Cyclic Group Actions On 4-Manifolds

In this dissertation, by using the method of A.L. Edmonds and J.H. Ewing which is on how to construct a locally linear group action on 4-manifold, Rohlin's theorem, G-index theorem(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. Nonmoothable involutions on spin 4-manifolds;2. Nonsmoothable Zp-actions on spin 4-manifolds (p is an odd prime);3. Symplectic cyclic group actions on elliptic surfaces.In Chapter 1, we give a review about the 4-manifolds theory, and we also introduce the main achievements obtained by some mathematicians in this research field.In Chapter 2, we give some preparations with an emphasis on the basic theory of symplectic 4-manifolds, spin geometry, finite groups actions on 4-manifolds. G-index theorem and Lefschetz fixed points formula, and we also introduce the method of constructing locally linear group actions on 4-manifolds which is given by Edmonds and Ewing.In Chapter 3, we construct a locally linear involution on a 4-manifold by using the equiv-ariant handles surgery. We also give a constraint on smooth involutions on spin 4-manifolds by considering Rohlin's theorem. The locally linear involution we construct violates the constraint. By considering the relation between spin types and sign assignment of fixed points, we can see that the involution is nonsmoothable with respect to any possible smooth structure on the 4-manifold. The main result is as follows:Let X be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to n(-E8)⊕mH, where H is the hyperbolic form. If n=2 mod 4, then there exists a locally linear pseudofree Z2-action on X which is nonsmoothable with respect to any possible smooth structure on X.Freedman's theorem says that the homeomorphism type of a simply-connected 4-manifold is uniquely determined by its intersection form if the intersection form is even. Thus our result can be applied to many spin 4-manifolds incluing certain elliptic surfaces E(n).For a closed, simply-connected, spin topological 4-manifold except S4 and S2 x S2, we give a interger such that for any prime p larger than this integer there exists a homologically trivial, pseudofree, nonsmoothable locally linear action of Zp for any possible smooth structure on X. Our result is an improvement of Kiyono's result.In Chapter 4, we use the G-signature formula, G-Spin theorem and Lefschetz fixed points formula to study pseudofree, homologically trivial, symplectic Zp-actions on elliptic surfaces E(n), where p takes 2,3,5,7. It can be seen that a homologically trivial(over Q coefficients), pseudofree, symplectic Z2-action must be trivial, and there is no homologically trivial(over Q coefficients),pseudofree, symplectic Z3-action on E(n). When p=5, we discribe the fixed point set completely. Also, we discribe the fixed point set of a symplectic Z7-action on E(n) by an example.