A note on the Nielsen realization problem for K3 surfaces

Abstract

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a K3 surface. The first theorem is that the natural map π₀ (Diff(K3)) → Aut(H²(K3;Z)) has a section over its image. The second is that there exists a subgroup G of π₀ (Diff(K3)) of order two over which there is no splitting of the map (Diff(K3) → π₀ (Diff(K3)), but there is a splitting of Homeo(K3) → π₀(Homeo(K3)) over the image of G in π₀(Homeo(K3)), which is non-trivial. The third is that the map π₁(Diff(K3)) → π₁(Homeo(K3)) is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K3 surfaces.