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Link maps in the four sphere. (English) Zbl 0662.57012

Differential topology, Proc. 2nd Topology Symp., Siegen/FRG 1987, Lect. Notes Math. 1350, 31-43 (1988).
[For the entire collection see Zbl 0645.00006.]
This paper studies the abelian semigroup \(LM^ 4_{2,2}\) of link homotopy classes of link maps f: \(S^ 2\cup S^ 2\to S^ 4\). It is known for some time that the Fenn-Rolfsen link map represents a nontrivial element [R. Fenn and D. Rolfsen, J. Lond. Math. Soc., II. Ser. 34, 177-184 (1986; Zbl 0603.57011)]. The author defines a homomorphism \(\sigma =(\sigma_+,\sigma_-): LM^ 4_{2,2}\to {\mathbb{Z}}[t]\oplus {\mathbb{Z}}[t]\) and determines the image, which surprisingly turns out to be an infinitely generated proper subgroup.
In particular he gives a construction of link maps, which widely generalizes the Fenn-Rolfsen link map. The following facts are proved:
(1) \(\sigma_+\) resp. \(\sigma_-\) is an obstruction (the only known!) to homotope \(f: S^ 2_+\cup S^ 2_-\to S^ 4\) to a link map which embeds \(S^ 2_+\) resp. \(S^ 2_-\). Indeed, \(\sigma_+\) measures the linking of \(f| S^ 2_-\) with a zero bordism of the restriction of f to the double points of a generic \(f| S^ 2_+\) inside \(f(S^ 2_+).\)
(2) \(\sigma\) is an obstruction to homotope f to a semiboundary link map (f is semiboundary if it can be extended to a map of 3-manifolds \(F: V_+\cup V_-\to S^ 4\) satisfying \(F(S^ 2_+)\cap f(S^ 2_- )=f(S^ 2_+)\cap F(V_-)=\emptyset).\)
(3) Any boundary link map (i.e. F in (2) is a link map) can be homotoped to the trivial link map.
Finally, a suspension construction is sketched which relates classical links to elements of \(LM^ 4_{2,2}\). The \(\sigma_+\)-invariant has been generalized by the author to link maps \(S^ p\cup S^{m-2}\to S^ m\), \(p\leq m-2\), and by U. Koschorke to arbitrary link maps \(S^ p\cup S^ q\to S^ m\), \(p\leq m-2\).
Reviewer: U.Kaiser

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R40 Embeddings in differential topology
57R90 Other types of cobordism