How not to study low-dimensional topology? 

Stallings gave a group-theoretic approach to the 3-dimensional Poincaré conjecture that was later turned into a group-theoretic statement equivalent to the Poincaré conjecture by Jaco and Hempel and then proven by Perelman. Together with Blackwell, Kirby, Longo, and Ruppik, we have extended Stallings's approach to give group-theoretically defined sets that are in bijection with (i) closed 3-manifolds, (ii) closed 3-manifolds with a link, (iii) closed 4-manifolds, and (iv) closed 4-manifolds with a link (of surfaces). I will explain these bijections and how this results in an algebraic formulation of the unknotting conjecture and a group-theoretic characterization of 4-dimensional knot groups.