A note on the Nielsen realization problem for K3 surfaces

We will show the following three theorems on diffeomorphism and homeomorphism groups of a K 3 K3 surface. The first theorem is that natural map alttext="pi 0 left-parenthesis upper D i f 3 right-parenthesis right-arrow A u t H squared semicolon double-struck Z right-parenthesis"> ? 0 ( D i f stretchy="false">) stretchy="false">? A u t H 2 ; Z encoding="application/x-tex">\pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z})) has section over its image. second there exists subgroup G"> G encoding="application/x-tex">G _{0}(Diff(K3)) order two which no splitting pi encoding="application/x-tex">Diff(K3) \pi , but o m e o m e encoding="application/x-tex">Homeo(K3) _{0}(Homeo(K3)) image in non-trivial. third 1 1 _{1}(Diff(K3)) _{1}(Homeo(K3)) not surjective. Our proof these results based Seiberg-Witten theory global Torelli for surfaces.