Mathematical Sciences: Algebraic Surfaces and 4-Manifold Topology

本文将研究单连通代数曲面和其他光滑4流形的拓扑结构。将讨论以下问题：代数曲面在连通和下的表现如何？哪些4-流形是代数曲面的和？Donaldson不变量如何从拓扑上定义？代数曲面是由多项式方程定义的，这是人们可以使用的最基本的方程。关于这种表面的非常自然和容易表述的问题仍然没有得到解答，这对拓扑学领域来说是一个挑战，几乎是一种谴责。

The topology of simply-connected algebraic surfaces and other smooth 4-manifolds will be studied. Questions such as the following will be addressed: How do algebraic surfaces behave under connected sum? Which 4-manifolds are sums of algebraic surfaces? How can Donaldson's invariants be defined topologically? Algebraic surfaces are defined by polynomial equations, the most elemetary equations one could use. That very natural and easily formulated questions about such surfaces still remain unanswered stands as a challenge and almost a rebuke to the field of topology.