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Bordism of links in codimension 2. (English) Zbl 0622.57026

Author’s summary: ”We first consider oriented and framed links of closed smooth manifolds embedded in Euclidean space. There is a simple relation between the oriented and framed cases. The bordism groups and the forgetful map from framed to oriented links are explicitly computed for the case of two surfaces in \({\mathbb{R}}^ 4\). For the oriented case the group is \({\mathbb{Z}}_ 2\) and for the framed case the group can be identified with \[ \pi_ 4(S^ 2\vee S^ 2)\cong \pi_ 4(S^ 2)\times \pi_ 4(S^ 2)\times \pi_ 4(S^ 3)\times \pi_ 4(S^ 4)\times \pi_ 4(S^ 4). \] The third factor is picked out by a geometrically defined ’twist invariant’. This was inspired by work of Fenn and Rolfsen on links of 2-spheres in \({\mathbb{R}}^ 4\) up to homotopy. It also desuspends the \(\mu_ 2\) invariant of Boardman and Steer. The forgetful map is trivial on the first and second factors of the above Hilton-Milnor decomposition and is nontrivial on each of the remaining factors.
We also consider oriented and framed links of manifolds immersed in Euclidean space. The bordism groups and the forgetful map from framed to oriented links are explicitly computed for the case of surfaces in \({\mathbb{R}}^ 4\). The forgetful maps from embedded to immersed links are also computed.”
{Reviewer’s remark: Some of this material also allows a direct geometric treatment based on the notion of a generalized Seifert surface for any map \(f: M\to {\mathbb{R}}^ n\) where M is a closed oriented \((n-2)\)- manifold.}
Reviewer: U.Koschorke

MSC:

57R90 Other types of cobordism
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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