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


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.

Alexander duality, 4–manifolds, gropes, link homotopy, Milnor group, Dwyer filtration
Mathematical Subject Classification
Primary: 55M05, 57M25
Secondary: 57M05, 57N13, 57N70
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Received: 17 June 1997
Revised: 17 October 1997
Published: 26 October 1997
Proposed: Robion Kirby
Seconded: Michael Freedman, Ronald Stern
Vyacheslav S Krushkal
Department of Mathematics, Michigan State University
East Lansing, Michigan 48824-1027, USA
Max-Planck-Institut für Mathematik
Gottfried-Claren-Strasse 26
D-53225 Bonn
Peter Teichner
Department of Mathematics
University of California in San Diego
La Jolla
California 92093-0112