A NOTE ON THE NIELSEN REALIZATION PROBLEM FOR K3 SURFACES

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 ir0(Diff (K3))-+ Aut(H2(K3;Z)) has a section over its im-age. The second is that there exists a subgroup G of ir0(Dif f (K3)) of order two over which there is no splitting of the map Dif f (K3)-+ ir0(Dif f (K3)), but there is a splitting of Homeo(K3)-+ ir0(Homeo(K 3)) over the image of G in ir0(Homeo(K3)), which is non-trivial. The third is that the map ir1(Dif f (K3))-+ ir1(Homeo(K 3)) is not surjective. Our proof of these re-sults is based on Seib erg-Witten theory and the global Torelli theorem for K3 surfaces.