Pulling apart 2-spheres in 4-manifolds
Let \(\Sigma=\coprod_{i}\Sigma_{i}\) be a surface decomposed into connected components, let \(V\) be a \(4\)-dimensional manifold and let \(f: \Sigma \rightarrow V\) be a map. The paper under review concerns the question of whether the map \(f\) is homotopic to a map \(g: \Sigma \rightarrow V\) such that the \(g(\Sigma_{i})\) are pairwise disjoint subsets of \(V\) (in this case we say that \(f\) can be pulled apart). This question which arises as a precursor to the problem of whether or not \(f\) is homotopic to an embedding, which also arises in the study of homotopy groups configuration spaces, has been studied in many particular cases (see [\textit{C. T. C. Wall}, Surgery on compact manifolds. 2nd ed. Providence, RI: AMS (1999; Zbl 0935.57003)]; [\textit{K. Kobayashi}, in: Math. Semin. Notes, Kobe Univ. 5, No. 1, 109--116 (1977; Zbl 0361.57015)]; [\textit{R. Schneiderman} and \textit{P. Teichner}, Geom. Topol. Monogr. 7, 101--134 (2004; Zbl 1090.57013)]).NEWLINENEWLINEIn this paper, the authors extend the work in [Zbl 1090.57013] to a surface which is a disjoint union of an arbitrary number of \(2\)-spheres. For this, they use a variation of the theory of Whitney towers as developed in [\textit{J. Conant} et al., Geom. Topol. 16, No. 3, 1419--1479 (2012; Zbl 1257.57005)].
scientific article; zbMATH DE number 6359325