【摘要】 首先讨论更为熟悉的三维Poincaré猜测,它的解是现代几何分析的一个主要成果,这启发了本文中给出的光滑四维Poincaré猜测的精确陈述.然后解释流形上的拓扑结构与光滑结构之间的差异以及Donaldson利用Yang-Mills方程瞬子解定义四维流形微分不变量的开创性工作.最后,给出了尝试解决该猜想的一些新方法.更多还原

【Abstract】 We start with a discussion of the more familiar 3-dimensional Poincaré conjecture whose solution constitutes a major achievement of modern geometric analysis. This motivates the precise formulation of the smooth 4 dimensional Poincaré conjecture given in this article. Then we explain the difference between the topological and smooth structures on a manifold as well as the groundbreaking work of Donaldson in defining differential invariant for 4-dimensional manifolds using Yang-Mills instanton. At the end, we give some recent approaches in the attack of the conjecture更多还原