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 K 3 K3 surface. The first theorem is that the natural map π 0 ( D i f f ( K 3 ) ) → A u t ( H 2 ( K 3 ; Z ) ) \pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z})) has a section over its image. The second is that there exists a subgroup G G of π 0 ( D i f f ( K 3 ) ) \pi _{0}(Diff(K3)) of order two over which there is no splitting of the map D i f f ( K 3 ) → π 0 ( D i f f ( K 3 ) ) Diff(K3) \to \pi _{0}(Diff(K3)) , but there is a splitting of H o m e o ( K 3 ) → π 0 ( H o m e o ( K 3 ) ) Homeo(K3) \to \pi _{0}(Homeo(K3)) over the image of G G in π 0 ( H o m e o ( K 3 ) ) \pi _{0}(Homeo(K3)) , which is non-trivial. The third is that the map π 1 ( D i f f ( K 3 ) ) → π 1 ( H o m e o ( K 3 ) ) \pi _{1}(Diff(K3)) \to \pi _{1}(Homeo(K3)) is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K 3 K3 surfaces.