Mathematical Sciences: Low-Dimensional Topology and Gauge Theory

9704204 Saveliev这项研究涉及到光滑4维流形理论的一个中心主题，即积分同调3-球面的同调余边群。长期的目标是在解决R.Kirby关于这个群中存在带有Rohlin不变量的二阶元素的长期问题方面取得进展。研究人员将注意力集中在由代数奇点链接所表示的元素上，特别是塞弗特纤维同调球面。他的方法是应用现代规范理论的方法来研究W.Neumann和L.Siebenmann在70年代由W.Neumann和L.Siebenmann为所谓的图同调三球面引入的不变量。这个应用程序有三个方面。首先，他利用一些特殊的奇点分解来构造具有规定交点形式的代数链环的4-余边，然后应用规范理论的结果，如果该链环限定了同调球，则禁止某些积分双线性形式作为这种分解的交点形式。进一步应用这一结果证明了Rohlin不变一代数环在同调余边群中不可能有二阶。其次，作者利用瞬子Floer同调引入了任意同调3-球面的一个新的不变量，并证明了当Neumann和Siebenmann的不变量被定义(图流形)时，它与Neumann和Siebenmann的不变量一致。这一结果还给出了对Floer同调的新见解，特别是肯定地回答了M.Atiyah关于某些代数链的Floer同调是否有Milnor纤维描述，以及它是否与Milnor纤维同调的复共轭作用有关的问题.最后，作者应用新的不变量研究了一般同调球面上的Kirby问题。这个方向的第一步是证明他的不变量应该在仅用一个1手柄(以及0和2手柄)约束同调球的所有同伦3-球面上消失。他的技巧是使用Floer精确三角形来跟踪沿Cobordism的Floer同调。光滑的n维流形是数学和理论物理研究的中心对象之一。虽然这个物体局部看起来像一个n维的欧几里德空间，但它的全局结构可能仍然非常丰富和复杂。关于二维流形的大多数主要结果都是在19世纪得到的。S在20世纪60年代成功地对维度大于或等于5的流形进行了分类。虽然关于3-流形的一些主要问题仍未得到回答，但在流形层次中占据最特殊位置的是4维的流形，它是相对论的维度。一方面，他们“不够大”，不能将事实证明在更高维度上如此有用的论点应用到他们身上。另一方面，他们的维度太大，无法应用更直观的方法，这些方法在较低的维度有效。几十年来，这里的进展一直很缓慢，直到最近的发展，涉及到规范理论物理学思想的应用。主要成果归功于1980年代初S发起整个项目的S.Donaldson，以及最近的N.Seiberg和E.Witten。规范理论方法被证明是非常卓有成效的，并导致了许多困难问题的解决。研究人员正在利用这些现代方法在拓扑学中的另一个长期存在的问题上取得进展，即同调三维球面的同调余边群的结构，这将为流形结构提供新的见解。***

9704204 Saveliev The research is concerned with a topic which is central to the theory of smooth 4-dimensional manifolds, namely the homology cobordism group of integral homology 3-spheres. The long-term goal is to make progress in solving the long-standing problem of R. Kirby about the existence of elements of order two in this group carrying the Rohlin invariant. The investigator concentrates his attention at the elements represented by the links of algebraic singularities, in particular, Seifert fibered homology spheres. His approach is based on application of the methods of modern gauge theory to investigation of the invariant introduced in the 70's by W. Neumann and L. Siebenmann for the so-called graph homology 3-spheres. This application is threefold. First, he uses some special resolutions of singularities to construct a 4-cobordism of an algebraic link with the prescribed intersection form, and he then applies the gauge-theoretical results which prohibit certain integral bilinear forms as intersection forms of such a resolution if the link bounds a homology ball. The result of this investigation is further applied to show that Rohlin invariant one algebraic links cannot have order two in the homology cobordism group. Second, the investigator uses the instanton Floer homology to introduce a new invariant for arbitrary homology 3-spheres, and he proves that it agrees with the invariant of Neumann and Siebenmann when the latter is defined (graph manifolds). This result also gives new insights into Floer homology, in particular, answers positively M. Atiyah's question whether there is a Milnor fiber description of Floer homology of certain algebraic links, and whether it is related to the complex conjugation action on the homology of the Milnor fiber. Finally, the investigator applies his new invariant to investigate Kirby's problem for general homology spheres. The first step in this direction is to prove that his invariant should vanish on all hom ology 3-spheres that bound a homology ball with just one 1-handle (and a 0- and 2-handle). His technique is to use the Floer exact triangle to keep track of the Floer homology along the cobordism. One of the central objects of investigation in both mathematics and theoretical physics is a smooth n-dimensional manifold. Although this object looks locally like an n-dimensional Euclidean space, its global structure may still be very rich and complicated. Most major results about 2-dimensional manifolds were obtained in the 19th century. Manifolds in dimensions greater than or equal to 5 were successfully classified in the 1960's. Though some major questions about 3-manifolds remain unanswered, it is manifolds of dimension 4, the dimension of relativity theory, that occupy the most special place in the manifold hierarchy. On the one hand, they are not "big enough" to apply to them the arguments that proved to be so useful in higher dimensions. On the other hand, their dimension is too big to apply more intuitive methods that work effectively in lower dimensions. Progress has been slow here for a few decades until recent developments that involved the application of ideas from the physics of gauge theories. The main results are due to S. Donaldson, who initiated the whole program in the early 1980's, and most recently to N. Seiberg and E. Witten. The gauge-theoretical approach proved to be very fruitful and led to the solution of many hard problems. The investigator is applying these modern methods to make progress on yet another long-standing problem in topology, the structure of the homology cobordism group of homology 3-spheres, which would provide new insight into manifold structure. ***