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\(\Gamma\)-type-invariants associated to PU(2)-bundles and the differentiable structure of Barlow’s surface. (English) Zbl 0691.57007
In 1987 S. K. Donaldson [J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010)] introduced the \(\Gamma_ X\)-invariant for compact, oriented smooth 4-manifolds X with \(b_ 1(X)=0\) and \(b_+(X)=1\). There are certain technical difficulties which make the definition of \(\Gamma_ X\) somewhat complicated. The purpose of the paper under review is to introduce similar but simpler invariants using moduli spaces of PU(2)- connections instead of SU(2)-connections. As an application, the authors study the Dolgachev surfaces, the Barlow surface [R. Barlow, Invent. Math. 79, 293-301 (1985; Zbl 0561.14015)], blown up in one point, and the complex projective plane, blown up in 9 points. They show that surfaces of different Kodaira dimension cannot be diffeomorphic.
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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References:
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