Pulling apart 2-spheres in 4-manifolds
type: Item
instance of: scholarly article
title: Pulling apart 2-spheres in 4-manifolds
zbMATH Open document ID: 1302.57057
author: Rob Schneiderman
author: Peter Teichner
published in: Documenta Mathematica
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.5534
full work available at URL: 31.html
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 6359325
zbMATH Keywords: surface
zbMATH Keywords: 4-manifold
zbMATH Keywords: disjoint immersion
zbMATH Keywords: Whitney towers.
reviewed by: Saïd Zarati
MaRDI profile type: Publication
arXiv ID: 1210.5534
review text: 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)].
description: scientific article; zbMATH DE number 6359325