Geodesic
Alexander duality, gropes and link homotopy
Krushkal, Vyacheslav S1 ; Teichner, Peter2
1 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
2 Department of Mathematics, University of California in San Diego, La Jolla, California 92093-0112, USA
Geometry & topology, Tome 1 (1997) no. 1, pp. 51-69.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.1997.1.51
Keywords: Alexander duality, 4–manifolds, gropes, link homotopy, Milnor group, Dwyer filtration

Krushkal, Vyacheslav S 1 ; Teichner, Peter 2

1 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
2 Department of Mathematics, University of California in San Diego, La Jolla, California 92093-0112, USA 
@article{GT_1997_1_1_a4,
     author = {Krushkal, Vyacheslav S and Teichner, Peter},
     title = {Alexander duality, gropes and link homotopy},
     journal = {Geometry & topology},
     pages = {51--69},
     publisher = {mathdoc},
     volume = {1},
     number = {1},
     year = {1997},
     doi = {10.2140/gt.1997.1.51},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.1997.1.51/}
}
TY  - JOUR
AU  - Krushkal, Vyacheslav S
AU  - Teichner, Peter
TI  - Alexander duality, gropes and link homotopy
JO  - Geometry & topology
PY  - 1997
SP  - 51
EP  - 69
VL  - 1
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.2140/gt.1997.1.51/
DO  - 10.2140/gt.1997.1.51
ID  - GT_1997_1_1_a4
ER  - 
%0 Journal Article
%A Krushkal, Vyacheslav S
%A Teichner, Peter
%T Alexander duality, gropes and link homotopy
%J Geometry & topology
%D 1997
%P 51-69
%V 1
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.2140/gt.1997.1.51/
%R 10.2140/gt.1997.1.51
%F GT_1997_1_1_a4
Krushkal, Vyacheslav S; Teichner, Peter. Alexander duality, gropes and link homotopy. Geometry & topology, Tome 1 (1997) no. 1, pp. 51-69. doi : 10.2140/gt.1997.1.51. http://geodesic.mathdoc.fr/articles/10.2140/gt.1997.1.51/

[1] W G Dwyer, Homology, Massey products and maps between groups, J. Pure Appl. Algebra 6 (1975) 177

[2] M H Freedman, X S Lin, On the $(A,B)$–slice problem, Topology 28 (1989) 91

[3] M H Freedman, F Quinn, The Topology of 4–manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)

[4] M H Freedman, P Teichner, 4–manifold topology I: Subexponential groups, Invent. Math. 122 (1995) 509

[5] M H Freedman, P Teichner, 4–manifold topology II: Dwyer's filtration and surgery kernels, Invent. Math. 122 (1995) 531

[6] C H Giffen, Link concordance implies link homotopy, Math. Scand. 45 (1979) 243

[7] D L Goldsmith, Concordance implies homotopy for classical links in $S^{3}$, Comment. Math. Helv. 54 (1979) 347

[8] V S Krushkal, Additivity properties of Milnor's $\bar\mu$–invariants

[9] X S Lin, On equivalence relations of links in 3–manifolds

[10] J Milnor, Link groups, Ann. of Math. $(2)$ 59 (1954) 177

[11] J Milnor, Isotopy of links. Algebraic geometry and topology, from: "A symposium in honor of S. Lefschetz", Princeton University Press (1957) 280

[12] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170

[13] P Teichner, Stratified Morse Theory and Link Homotopy, Habilitationsschrift

Cité par Sources :