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Graph homology: Koszul and Verdier duality

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Graph homology: Koszul and Verdier duality. / Lazarev, Andrey; Voronov, Alexander.
In: Advances in Mathematics, Vol. 218, No. 6, 2008, p. 1878-1894.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Lazarev, A & Voronov, A 2008, 'Graph homology: Koszul and Verdier duality', Advances in Mathematics, vol. 218, no. 6, pp. 1878-1894. https://doi.org/10.1016/j.aim.2008.03.022

APA

Lazarev, A., & Voronov, A. (2008). Graph homology: Koszul and Verdier duality. Advances in Mathematics, 218(6), 1878-1894. https://doi.org/10.1016/j.aim.2008.03.022

Vancouver

Lazarev A, Voronov A. Graph homology: Koszul and Verdier duality. Advances in Mathematics. 2008;218(6):1878-1894. doi: 10.1016/j.aim.2008.03.022

Author

Lazarev, Andrey ; Voronov, Alexander. / Graph homology : Koszul and Verdier duality. In: Advances in Mathematics. 2008 ; Vol. 218, No. 6. pp. 1878-1894.

Bibtex

@article{bb1f015a7cae4e1daba86576df523b89,
title = "Graph homology: Koszul and Verdier duality",
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.",
keywords = "Graph homology, Cyclic operad, Koszul duality, Constructible sheaf, Verdier duality, Simplicial complex",
author = "Andrey Lazarev and Alexander Voronov",
year = "2008",
doi = "10.1016/j.aim.2008.03.022",
language = "English",
volume = "218",
pages = "1878--1894",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",
number = "6",

}

RIS

TY - JOUR

T1 - Graph homology

T2 - Koszul and Verdier duality

AU - Lazarev, Andrey

AU - Voronov, Alexander

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Graph homology

KW - Cyclic operad

KW - Koszul duality

KW - Constructible sheaf

KW - Verdier duality

KW - Simplicial complex

U2 - 10.1016/j.aim.2008.03.022

DO - 10.1016/j.aim.2008.03.022

M3 - Journal article

VL - 218

SP - 1878

EP - 1894

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -