#### Volume 1, issue 1 (1997)

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Alexander duality, gropes and link homotopy

### Vyacheslav S Krushkal and Peter Teichner

Geometry & Topology 1 (1997) 51–69
 arXiv: math.GT/9705222
##### Abstract

We prove a geometric refinement of Alexander duality for certain 2–complexes, the so-called gropes, embedded into 4–space. This refinement can be roughly formulated as saying that 4–dimensional Alexander duality preserves the disjoint Dwyer filtration.

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.

##### Keywords
Alexander duality, 4–manifolds, gropes, link homotopy, Milnor group, Dwyer filtration
##### Mathematical Subject Classification
Primary: 55M05, 57M25
Secondary: 57M05, 57N13, 57N70