On the Asphericity of Knot Complements

Liem, Vo Thanh ; Venema, Gerard A.

Canadian journal of mathematics, Tome 45 (1993) no. 2, pp. 340-356

Résumé

Two examples of topological embeddings of S 2 in S 4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.Two theorems are proved. The first answers the question of just when good π 1 implies the vanishing of the higher homotopy groups for knot complements in S 4. The second theorem characterizes local flatness for 2-spheres in S 4 in terms of a local π 1 condition.

Détail

DOI : 10.4153/CJM-1993-016-5

Mots-clés : 57Q45, 57N45, knot complement, fundamental group, asphericity, 1 –alg, local flatness, 2–sphere

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