Grothendieck-Teichmueller group, operads and graph complexes: a survey

This paper attempts to provide a more or less self-contained introduction into theory of the Grothendieck-Teichmueller group and Drinfeld associators using the theory of operads and graph complexes.Comment: One reference is adde

Deformation theory of representations of prop(erad)s

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative) Frobenius bialgebras fixed in Section 4. [82 pages

On interrelations between graph complexes

We study Maxim Kontsevich's graph complex $GC_d$ for any integer $d$ as well as its oriented and targeted versions, and show new short proofs of the theorems due to Thomas Willwacher and Marko Zivkovic which establish isomorphisms of their cohomology groups. A new result relating the cohomology of the sourced-targeted graph complex in dimension $d+1$ with the direct sum of two copies of the cohomology group of Maxim Kontsevich's graph complex $GC_d$ in dimension $d$ is obtained. We introduce a new graph complex spanned by purely trivalent graphs and show that its cohomology is isomorphic to $H(GC_d)$.Comment: 26p, typos correcte