On the diffeomorphism types of elliptic surfaces with multiple fibers
type: Item
instance of: scholarly article
title: On the diffeomorphism types of elliptic surfaces with multiple fibers
zbMATH Open document ID: 0595.14028
DOI: 10.1007/BF01388750
author: Masaaki Ue
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143355
review text: The subject of this paper is to study the diffeomorphism types of the relatively minimal (and closed) elliptic surfaces and the following theorem is proved (C.T.C.~Wall also proved the same result independently): The diffeomorphism type of an elliptic surface \(\pi : M\to S\) is completely determined by its Euler number e(M) and its fundamental group \(\pi_ 1M\) if \(e(M)>0\) and \(\pi_ 1M\) is not cyclic. This generalizes \textit{Y. Matsumoto's} theorem [Proc. Japan Acad., Ser. A 61, 55-58 (1985; Zbl 0586.14035)] which states that the diffeomorphism type of the elliptic surface without multiple fibers is determined by its Euler number and the genus of its base Riemann surface. (The case with base Riemann surface is \(C{\mathbb{P}}^ 1\) is already proved by A. Kas and B. Moishezon.) The assumption that \(\pi_ 1M\) is not cyclic is crucial in the above theorem. In fact the Dolgachev surface is homeomorphic but not diffeomorpic to the rational elliptic surface in general as was proved by Donaldson, Morgan and Friedman. A short comment on the case when \(e(M)=0\) is added.
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 3957299
zbMATH Keywords: diffeomorphism type of an elliptic surface
zbMATH Keywords: Euler number
MaRDI profile type: Publication
cites work: Q3706954
cites work: Q3739910
cites work: Q4120674
cites work: On compact analytic surfaces. II
cites work: Q3941524
cites work: Diffeomorphism types of elliptic surfaces
cites work: Complex surfaces and connected sums of complex projective planes
cites work: On irreducible 3-manifolds which are sufficiently large
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description: scientific article; zbMATH DE number 3957299