Using Heegaard splittings and trisections respectively, the sets of closed 3- and 4-manifolds can be shown to be in bijection with certain equivalence classes of pairs (resp. triples) of homomorphisms from surface groups to free groups. I'll talk about an extension of this to manifolds containing links in the 3D case and surfaces in 4D, due to Blackwell-Kirby-Klug-Longo-Ruppik. Expect topology disguised as group theory, but mostly group theory disguised as topology.