On four-dimensional Poincaré duality cobordism groups
type: Item
instance of: scholarly article
title: On four-dimensional Poincaré duality cobordism groups
zbMATH Open document ID: 1394.57019
author: Alberto Cavicchioli
author: Friedrich Hegenbarth
author: Fulvia Spaggiari
published in: Mediterranean Journal of Mathematics
publication date: 2018-06-01T00:00:00Z
review text: The Poincaré duality spaces in this paper are finite CW-complexes and the paper considers the bordism groups \(\Omega^{PD}(P)\) where \(P\) is a Poincaré duality space. The main result is to show that there is an exact sequence \[ 0 \;\rightarrow \;L_{4}(\pi_{1}(P)) / A_{4}(H_{2}(B\pi_{1}(P), L_{2})) \;\rightarrow \;\tilde{\Omega}^{PD}_{4}(P) \;\rightarrow \;{\mathbb Z}_{8} \;\rightarrow \;0 \] where \(A_{4}\) is the assembly map and \(\tilde{\Omega}^{PD}_{4}(P)\) is the kernel of the canonical map \(\Omega^{PD}_{4}(P) \;\rightarrow \;H_{4}(P, {\mathbb Z}) \cong {\mathbb Z}\). The arguments make use of spectral sequences and specific information on homotopy groups of the Thom space \(\mathbf{MSG}\). In addition the authors establish the exactness of the following sequence \[ \Omega_{4}^{TOP} (P) \;\longrightarrow \;\Omega_{4}^{PD}(P) \;\overset{s}{\longrightarrow} \;{\mathcal S}_{4} \;\longrightarrow \;0 \] \noindent where \(s\) is Ranicki's total surgery obstruction map.
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 57Q10
zbMATH DE Number: 6879901
zbMATH Keywords: Poincaré complexes
zbMATH Keywords: four-manifold
zbMATH Keywords: cobordism group
zbMATH Keywords: homotopy type
zbMATH Keywords: Wall group
zbMATH Keywords: spectral sequence
zbMATH Keywords: obstruction theory
zbMATH Keywords: homology with local coefficients
zbMATH Keywords: total surgery obstruction
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: s00009-018-1102-3
OpenAlex ID: W2793321405
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DOI: 10.1007/S00009-018-1102-3
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description: scientific article; zbMATH DE number 6879901