Freedman, Michael H. Poincaré transversality and four-dimensional surgery. (English) Zbl 0654.57007 Topology 27, No. 2, 171-175 (1988). 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 Cited in 1 ReviewCited in 2 Documents 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 Keywords:topological surgery in dimension 4; 4-dimensional surgery problem; atomic surgery problems; Poincaré transversality PDFBibTeX XMLCite \textit{M. H. Freedman}, Topology 27, No. 2, 171--175 (1988; Zbl 0654.57007) Full Text: DOI