Classification of homotopy Wall's manifolds
type: Item
instance of: scholarly article
title: Classification of homotopy Wall's manifolds
zbMATH Open document ID: 1115.55004
author: Himadri Kumar Mukerjee
published in: Topology and its Applications
publication date: 2007-01-09T00:00:00Z
review text: If \(X\) is of a compact, connected, smooth, PL or topological manifold with or without boundary \(\partial X\), a homotopy smoothing (resp., homotopy PL or TOP triangulation) of \(X\) is a pair \((M,f)\), where \(M\) is a smooth, PL or topological manifold and \(f \colon (M, \partial M) \to (X, \partial X)\) is a simple homotopy equivalence of pairs, for which \(f| _{\partial M} \colon \partial M \to \partial X\) is a diffeomorphism (resp. a PL or TOP homeomorphism). Two homotopy smoothings (resp. PL or TOP triangulations) \((M,f)\) and \((M',f')\) are said to be equivalent if there is a diffeomorphism (resp. PL or TOP homeomorphism) \(h:(M, \partial M) \to (M', \partial M')\) such that the maps \(f'\circ h\) and \(f\) are homotopic relative to the boundary \(\partial M\). The set of equivalence classes of homotopy smoothings (resp. PL or TOP triangulations) of the manifold \(X\) is denoted by \(hS(X)\) (resp. \(hT_{\text{PL}}(X)\) or \(hT_{\text{TOP}}(X)\)). The aim of this paper is to determine completely the sets \(hT_{\text{PL}}(X)\) and \(hT_{\text{TOP}}(X)\), and partially the sets \(hS(X)\) for \(X=D^m \times Q(r,s)\), \(r,s >1\), where \(D^m\) is the disk of dimension \(m\geq 0\) and \(Q(r,s)\) is \textit{C. T. C. Wall}'s manifold defined in [Ann. Math. (2) 72, 292--311 (1960; Zbl 0097.38801)] as a mapping torus of a Dold manifold \(P(r,s)\), where \(P(r,s)\) can be seen as the total space of a fibre bundle over \(\mathbb RP^r\) with fibre \(\mathbb CP^s\). The method for determining the sets \(hS(X)\) (resp. \(hT_{\text{PL}}(X)\) or \(hT_{\text{TOP}}(X)\)) is the study of the Sullivan-Wall surgery exact sequence and, consequently, the determination of Wall's surgery obstruction groups \(L_n(\pi_1(X), \omega(X))\) and the sets of normal invariants of \(X\) \([X/\partial X, G/\text{CAT}]\), where \(\text{CAT}=\text{O,PL,TOP}\).
reviewed by: Fulvia Spaggiari
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 57R67
zbMATH DE Number: 5083216
zbMATH Keywords: normal invariant
zbMATH Keywords: surgery invariant
zbMATH Keywords: Browder-Livesay invariant
zbMATH Keywords: Arf invariant
MaRDI profile type: Publication
full work available at URL: j.topol.2006.03.002
OpenAlex ID: W2087948664
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DOI: 10.1016/J.TOPOL.2006.03.002
description: scientific article; zbMATH DE number 5083216