Stable bundles and differentiable structures on certain elliptic surfaces (Q1088743)

The authors extend a result of \textit{S. K. Donaldson} [C. R. Acad. Sci., Paris, Sér. I 301, 317-320 (1985; Zbl 0584.57010)] that shows that a rational elliptic surface obtained from the projective plane by blowing up 9 base points of a pencil of cubic curves is not diffeomorphic to its homogeneous space \(X_{2,q}\) with two multiple fibres of \(multiplicity^ 2\) and odd q. The fact that the both surfaces are simply-connected and homotopy equivalent was observed earlier by the reviewer. Using a similar technique based on computation of Donaldson's invariant, the authors show that for two prime numbers p and q the surfaces \(X_{2,q}\) and \(X_{2,p}\) are diffeomorphic if and only if \(p=q\). A more general result about the diffeomorphy of surfaces \(X_{p,q}\) was independently obtained by R. Friedman and J. Morgan (to appear).