Study On Some Problems For Four-Manifolds

In this dissertation, around the group actions on four-manifolds and some related prob-lems, we use Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem to discuss some problems on the four-manifolds, including the following topics:1. The genus of surfaces representing some homology classes on four-manifolds;2. The homologically trivial cyclic group actions on symplectic elliptic surfaces;3. The automorphisms of the E8(?) E84-manifold with prime periods.In Chapter1, we give a review on the background, importance and application of the four-manifold theory. We also introduce main recent achievements and open questions in the field about group actions on four-manifolds and other relative problems.In Chapter2, basic concepts, knoweledges and tools are given, such as intersection forms, classification theorem and lattices, as well as group actions on four-manifolds, integral rep-resentations, realizing theorem, Spin structure, symplectic structure, Seiberg-Witten theory, G-signature formula and Lefchetz fixed point theorem.In Chapter3, we use the results from Seiberg-Witten theory, especially10/8-theorem, and classical methods about branched covering, to study embedded surfaces in four-manifolds, which represent some2-homology classes. We obtain some lower bounds of the genus of some embedded surfaces representing some2-homology classes divisible by2r in certain four-manifolds with H1(X;Z) finite.In Chapter4, we consider all the possible integral representations of the E8(?)E84-manifold under Z7、Z5and Z3actions respectively with G-signature formula. We recognize those rep-resentations that can be realizable by locally linear pseudofree actions and exclude those that cannot be realizable by locally linear actions with two-dimensional fixed components.In Chapter5, we start from some basic Seiberg-Witten knowledge of Taubes, and focus on the question of homologically trivial actions on symplectic elliptic surfaces, using the results of Chen and Kwasik about the union UiCi of finite J-holomorphic curves and the structure of fixed point set under symplectic actions. We discuss the pseudofree Zp actions on the symplec-tic homotopy elliptic surfaces E(n) with c12=0, and obtain there are no homologically trivial symplectic Zp actions on symplectic homotopy elliptic surfaces when p=3, and when p=5, there is a nontrivial symplectic homologically trivial action with some fixed datum. Meanwhile, by disccussing defects in the G-signature formula when p=2,3,5and other primes, we prove that all the homologically trivial cyclic actions preserving symplectic structures on the minimal symplectic elliptic surfaces E(4) with a disturb c1(K)·[ω]<16are trivial.