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Zbl 0813.57019
Ebeling, Wolfgang; Okonek, Christian
Homology Hopf surfaces. (English)
[J] Compos. Math. 91, No.3, 277-304 (1994). ISSN 0010-437X; ISSN 1570-5846/e

A homology Hopf surface is a complex surface with the same rational homology groups as $S\sp 1 \times S\sp 3$. The classification of these was previously known, and the paper begins by reviewing it. Next the classification of Seifert ${\bold C}\sp x$-bundles over $P\sp 1$ is reviewed.\par Factoring out the action of $\langle \lambda \rangle \subset {\bold C}\sp x$ (with $\vert \lambda \vert > 1)$ on such a bundle gives a homology Hopf surface $M$, diffeomorphic to $S\sp 1 \times \Sigma$ where $\Sigma$ is a Seifert 3-manifold: it is deduced that any such $\Sigma$ may be obtained. The place of $M$ in the classification is determined. From this the authors deduce their main results: an identification of those homology Hopf surfaces diffeomorphic to a product $S\sp 1 \times \Sigma$; a proof that such $\Sigma$ are all Seifert manifolds and (when $\Sigma$ is also an integral homology sphere) a classification of complex structures on $S\sp 1 \times \Sigma$.
[C.T.C.Wall (Liverpool)]

MSC 2000:: *57N13 Topology of Euclidean 4-space, 4-manifolds
14J15 Analytic moduli, classification (surfaces)
32C15 Complex spaces

Keywords: product of the circle with a Seifert 3-manifold; rational homology 3- sphere; homology Hopf surface; complex surface; rational homology groups; diffeomorphic; Seifert 3-manifold; complex structures

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