On the diffeomorphism groups of elliptic surfaces (Q1377075)

It is well known that in dimension four corresponding notions from topological and smooth manifold theory differ much more than in in any other dimension. It may suffice to mention the results on non-existence for smooth structures on some topological four-manifolds obtained by \textit{S. K. Donaldson} [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)] and multitudes of them on others obtained among others by \textit{R. Friedman} and \textit{J. W. Morgan} [J. Differ. Geom. 27, No. 2, 297-369 (1988; Zbl 0669.57016)]. Along another line one can mention work by various authors on the respective groups of self-equivalences, homeomorphisms and diffeomorphisms of four-manifolds, studied via their representations on the second homology group. The present author makes a contribution to this latter line of research. For relatively minimal elliptic surfaces with positive Euler number, he determines the image of the natural representation of the group of orientation preserving self-diffeomorphisms on the second homology group reduced modulo torsion.