在四维流形的研究中,有限群作用的同调刚性以及奇异光滑结构的存在性是两个重要内容。在引进了Seiberg-Witten理论以后，这两个方面都取得了突破性的进展，但仍有一些问题值得深入研究。有鉴于此,本项目将研究如下三方面内容。①运用Seiberg-Witten理论，G-Spin定理 以及Adams算子等工具，寻找光滑四维流形上同调平凡的奇素数阶周期微分同胚作用存在的限制条件，由此研究一些光滑四维流形上同调平凡群作用的存在性。②研究辛四维流形上同调平凡群作用存在的限制条件，并对同调平凡的非平凡奇素数阶循环群作用给出其不动点集的实现形式。③研究辛四维流形上奇异光滑结构的存在性。具体地，通过一些辛K3群(如交错群A_5 ，对称群S_5等)的作用，并结合Lefschetz 不动点定理，G-signature 定理等工具，对辛同伦K3群上奇异光滑结构的存在性进行估计。

The homological rigidity of finit group actions and existence of exotic smooth structure are two important aspects in the study of 4-dim manifolds. After the introduction of the Seiberg-Witten theory, we make remarkable progress in these two aspects. But there are still somes questions need to be studied deeply. For this reason, we will study the following 3 questions in our project. ① By using the Seiberg-Witten theory, G-Spin theorem and Adams operator and so on, we look for the restractions for the existence of homologically trivial diffeomophisms of odd prime order on smooth 4-manifolds. ② We study the restrictions for homologically trivial group actions on symplectic 4-dim manifolds. Furthermore, we give the realization results of the fixed point set for homologically trivial but nontrivial group actions of odd prime order. ③ We study the existence of exotic smooth structure on symplectic 4-dim manifolds. Concretely, under the actions of symplectic K3 group (such as alternative group A_5 and symmetric group S_5 and so on), and combining the Lefschetz fixed point theorem and G-signature theorem and so on, we give an estimate about the existence of exotic smooth structure.