CASSON-TYPE INVARIANTS IN DIMENSION FOUR
CASSON-TYPE INVARIANTS IN DIMENSION FOUR
复制标题
维度四中的 CASSON 型不变量
DOI：
10.1090/fic/047/18
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
N. Saveliev
中科院分区：
文献类型：
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作者：
Daniel Ruberman;N. Saveliev
这篇文章调查我们正在进行的项目之间的关系不变量扩展的经典Rohlin不变量的同调球和那些来自4维（杨米尔斯）规范理论;它将出现在会议的菲尔兹-麦克马斯特几何和拓扑流形。我们主要关注的是一类特殊的流形，这两种类型的不变量的定义和比较。这类包含，特别是，流形具有的同调S1 × S3。关于闭光滑自旋4-流形签名的罗林定理给出了一个同调S1 × S3的模2不变量。在规范理论方面，不变量是通过计算适当丛上的平坦联络而得到的。这个计数受到唐纳森[13]关于SU（2）丛上的反自对偶连接计数的启发;它的平坦类似物首先由Furuta和Ohta [21]研究。主要的猜想对这一项目是直接的是，洛林不变量和规范理论不变量一致的同调S1 × S3。整个讨论的模型是卡森关于他的不变量（在其规范理论的表现形式中，如Taubes [48]所描述的）和Rohlin的同调3-球面不变量的美丽定理。我们将讨论这一猜想对低维拓扑中一些经典问题的影响，以及我们在证明这一猜想方面所取得的进展。这一进展包括验证的猜想在一些特殊情况下，一个'手术'计划接近猜想扩大其范围，包括更广泛的类别的流形，并验证这个扩大猜想同源4-环面。大部分材料包含在我们的三篇论文中[36，37，38]，但我们包括了更广泛的概述以及一些额外的例子。致谢。我们要感谢Scott Baldridge指出第8节中描述的流形，以及Liviu Nicolaescu在计算这些流形上平坦模空间的方向上的投入。我们也
This article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4–dimensional (Yang-Mills) gauge theory; it will appear in the Proceedings of the Fields-McMaster Conference on Geometry and Topology of Manifolds. We are mainly concerned with a special class of manifolds for which the two types of invariants are defined and can be compared. This class contains, in particular, manifolds having the homology of S 1 × S 3 . Rohlin’s theorem about the signature of closed smooth spin 4–manifolds gives rise to a mod–2 invariant of a homology S 1 × S 3 . On the gauge theoretic side, the invariant is obtained by counting flat connections on appropriate bundles. This count is inspired by Donaldson’s [13] count of anti-self-dual connections on SU(2) bundles; its flat analogue was first studied by Furuta and Ohta [21]. The main conjecture towards which this project is directed is that the Rohlin invariant and the gauge theoretic invariant coincide for homology S 1 × S 3 . The model for the whole discussion is Casson’s beautiful theorem relating his invariant (in its gauge theoretic manifestation as described by Taubes [48]) and Rohlin’s invariant of homology 3–spheres. We will discuss the implications of this conjecture for some classical problems in low-dimensional topology, and progress we have made towards proving the conjecture. This progress includes the verification of the conjecture in some special cases, a ‘surgery’ program for approaching the conjecture by expanding its scope to include a wider category of manifolds, and the verification of this expanded conjecture for homology 4–tori. Much of this material is contained in our three papers [36, 37, 38] but we have included a broader overview as well as some additional examples. Acknowledgments. We would like to thank Scott Baldridge for pointing out the manifolds described in Section 8, and Liviu Nicolaescu for his input on computing orientations of flat moduli spaces over these manifolds. We also