Altenating Group A₅ Actions On K3 Surfaces

This dissertation investigates the spin alternating group A₅ actions on K3 surfaces. Beginning with some A₅ action on a special K3 surface,we obtain the fixed points set of this action and some invariants of the group action,as well as we get some topological properties.Furthermore,we generalize some results to homotopy K3 surfaces.The content of this paper is summarized as follows:1.We introduce the development of this research field and some recent achievements given by researchers at home and abroad.The main results of this paper are presented in the end.2.We present the fundamental knowledge of 4-manifolds,the smooth closed spin 4-manifolds, actions and linear representations of finite groups,as well as some properties and the character table for alternating group A₅.3.We discuss a specific action of A₅ on a special K3 surfaces and compute the numbers of fixed points for conjugate classes of A₅.We also give some invariants of the group action.Topological properties are consequently obtained at last.4.We generalize the results obtained to homotopy K3 surfaces and give some restriction of G-equivariant index.