Alternating Group A₄ Actions On K3 Surfaces

In this paper, we mainly study the alternating group A₄ actions on spin 4-manifolds. First we discuss the special action of A₄ on K3 surfaces of Fermat type, we compute the fixed points set of this action and some invariants of the group action, also we get some topological properties. Then we study the A₄ actions on general K3 surfaces and obtain some similar results.In the first Chapter, we give a review of this research field, especially some beautilul results about this research in recent years, some great achievements on this subject are listed, which include domestic and abroad. In the end of this chapter, the main results of this paper are given.In Chapter 2, we give some preparations of this paper. In the first part, the fundamental knowledge of 4-manifold is mainly introduced. And then, in the second part we introduce some knowledge about linear representations of finite groups. At last, in the third part we discuss the alternating group A₄ and give the character table for it.In Chapter 3, we mainly discuss a specific action of A₄ on K3 surfaces of Fermat type. In this part, the numbers of fixed points for conjugate classes of A₄ are computed and relative topology properties are discussed.In Chapter 4, we get some topology properties when A₄ acts on homotopy K3 surfaces with #Fix(t) = 6 from the specific action of A₄ mentioned in Chapter 3.In Chapter 5, we go forward extend the results we have obtained to homotopy K3 surfaces of general type and give some restriction of G-equivariant index.In the last part, we make a conclusion of this paper and give some problems or conjuctures which should be studied in the future.