关于K3曲面的Nielsen实现问题的一个注记
A note on the Nielsen realization problem for K3 surfaces
AI总结：
本文利用Seiberg-Witten理论与整体Torelli定理，证明了K3曲面微分同胚群与同胚群之间三个关于截面存在性、二阶子群分裂及基本群满射性的结果。
摘要：
我们将证明关于$K3$曲面的微分同胚群和同胚群的以下三个定理。第一个定理是，自然映射$π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$在其像上存在截面。第二个定理是，存在$π_{0}(Diff(K3))$的一个二阶子群$G$，使得映射$Diff(K3) \to π_{0}(Diff(K3))$在$G$上没有分裂，但映射$Homeo(K3) \to π_{0}(Homeo(K3))$在$G$于$π_{0}(Homeo(K3))$中的像上存在分裂，且该分裂是非平凡的。第三个定理是，映射$π_{1}(Diff(K3)) \to π_{1}(Homeo(K3))$不是满射。我们对这些结果的证明基于Seiberg-Witten理论和$K3$曲面的整体Torelli定理。
英文摘要：
We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map $π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which there is no splitting of the map $Diff(K3) \to π_{0}(Diff(K3))$, but there is a splitting of $Homeo(K3) \to π_{0}(Homeo(K3))$ over the image of $G$ in $π_{0}(Homeo(K3))$, which is non-trivial. The third is that the map $π_{1}(Diff(K3)) \to π_{1}(Homeo(K3))$ is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for $K3$ surfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
