Graph complexes and spaces of metric graphs

Ribbon graphs