Embedding up to homotopy type - the first obstruction
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.
scientific article; zbMATH DE number 3872293
scientific article