Geometry & Topology, Vol. 1 (1997) Paper no. 5, pages 51-69.

Alexander duality, gropes and link homotopy

Vyacheslav S Krushkal and Peter Teichner

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

AMS subject classification. Primary: 55M05, 57M25. Secondary: 57M05, 57N13, 57N70.

DOI: 10.2140/gt.1997.1.51

E-print: arXiv:math.GT/9705222

Submitted to GT on June 17, 1997; revised October 17, 1997. Paper accepted October 26, 1997.

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Vyacheslav S Krushkal and Peter Teichner
Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027, USA
Current address:
Max-Planck-Institut fur Mathematik
Gottfried-Claren-Strasse 26
D-53225 Bonn, Germany
and
Department of Mathematics
University of California in San Diego
La Jolla, CA, 92093-0112, USA
Email: krushkal@math.msu.edu and teichner@euclid.ucsd.edu

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