Mathematical Sciences: Smooth 4-Manifolds

9704927 Fintushel这个项目将通过构造由纽结理论不变量来区分的4-流形，来探索光滑单连通4-流形的分类与纽结理论不变量之间的关系。这位研究人员已经与R·斯特恩一起完成了亚历山大多项式的这一过程。任何绝对值为P(1)=1的对称洛朗多项式P(T)都是3-球面上某个纽结的Alexander多项式。例如，R.Stern和主要研究者已经构造了一族不同的4-流形，每个流形都同胚于K3-曲面，由纽结构造，并且通过它们的Seiberg-Witten不变量是纽结的Alexander多项式的事实来区分。还有其他纽结不变量应该与其他构造相对应。一个相关的问题是单连通不可约4-流形的地理问题。每个这样的流形都可以在平面上被分配一个与其特征数相对应的格点。问题是要研究实现了哪些点。已经取得了显著的进展，但仍有许多工作要做，主要研究者计划寻找新的方法来构造正签名的不可约单连通4-流形。他和R·斯特恩还猜想了某些推广了“一般类型”概念的辛四维流形的Noether不等式的一个替代。这位调查员有一种很有希望的技术来证明它，他计划继续研究。光滑四维流形理论的重要性在于它位于低维和高维拓扑之间的中心位置，以及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。拓扑学和物理学之间的相互作用刺激了不变量的构造--首先是Donaldson不变量，然后是Seiberg和Witten-tha t的不变量在区分(微分同态)类型的4-流形中是有用的。这些都在分类问题上取得了重大进展。具体地说，已经构建了可能新的4-流形的族的研究人员可以使用这些不变量来确认它们的现象确实是“新的”。近年来，单连通(不可约)光滑4-流形不允许有复杂结构(甚至直到同伦)，然后出现了不允许有辛结构的流形。这些例子混淆了分类的问题，以至于我们甚至没有一个猜测的分类，但它们也活跃了理论，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. ***