A fake compact contractible 4-manifold
type: Item
instance of: scholarly article
title: A fake compact contractible 4-manifold
zbMATH Open document ID: 0839.57015
author: Selman Akbulut
published in: Journal of Differential Geometry
publication date: 1992-06-25T00:00:00Z
review text: Two smooth contractible 4-manifolds \(V,W\) are constructed with the same boundary so that they are homeomorphic but are not diffeomorphic relative to the boundary. The differential structure on \(V\) arises from that of \(W\) as the pullback under a self-homeomorphism \(F\) on \(W\) which extends a diffeomorphism \(f\) of the boundary. That \(f\) does not extend to a diffeomorphism plays a key role and is related to the fact that a knot \(\alpha\) in \(\partial W\) is not slice in \(W\). \(W\) is a Mazur manifold formed by attaching a 2-handle to \(S^1 \times B^3\) and \(\alpha\) is the loop given by \(S^1 \times 1 \subset S^1 \times S^2 = \partial (S^1 \times B^3)\). \textit{E. C. Zeeman} [Topology 2, 341-358 (1963; Zbl 0116.40801)] raised the question as to whether \(\alpha\) is slice, and this paper finally settles this question negatively. This can be contrasted with an earlier result of the author and \textit{R. Kirby} [Mich. Math. J. 26, 259-284 (1979; Zbl 0443.57011)] that \(\alpha\) is slice in another smooth contractible manifold with the same boundary. The manifold \(W\) arises as a piece of two homotopy equivalent closed manifolds \(M \#\) and \(\widetilde M\). If \(f\) were to extend to a diffeomorphism, these manifolds would be diffeomorphic. \(\widetilde M\) is shown to be diffeomorphic to (3)\(\# (20)\), and thus has vanishing Donaldson polynomials by \textit{S. K. Donaldson}'s connected sum theorem [Topology 29, 257-315 (1990; Zbl 0715.57007)]. However, the fact that \(M\) is a homotopy \(K3\) surface containing an embedded \(\Sigma (2, 3, 7)\) is used to show that there is a nonvanishing Donaldson polynomial for \(M \#\) by an argument due to \textit{R. Fintushel} and \textit{R. Stern} [J. Differ. Geom. 34, 255-265 (1991; Zbl 0754.14021)]. A key role in the proof is played by Kirby calculus constructions of diffeomorphisms of various 3- and 4-manifolds. There is also a description of a nontrivial \(h\)-cobordism between \(V\) and \(W\) constructed …
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 11236
zbMATH Keywords: handle
zbMATH Keywords: \(h\)-cobordism
zbMATH Keywords: 4-manifolds
zbMATH Keywords: knot
zbMATH Keywords: diffeomorphism
zbMATH Keywords: Donaldson polynomials
zbMATH Keywords: Kirby calculus
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214446320
OpenAlex ID: W1589415759
Wikidata QID: Q115181672
DOI: 10.4310/JDG/1214446320
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description: scientific article; zbMATH DE number 11236