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Zbl 0663.14027
Lübke, Martin; Okonek, Christian
Differentiable structures of elliptic surfaces with cyclic fundamental group. (English)
[J] Compos. Math. 63, 217-222 (1987). ISSN 0010-437X; ISSN 1570-5846/e

Let $X\sb{p,q}$ be elliptic surfaces over ${\bbfP}\sb 1$ with at most 2 multiple fibres of multiplicities p and q. It is known that $\pi\sb 1(X\sb{p,q})={\bbfZ}/k$ where $k=g.c.d.(p,q)$. In the special case where $k=1$ and $p\sb g=0$, one has the following theorem: \par The surfaces $X\sb{2,q}$ with $q=2n+1$ are pairwise differentiably inequivalent [see e.g. {\it C. Okonek} and {\it A. Van de Ven}, Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)]. \par This result is interesting for two reasons; first of all, the surfaces $X\sb{p,q}$ are all homeomorphic to ${\bbfP}\sb 2$ with 9 points blown up [see {\it M. H. Friedman}, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. Furthermore, this theorem is in sharp contrast with a recent result by {\it M. Ue} [Invent. Math. 84, 633-643 (1986; Zbl 0595.14028)] who shows that for elliptic surfaces with at least 3 multiple fibres or elliptic surfaces over base curves S with $g(S)>1$, their diffeomorphism type is completely determined by their homeomorphism type. \par In this paper, using their previous work and by computing Donaldson's invariants, the authors generalize the previous theorem to the cases where $k>1$. Also notice that, at least for odd k, the $X\sb{p,q}$ are homeomorphic in view of recent results by Hambleton and Kreck.
[Vo Van Tan]

MSC 2000:: *14J27 Elliptic surfaces
14E20 Coverings, fundamental group (mappings)
57R50 Diffeomorphisms

Keywords: cyclic fundamental group; elliptic surfaces; diffeomorphism type; homeomorphism type

Citations: Zbl 0613.14018; Zbl 0528.57011; Zbl 0595.14028

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