Alexander duality, gropes and link homotopy

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 μ-invariants as in the original proofs.

19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol1/paper5.abs.html

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• Whitney tower concordance of classical links
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• Particle Spaces on Manifolds and Generalized Poincaré Dualities
• Jacobi identities in low-dimensional topology
• Simple Whitney towers, half-gropes and the Arf invariant of a knot
• Whitney towers and gropes in 4--manifolds
• Link groups of 4-manifolds
• Robust 4-manifolds and robust embeddings