Vyacheslav S. Krushkal and Peter Teichner
We prove a geometric refinement of Alexander duality for certain $2$-complexes, the so-called gropes, embedded into $4$-space. In addition, we give new proofs and extended versions of two lemmas of Freedman and Lin which are of central importance in the A-B-slice problem, the main open problem in the classification theory of topological $4$-manifolds. Our methods are group theoretical, rather than using Massey products and Milnor $\mu$-invariants as in the original proofs.