Orientation-reversing homeomorphisms in surface geography
It is proved that all known simply-connected algebraic surfaces of general type with positive signature and even first Chern number can be used to give examples of non-smoothable orientation-reserving homeomorphisms between algebraic surfaces. This leads to: Theorem. There are infinitely many pairs of simply-connected algebraic surfaces of general type which are orientation-reversing homeomorphic, but not diffeomorphic. The author conjectures that the non-smoothability of these homeomorphisms is a special case of the following: Conjecture 1. If two algebraic surfaces with finite fundamental groups are orientation-reversing diffeomorphic, then they are homeomorphic to a geometrically ruled rational surface. In particular they are simply-connected. Conjecture 2. There is no connected and simply-connected orientable smooth 4-manifold for which Donaldson's polynomial invariants are defined and nontrivial for both choices of orientation. In fact, it is shown in this paper that the above conjectures would follow from certain standard conjectures about 4-manifolds, made independently of considerations of orientation-reversing homeomorphisms, cf. \textit{R. Mandelbaum} in Bull. Am. Math. Soc., New Ser. 2, 1-159 (1980; Zbl 0476.57005). It follows that there are hardly any smooth closed 4- manifolds with finite fundamental groups which admit complex structures compatible with both choices or orientation, in contrast with the case of almost complex manifolds and with the case of infinite fundamental groups. This answers a question raised by \textit{A. Beauville} [in Appendix B (p. 41-43) to the article by \textit{M. Demazure} in Géométrie de surfaces K3: Modules et périodes, Sémin. Palaiseau 1981-1982, Astérisque 126, 29-43 (1985; Zbl 0574.14032)].
non-smoothable orientation-reserving homeomorphisms between algebraic surfaces
scientific article; zbMATH DE number 56288