Cyclic operads and homology of graph complexes.

*(English)*Zbl 0970.18011
Slovák, Jan (ed.) et al., Proceedings of the 18th winter school “Geometry and physics”, Srní, Czech Republic, January 10-17, 1998. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 59, 161-170 (1999).

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 \(\mathcal P\) for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for \(\Omega({\mathcal P}^*)\)-algebras with trivial differential over the (dual) cobar construction \(\Omega({\mathcal P}^*)\). If \(\mathcal P\) is Koszul the explicit form of the universal cycle will be presented. In a second step the author then considers general \(\mathcal P\)-coloured graphs over cyclic operads \(\mathcal P\). 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 theory which describes cyclic \(\Omega({\mathcal P}^*)\)-algebras with trivial differential. Based on these considerations some results of M. Penkava and A. Schwarz on the construction of an invariant coming from a cyclic deformation of a cyclic algebra will be explained and generalized to algebras over arbitrary cyclic operads.

For the entire collection see [Zbl 0913.00039].

In a first step the author presents a toy model of tree graphs coloured by an operad \(\mathcal P\) for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for \(\Omega({\mathcal P}^*)\)-algebras with trivial differential over the (dual) cobar construction \(\Omega({\mathcal P}^*)\). If \(\mathcal P\) is Koszul the explicit form of the universal cycle will be presented. In a second step the author then considers general \(\mathcal P\)-coloured graphs over cyclic operads \(\mathcal P\). 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 theory which describes cyclic \(\Omega({\mathcal P}^*)\)-algebras with trivial differential. Based on these considerations some results of M. Penkava and A. Schwarz on the construction of an invariant coming from a cyclic deformation of a cyclic algebra will be explained and generalized to algebras over arbitrary cyclic operads.

For the entire collection see [Zbl 0913.00039].

Reviewer: Bernhard Drabant (Walldorf)

##### MSC:

18D50 | Operads (MSC2010) |

05C15 | Coloring of graphs and hypergraphs |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

18C10 | Theories (e.g., algebraic theories), structure, and semantics |