Stable diffeomorphism of compact 4-manifolds
type: Item
instance of: scholarly article
title: Stable diffeomorphism of compact 4-manifolds
zbMATH Open document ID: 0589.57017
DOI: 10.1016/0166-8641(84)90004-X
author: Robert E. Gompf
published in: Topology and its Applications
publication date: 1984-01-01T00:00:00Z
review text: The author proves that if two closed smooth orientable 4-manifolds M and N are homeomorphic they are stably diffeomorphic i.e. there exists \(r\in {\mathbb{N}}\) such that M{\#}r(S\({}^ 2\times S^ 2)\) is diffeomorphic to N{\#}r(S\({}^ 2\times S^ 2)\). The same result holds for nonorientable manifolds if M and N are replaced by the connected sum with the nontrivial \(S^ 2\)-bundle over \(S^ 2\). The proof is based on the construction of ''bad'' smoothings of some simple manifolds which together with results of Quinn reduces the problem to stable smoothing theory of Lashof and Shaneson. The ''bad'' smoothings are constructed with the help of Freeman's classification of 1-connected 4-manifolds. (The reviewer has in 1982 independently obtained a slight generalization of these results. This will be published in a book with the title ''Surgery and duality''.)
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 57R10
zbMATH DE Number: 3947229
zbMATH Keywords: smooth orientable 4-manifolds
zbMATH Keywords: homeomorphic
zbMATH Keywords: stably diffeomorphic
zbMATH Keywords: ''bad'' smoothings
zbMATH Keywords: stable smoothing theory
MaRDI profile type: Publication
cites work: Some new four-manifolds
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description: scientific article; zbMATH DE number 3947229