Elliptic surfaces and some simple exotic \(\mathbb{R}^ 4\)'s
type: Item
instance of: scholarly article
title: Elliptic surfaces and some simple exotic \(\mathbb{R}^ 4\)'s
zbMATH Open document ID: 0868.57023
author: Žarko Bižaca
author: Robert E. Gompf
published in: Journal of Differential Geometry
publication date: 1996-12-08T00:00:00Z
review text: The main purpose of the present paper is to exhibit some exotic \(R^4\)'s which admit simple, explicit description. The authors focus their attention on those examples of exotic \(R^4\)'s which arise from the \(h\)-cobordism theorem. They use a generalization of \textit{S. Akbulut}'s construction [J. Differ. Geom. 33, 335-356 (1991; Zbl 0839.57015)] to obtain a sequence of non-product \(h\)-cobordisms. The 4-manifolds involved in these constructions are minimal elliptic surfaces, \(E(n)\), and their logarithmic 0-transforms, \(E(n;0)\). Akbulut's \(h\)-cobordism is between \(X\#k \overline {CP}^2\) and \(E(2;0) \#k \overline {CP}^2\), where \(X\) is a homotopy \(K3\) surface and \(k\) is an arbitrary positive integer. In the present paper the authors show that the homotopy \(K3\) surface used by Akbulut is the standard \(K3\) surface, that is \(X=E(2)\). Furthermore, Akbulut had described the surgery induced by his \(h\)-cobordism as the result of deleting a Mazur manifold from \(X\# \overline{CP}^2\) and then gluing it back by an identification of the boundaries that differs from the initial one by an involution. The authors generalize Akbulut's construction to obtain an explicit \(h\)-cobordism between \(E(n)\#2 \overline {CP}^2\) and \(E(n;0) \# \overline{CP}^2\), for any integer \(n\geq 2\).
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 955317
zbMATH Keywords: exotic smooth structures on 4-manifolds
zbMATH Keywords: Casson handle
zbMATH Keywords: knots
zbMATH Keywords: \(h\)-cobordism theorem
zbMATH Keywords: elliptic surfaces
Wikidata QID: Q115177589
reviewed by: Viktor Abramov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214458322
OpenAlex ID: W1542739030
DOI: 10.4310/JDG/1214458322
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description: scientific article; zbMATH DE number 955317