Embedding up to homotopy type - the first obstruction
type: Item
instance of: scholarly article
title: Embedding up to homotopy type - the first obstruction
zbMATH Open document ID: 0547.57015
DOI: 10.1016/0166-8641(84)90037-3
author: Nathan Habegger
published in: Topology and its Applications
publication date: 1984-01-01T00:00:00Z
review text: It is a theorem of several authors [e.g. \textit{C. T. C. Wall}, Topology 5, 73-94 (1966; Zbl 0149.205)] that a map f: \(K\to M\) where K is a CW- complex of dimension n and M a PL-manifold of dimension m, is homotopic to a ''homotopy imbedding'', if \(m\geq n+3\) and f is \((2n-m+1)-\)connected. A homotopy imbedding is a simple homotopy equivalence e: \(K\to N\), where N is a compact PL-submanifold of M (a thickening of K). The main aim of this paper is to define a homological obstruction whose vanishing is necessary and sufficient for f to be homotopic to a homotopy imbedding when f is \((2n-m)-\)connected. The solution of this problem uses the author's previous work (see the preceding review) in which he determines the obstruction to homotoping a map of several disks \(\coprod (D^ n,S^{n-1})\to (W,\partial W)\) into an m-manifold W, to an imbedding, when \(m\geq n+3, 2n\geq m+2,\) and (W,\(\partial W) (2n-m)- \)connected. The proof begins with a homotopy of \(F| K^{n-1}\) to a homotopy imbedding, by Wall's theorem. If N is the thickening of \(K^{n-1}\) thus defined then there remains the problem of thickening the n-cells of K into handles on \(\partial N\) in \(\overline{M-N}\). The obstructions to this thickening are represented as a suitable cocycle - different choices of mappings of these cells into \((\overline{M-N},\partial N)\) determine cohomologous cocycles. In the case \(M={\mathbb{R}}^ m\), this obstruction is shown to vanish if and only if certain homology operations, first introduced by Thom as imbedding obstructions, vanish. Finally, a companion theorem classifies the homotopy imbeddings corresponding to a given f which is \((2n-m+1)-\)connected.
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 3872293
zbMATH Keywords: highly connected maps
zbMATH Keywords: homotopy imbedding
zbMATH Keywords: simple homotopy equivalence
zbMATH Keywords: thickening
zbMATH Keywords: homology operations
reviewed by: Jerome Levine
MaRDI profile type: Publication
full work available at URL: 0166-8641(84)90037-3
OpenAlex ID: W2073368040
cites work: Classification problems in differential topology. IV: Thickenings
cites work: Q4766257
cites work: Obstructions to embedding disks. II: A proof of a conjecture of Hudson
cites work: Q5594673
cites work: Q5584509
cites work: Q3879963
cites work: Obstructions to embedding and isotopy in the metastable range
cites work: Q3902182
cites work: Espaces fibrés en sphères et carrés de Steenrod
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description: scientific article; zbMATH DE number 3872293