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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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