Peter Zvengrowski - Diagonal formulae in group cohomology

For any group G, its cohomology can be defined to be the cohomology of the Eilenberg-MacLane space K(G,1). From this definition, or otherwise, one sees that the cohomology with coefficients in a ring R forms a graded algebra under the cup product operation. One method to find these cup products is to start with a projective resolution of Z over the group ring ZG and then find a diagonal map . We will concern ourselves with the existence of explicit diagonal formulae of this type for certain groups arising as the fundamental group of a 3-manifold, with emphasis on the Seifert manifolds, and also mention some of the applications of the cup products.