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Poincaré transversality and four-dimensional surgery. (English) Zbl 0654.57007

The author shows that the conjectural topological surgery theorem in dimension 4 is equivalent to a certain question in homotopy theory. The reduction goes in two steps. First, a sufficient criterion is presented which guarantees that a 4-dimensional surgery problem \(f: (M,\partial M)\to (X,\partial X)\) has a topological solution. The criterion says that if f induces an isomorphism on \(\pi_ 1\) and if \(Ker_ 2(f)\) is a direct sum of standard planes represented by a mapping \(h: \coprod(S^ 2\vee S^ 2)\to M\) such that all loops in \(h(\coprod(S^ 2\vee S^ 2))\) are contractible in M then f is normally cobordant to a homotopy equivalence. Second, this criterion is applied to atomic surgery problems which gives their equivalent reformulation in terms of the so-called Poincaré transversality.
Reviewer: V.Turaev

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N70 Cobordism and concordance in topological manifolds
57N65 Algebraic topology of manifolds
57M05 Fundamental group, presentations, free differential calculus
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