On the diffeomorphism types of elliptic surfaces with multiple fibers
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.
diffeomorphism type of an elliptic surface
scientific article; zbMATH DE number 3957299
scientific article