Mathematical Sciences: Smooth 4-Manifolds

小行星9704927 本项目将通过构造由纽结理论不变量区分的4-流形族，探索光滑单连通4-流形的分类与纽结理论不变量之间的关系。 已经是研究者，与R. Stern，已经为亚历山大多项式完成了这一点。 任何绝对值为P（1）=1的对称Laurent多项式P（t）都是三维球面中某个纽结的亚历山大多项式。 例如，R.斯特恩和首席研究员已经构建了一个家庭的不同4-流形，每个同胚的K3-曲面，从结，并区分的事实，他们的塞伯格-威滕不变量是亚历山大多项式的结。 还有其他的纽结不变量应该对应于其他的构造。 一个相关的问题是单连通不可约4-流形的地理问题。 每一个这样的流形可以被分配一个格点在平面上对应于它的特征数。 问题是要研究哪些点是实现的。 已经取得了显着的进展，但仍有许多工作要做，主要研究人员计划寻求新的方法来构建不可约的单连通4-流形的正签名。 他和R.斯特恩还猜想一个替代诺特不等式的某些辛4-流形，推广了概念的“一般类型”。 调查人员有一个很有前途的技术来证明它，他计划继续下去。 光滑四维流形理论的重要性不仅在于它在低维和高维拓扑之间的中心位置，而且在于它与高能物理的密切相互作用。 这一领域的主要问题是光滑单连通4-流形的分类。 拓扑学与物理学之间的相互作用促进了不变量的构造--首先是唐纳森不变量，然后是Seiberg和维滕不变量--它们对区分四维流形的（同形）类型很有用。 这导致了分类问题的重大进展。 具体来说，那些已经构建了可能是新的4-流形家族的研究人员可以使用这些不变量来确认他们的现象确实是“新的”。 近年来，出现了单连通（不可约）光滑4-流形，不承认复杂的结构（甚至同伦），然后到那些不承认辛结构。 这些例子混淆了分类的问题，以至于我们甚至没有一个理论上的分类，但它们也简化了理论，加强了它的丰富性。 ***

9704927 Fintushel This project will explore the relationship between the classification of smooth simply connected 4-manifolds and invariants of knot theory, by constructing families of 4-manifolds that are distinguished by knot theoretic invariants. Already the investigator, jointly with R. Stern, has accomplished this for the Alexander polynomial. Any symmetric Laurent polynomial P(t) with absolute value of P(1)=1 occurs as the Alexander polynomial of some knot in the 3-sphere. For example, R. Stern and the principal investigator have constructed a family of distinct 4-manifolds, each homeomorphic to the K3-surface, constructed from knots, and distinguished by the fact that their Seiberg-Witten invariants are the Alexander polynomials of the knots. There are other knot invariants that should correspond to other constructions. A related issue is the geography problem for simply connected irreducible 4-manifolds. Each such manifold can be assigned a lattice point in the plane corresponding to its characteristic numbers. The problem is to study which points are realized. There has been notable progress, but much work still remains, and the principal investigator plans to seek new methods for constructing irreducible simply connected 4-manifolds of positive signature. He and R. Stern also conjecture a replacement for the Noether inequality for certain symplectic 4-manifolds that generalize the notion of `general type'. The investigator has a promising technique for its proof, which he plans to pursue. The theory of smooth 4-manifolds gains its importance both from its central location between low and high-dimensional topology, and from its close interaction with high energy physics. The major problem in this field is the classification of smooth simply connected 4-manifolds. The interaction between topology and physics has stimulated the construction of invariants - at first Donaldson's invariant, and then the invariant of Seiberg and Witten - tha t are useful in distinguishing the (diffeomorphism) types of 4-manifolds. These have led to major advances in the classification problem. Specifically, researchers who have constructed possibly new families of 4-manifolds can use these invariants to confirm that their phenomena are indeed `new'. Recent years have given rise to simply connected (irreducible) smooth 4-manifolds that admit no complex structure (even up to homotopy), then to those that admit no symplectic structure. These examples have confused the issue of classification to the point that we are left without even a conjectural classification, but they have also invigorated the theory and reinforced its richness. ***