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On manifolds homeomorphic to \({\mathbb{C}}P^ 2\#8\overline{{\mathbb{C}}P}^ 2\). (English) Zbl 0691.57008
The following problem has been of considerable interest recently: Which is the smallest value of k (denote it by \(k_ 0)\) such that there is a complex surface homeomorphic but not diffeomorphic to \({\mathbb{C}}P^ 2\#k\overline{{\mathbb{C}}P}^ 2\), equipped with the standard smooth structure? It was shown earlier by R. Friedman and J. W. Morgan [J. Differ. Geom. 27, 297-369 (1988; Zbl 0669.57016)] and, independently, by C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)] that \(k_ 0\leq 9\). The main result of the paper under review is that \(k_ 0\leq 8\). Namely, it is demonstrated that the Barlow surface is not diffeomorphic to \({\mathbb{C}}P^ 2\#8\overline{{\mathbb{C}}P}^ 2\) (although these two 4-manifolds are homeomorphic by M. H. Freedman’s classification theorem). A key ingredient of the proof is a new invariant \(\phi\) defined for oriented Riemannian 4-manifolds X such that \(b_ 1(X)=0\), \(b^+_ 2(X)=1\), and \(b^-_ 2(X)=8k\), which is simpler than S. K. Donaldson’s \(\Gamma_ X\)-invariant and polynomial invariants.
Reviewer: D.Repovš

MSC:
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
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