Gravity properad and moduli spaces Mg,n${\mathcal {M}}_{g,n}$

Abstract Let be the moduli space of algebraic curves of genus with boundaries and marked points, and its compactly supported cohomology group. We prove that the collection of ‐modules has the structure of a properad (called the gravity properad ) such that it contains the Getzler's gravity operad as the sub‐collection . The properadic structure in is highly nontrivial and generates higher genus cohomology classes from lower ones (which is demonstrated on infinitely many nontrivial examples producing higher genus cohomology classes from just zero genus ones). Moreover, we prove that the generators of the 1‐dimensional cohomology groups , and satisfy with respect to this properadic structure the relations of the (degree shifted) quasi‐Lie bialgebra, a fact making the totality of cohomology groups into a complex with the differential fully determined by the just mentioned three cohomology classes. It is proven that this complex contains infinitely many nontrivial cohomology classes, all coming from Kontsevich's odd graph complex. The prop structure in is established with the help of Willwacher's twisting endofunctor (in the category of properads under the operad of Lie algebras) and Costello's theory of moduli spaces of nodal disks with marked boundaries and internal marked points.