Alexander duality, gropes and link homotopy

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.

Keywords

Alexander duality, 4–manifolds, gropes, link homotopy, Milnor group, Dwyer filtration

Mathematical Subject Classification

Primary: 55M05, 57M25

Secondary: 57M05, 57N13, 57N70

References

Forward citations

Publication

Received: 17 June 1997

Revised: 17 October 1997

Published: 26 October 1997

Proposed: Robion Kirby

Seconded: Michael Freedman, Ronald Stern

Authors

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 Germany

Peter Teichner
Department of Mathematics University of California in San Diego La Jolla California 92093-0112 USA

Volume 1, issue 1 (1997)

Vyacheslav S Krushkal and Peter Teichner