Assembly maps and realization of splitting obstructions
type: Item
instance of: scholarly article
title: Assembly maps and realization of splitting obstructions
zbMATH Open document ID: 1209.57026
author: Alberto Cavicchioli
author: Fulvia Spaggiari
author: Yuri V. Muranov
published in: Monatshefte für Mathematik
publication date: 2009-12-14T00:00:00Z
review text: Let \(X^n\) be a closed, connected topological manifold of dimension \(n\) (in this paper \(n\geq 5\) usually) with fundamental group \(\pi\). In the surgery theoretic classification of manifolds it is important to determine those elements in the Wall group \(L_n(\mathbb{Z}\pi)\) that can be realized as the normal maps of closed manifold surgery problems on \(X\) [see \textit{C. T. C. Wall}, ``Surgery on compact manifolds. 2nd ed.'', Mathematical Surveys and Monographs. 69. Providence, RI: American Mathematical Society (1999; Zbl 0935.57003)]. There is a geometric approach to this problem based on splitting simple homotopy equivalences to \(X\) along certain one-sided submanifolds of \(X\) of various codimensions. The paper under review continues this approach by first recasting results of \textit{A. F. Kharshiladze} [Math. Notes 41, 312--315 (1987); translation from Mat. Zametki 41, No.~4, 557--563 (1987; Zbl 0673.57026) and Russ. Math. Surv. 42, No.~4, 65--103 (1987); translation from Usp. Mat. Nauk 42, No.~4(256), 55--85 (1987; Zbl 0671.57020)] in \textit{A. A. Ranicki}'s algebraic theory of surgery [``Algebraic L-theory and topological manifolds''. Cambridge Tracts in Mathematics 102. Cambridge: Cambridge University Press (2008; Zbl 1143.57001)]. The authors then go further on the geometric front by using the Browder-Quinn surgery obstruction groups for filtered manifolds.
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 19G24
Mathematics Subject Classification ID: 58A35
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 5646274
zbMATH Keywords: topological manifolds
zbMATH Keywords: surgery theory
zbMATH Keywords: splitting obstructions
zbMATH Keywords: Wall groups
zbMATH Keywords: Browder--Livesay invariants
zbMATH Keywords: Browder--Quinn surgery obstructions
zbMATH Keywords: Ranicki's algebraic surgery theory
MaRDI profile type: Publication
full work available at URL: s00605-009-0129-8
OpenAlex ID: W2004956124
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DOI: 10.1007/S00605-009-0129-8
description: scientific article; zbMATH DE number 5646274