\(K_{-i}\) obstructions to factoring an open manifold
type: Item
instance of: scholarly article
title: \(K_{-i}\) obstructions to factoring an open manifold
zbMATH Open document ID: 0654.57009
DOI: 10.1016/0166-8641(88)90070-3
author: John L. Bryant
author: Peter S. Pacheco
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: 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.''
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57Q30
zbMATH DE Number: 4067880
zbMATH Keywords: negative K-theory
zbMATH Keywords: splitting off Euclidean factors
zbMATH Keywords: open PL manifold
zbMATH Keywords: bounded homotopy equivalence
zbMATH Keywords: radial engulfing
zbMATH Keywords: Siebenmann's twist- gluing
zbMATH Keywords: Siebenmann obstruction
reviewed by: Hans Jørgen Munkholm
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90070-3
OpenAlex ID: W1971939063
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description: scientific article; zbMATH DE number 4067880