The cohomology of moduli stacks of curves $\mathcal M_g$ and the diamond ribbon graph complex

The problem of understanding $H(\mathcal M_g)$ is undoubtedly classical but despite the great effort of a lot of mathematicians, we know a little about it. One source of classes (tautological classes) in $H(\mathcal M_g)$ comes from the elements of the Chow group of $\mathcal M_g.$ On another hand, the computation of the Euler characteristic of $\mathcal M_g$ by J. Harer and D. Zagier shows that tautological classes form a very small part of $H(\mathcal M_g).$ Namely there should be a lot of non-tautological classes and specifically a lot of odd degree classes for even $g.$ However the first cohomology class of odd degree was found only in 2005 by O. Tommasi.

After an overview of some known results about the cohomology of $\mathcal M_g$, I will explain the certain approach which should shed some light on where we should look for non-tautological classes in $H(\mathcal M_g).$ This approach is based on the diamond ribbon graph complex, which was introduced by S. Merkulov and T. Willwacher and has roots in the deformation theory of the involutive (diamond) Lie bialgebras. The talk is based on the work in progress.