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On the homotopy classification of pairs of linked maps of manifolds into a linear space. (English) Zbl 0563.57007

It is shown that if \(M^ m,N^ n\) are connected closed oriented smooth manifolds and \(E={\mathbb{R}}^{m+n+1}\), then pairs (f,g) of maps \(f: M\to E\), \(g: N\to E\) with disjoint images are classified up to homotopy (through such pairs) by the degree (in \({\mathbb{Z}})\) of the associated map \(\phi\) : \(M\times N\to S^{m+n}\) given by \(\phi (u,v)=(g(v)-f(u))/\| g(v)-f(u)\|\). This extends the homotopy classification of 2-component links in \(S^ 3\) by linking number, due to J. Milnor [Ann. Math., II. Ser. 59, 177-195 (1954; Zbl 0055.169)]. An analogous classification by means of a mod(2) degree (in \({\mathbb{Z}}/2{\mathbb{Z}})\) is given for the case when at least one of the manifolds M,N is assumed nonorientable.
Reviewer: J.Hillman

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R40 Embeddings in differential topology
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 0055.169
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