Reducible smooth structures on 4-manifolds with zero signature
type: Item
instance of: scholarly article
title: Reducible smooth structures on 4-manifolds with zero signature
zbMATH Open document ID: 1310.57042
author: Anar Akhmedov
author: Masashi Ishida
author: B. Doug Park
published in: Journal of Topology
publication date: 2014-09-30T00:00:00Z
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57R55
zbMATH DE Number: 6349714
zbMATH Keywords: exotic smooth structures
zbMATH Keywords: 4-manifolds
zbMATH Keywords: reducible manifolds
zbMATH Keywords: symplectic manifolds
zbMATH Keywords: signature zero
zbMATH Keywords: stable cohomotopy refinement of the Seiberg-Witten invariant
zbMATH Keywords: positive scalar curvature
MaRDI profile type: Publication
full work available at URL: jtt043
OpenAlex ID: W2020795286
DOI: 10.1112/JTOPOL/JTT043
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review text: In this paper, the authors deal with simply-connected, zero-signature 4-manifolds. Such 4-manifolds are homeomorphic to \(n(S^2\times S^2)\) or \(m({\mathbb C}P^2\#\overline{{\mathbb C}P^2})\). In [\textit{A. Akhmedov} and \textit{B. D. Park}, J. Gökova Geom. Topol. GGT 2, 1--13 (2008; Zbl 1184.57017)] and [Math. Res. Lett. 17, No. 3, 483--492 (2010; Zbl 1275.57039)], it was proven that if an odd integer \(n\), and \(m\) is sufficiently large, then the 4-manifolds above admit infinitely many irreducible exotic smooth structures. When \(n\) and \(m\) are even integers, whether or not the 4-manifolds admit any irreducible smooth structure is an open question.NEWLINENEWLINEThis article proves that \(n(S^2\times S^2)\) and \(m({\mathbb C}P^2\#\overline{{\mathbb C}P^2})\) admit reducible smooth structures when \(n,m\) obey one of following conditions: {\parindent=6mm \begin{itemize}\item{} \(n=2p\), \(p\geq 275\), and \(p\equiv 1(2)\). \item{} \(n=2p\), \(p\geq 550\), and \(p\equiv 2(4)\). \item{} \(m=2q\), \(q\geq 51\), and \(p\equiv 1(2)\). \item{} \(m=2q\), \(q\geq 102\), and \(q\equiv 2(4)\). \item{} \(n\geq 825\) and \(n\equiv 1(4)\) \item{} \(m\geq 153\) and \(m\equiv 1(4)\) NEWLINENEWLINE\end{itemize}} Furthermore, the last two classes admit vanishing Seiberg-Witten invariants. This implies that these classes are not obtained as blow-up of certain 4-manifolds with non-vanishing Seiberg-Witten invariant. The first four classes give infinitely many non-diffeomorphic reducible smooth structures not having any metrics with positive scalar curvature. These results are also interesting when compared with the famous and classical theorem by \textit{C. T. C. Wall} [J. Lond. Math. Soc. 39, 141--149 (1964; Zbl 0131.20701)].NEWLINENEWLINETo distinguish these smooth structures, the authors use the stable cohomotopy refinement of the Seiberg-Witten invariant (Bauer-Furuta invariant) and the non-vanishing theorem. As for the construction of manifolds, they use Akhmedov, Park, a…
reviewed by: Motoo Tange
description: scientific article; zbMATH DE number 6349714