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\(K_{-i}\) obstructions to factoring an open manifold. (English) Zbl 0654.57009
The authors give the following abstract which adequately describes the paper: “Let \(M^{n+k}\) be an open PL manifold of dimension \(n+k>5\), let X be a finite polyhedron, and suppose \(f: M^{n+k}\to X\times {\mathbb{R}}\quad k\) is a bounded homotopy equivalence. If \(k\geq 1\), we use radial engulfing and Siebenmann’s twist-gluing (twist \(=\) id.) to construct a manifold \(M_ 1\) with infinite cyclic cover M and a bounded homotopy equivalence \(f_ 1: M_ 1\to X\times S\quad 1\times {\mathbb{R}}^{k-1}.\) By iterating this construction we obtain a manifold \(M_{k-1}\) and a bounded homotopy equivalence \(f_{k-1}: M_{k-1}\to X\times S\quad 1\times \cdot \cdot \cdot \times S\quad 1\times {\mathbb{R}}.\) We show that the Siebenmann obstruction \(\sigma_ 1\) in \(\tilde K_ 0({\mathbb{Z}}\pi_ 1M_{k-1})\) to factoring \(M_{k-1}=N_ 1\times {\mathbb{R}}\) is an element of \(\tilde K_{-k+1}({\mathbb{Z}}\pi_ 1X)\). If \(\sigma_ 1=0\), let \(\beta\) be the automorphism of \(\tilde K_ 0\) induced by mapping the class of a projective P to the class of the projective \(\bar P\) consisting of antihomomorphisms from P into the coefficient ring \({\mathbb{Z}}\pi_ 1(X\times S\) \(1\times \cdot \cdot \cdot \times S\) 1) (k-2 S 1 factors). Then subsequent obstructions lie in the quotients \(\tilde K_{-1}/image (1+\beta).\) That is, if \(\sigma_ 1\) vanishes, there exists a sequence of obstructions \(\sigma_ i\) in \(\tilde K_{-k+i}({\mathbb{Z}}\pi_ 1X)/image (1+\beta),\) \(i=2,...,s\), such that \(M=N_ s\times {\mathbb{R}}\quad s,\) for some PL manifold \(N_ s\) if and only if each \(\sigma_ i\) vanishes.”
Reviewer: H.-J.Munkholm

MSC:
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57Q30 Engulfing
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