3-manifolds with prescribed z2-cohomology rings

The question of whether a graded Z2-algebra can be realized as a Z2-cohomology ring of a closed 3-manifold is settled by M. M. Postnikov in 1948. Recently in 2024 J. A. Hillman provided a new argument, giving explicit surgery descriptions for 3- manifolds that realize the given ring. We present Hillman's construction, provide illustrations and streamline the argument by appealing only to intersection theory to obtain the desired cup products. Central to both Postnikov's and Hillman's approaches is the Postnikov–Wu identity, which pertains to a distinguished element in the ring and its cup products. We also provide an alternative proof of this identity, again using intersection theory.