Mathematical Sciences: Smooth 4-Manifolds

本项目将通过构造以结理论不变量区分的4-流形族来探讨光滑单连通4-流形的分类与结论不变量之间的关系。研究者已经和R. Stern一起完成了Alexander多项式的这一过程。任意绝对值为P(1)=1的对称洛朗多项式P(t)作为3球中某个结点的亚历山大多项式出现。例如，R. Stern和首席研究员构造了一组不同的4流形，每个流形都与k3曲面同胚，由结构造而成，并以它们的Seiberg-Witten不变量是结的Alexander多项式这一事实来区分。还有其他的结不变量应该对应于其他结构。一个相关的问题是单连通不可约4流形的地理问题。每一个这样的流形都可以在平面上指定一个与它的特征数相对应的点阵点。问题是研究哪些点实现了。虽然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不可约的正签名单连通4流形。他和R. Stern还对某些推广“一般类型”概念的辛4流形提出了Noether不等式的替代。调查人员有一种很有前途的技术来证明这一点，他计划继续研究。光滑4流形理论的重要性在于它处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。该领域的主要问题是光滑单连通4流形的分类问题。拓扑和物理之间的相互作用刺激了不变量的构造——首先是Donaldson的不变量，然后是Seiberg和Witten的不变量——这对区分4流形的（微分同构）类型很有用。这些都导致了分类问题的重大进展。具体来说，那些可能构造了新的4流形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，出现了不承认复杂结构（甚至到同伦）的单连通（不可约）光滑4流形，然后出现了不承认辛结构的流形。这些例子混淆了分类问题，以至于我们甚至没有一个推测的分类，但它们也使理论充满活力，并增强了它的丰富性。* * *

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. ***