The aim of the article is to give a conceptual understanding of Kontsevich’s construction of the universal element of the cohomology of the coarse moduli space of smooth algebraic curves with given genus and punctures. In a first step the author presents a toy model of tree graphs coloured by an operad $𝒫$ for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for $\Omega \left({𝒫}^{*}\right)$-algebras with trivial differential over the (dual) cobar construction $\Omega \left({𝒫}^{*}\right)$. If $𝒫$ is Koszul the explicit form of the universal cycle will be presented. In a second step the author then considers general $𝒫$-coloured graphs over cyclic operads $𝒫$. The construction of the graph complex and the universal class in the cohomology of the graph complex resembles the previous constructions for tree graphs. The coefficients of the universal cohomology class are elements in the !

EUDML-ID : urn:eudml:doc:221280
Mots clés:

@article{702138,
     title = {Remarks on CR-manifolds of codimension $2$ in $C^4$
               },
     booktitle = {Proceedings of the 18th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1999},
     pages = {171-180},
     mrnumber = {MR1692268},
     zbl = {0960.32023},
     url = {http://dml.mathdoc.fr/item/702138}
}

Schmalz, Gerd. Remarks on CR-manifolds of codimension $2$ in $C^4$
               , dans Proceedings of the 18th Winter School "Geometry and Physics", GDML_Books,  (1999), pp. 171-180. http://gdmltest.u-ga.fr/item/702138/