Finite Group Actions On 4-Manifolds

In this dissertation, by using the Seiberg-Witten theory, equivariant K-theory, G-indextheorem(G-signature formula, G-Spin theorem) and Lefschetz fixed points formula and so on,we discuss finite group actions on 4-manifolds. The main research work consists of the following:1. Smooth finite group actions on 4-manifolds;2. Locally linear pseudofree group actions on elliptic surfaces.In Chapter 1, we give a review about the Seiberg-Witten theory and its applications,meanwhile we also introduce the main achievements in this research field obtained by somemathematicians.In Chapter 2, we give some preparations with an emphasis on the basic theory of Seiberg-Witten invariants, we also introduce the technique of the finite dimensional approximation ofSeiberg-Witten theory, basic knowledge of finite group actions on manifolds and K-theory.In Chapter 3, by using the technique of finite dimensional approximation of Seiberg-Wittentheory, the equivariant K-theory and some other methods, we study the problem of topologicalrestriction when there are smooth groups acting on Spin 4-manifolds such as the alternatinggroup A₅, cyclic group Z₆ and symmetric group S₃ with actions of odd type, consequently weimprove Furuta's 10/8 theorem under the condition of group actions. In particular, when thereis an alternating group A₅ action on a Spin 4-manifold X, we obtain: If X is smooth Spin4-manifold with non-positive signature and b₁(X)=0, denote k=-σ(X)/16 and m=b₂⁺(X),then 2k+3≤m if b₂⁺(X/)+b₂⁺(X/)≠2b₂⁺(Z/A₅) and b₂⁺(X/)≠0,where and are subgroups of A₅ generated by the elements s=(abcde)∈A₅and t=(abc)∈A₅ respectively. Besides, we get simple expression of the K-theory degree andthe G-index of the equivariant Dirac operator. At last, we discuss a concrete example of thealternating group A₅ action on K3 surfaces.We also study alternating group A₅ actions on the homotopy S²Ã—S² in this chapter. Undersome conditions, we simplify the G-index of the G equivariant Dirac operator to Ind_A5D_X=a(1-2ρ₁+ρ₄)+b(ρ₂-ρ₁), where a, b are integers. Furthermore, using the same methods, westudy alternating group A₅ actions on the connected sum of n copies of S²Ã—S².In Chapter 4, we apply the G-signature formula, G-Spin theorem and Lefschetz fixed pointsformula to get a totally topological classification of locally linear pseudofree Z₃ actions on elliptic surfaces, we prove that the locally linear pseudofree Z₃ action on elliptic surfaces E(4) belongsto ten types and nine of them can actually be realized by locally linear pseudofree Z₃-actionson elliptic surfaces E(4), we also give the realization theorem. Meanwhile, by using of the modp vanishing theorem of Seiberg-Witten invariants, we prove the existence of such actions whichcan not be realized as smooth actions on the standard smooth elliptic surfaces.