Seiberg-Witten invariants for 3-manifolds in the case
Seiberg-Witten invariants for 3-manifolds in the case
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本例中 3 流形的 Seiberg-Witten 不变量
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发表时间：
2000
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通讯作者：
Y. Lim
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作者：
Y. Lim
本文详细讨论了闭定向三维流形的SeibergWitten不变量，特别注意了b1 = 0和b1 = 1的情形。这些都是从模空间的解决方案SeibergWitten方程取决于选择的黎曼度量的基础流形以及某些扰动项的方程。在有利的情况下，这个模空间是有限的和自然定向的，我们可以形成点的代数和。给定任意两组度量和扰动的选择，它们由一个单参数族连接，我们详细分析了插值模空间中可能出现的奇异性。这就导致了对代数和如何变化的理解。在b1 = 0的情况下，拓扑不变量可以通过添加合适的反项来提取，我们确定了这一点（这一想法归功于唐纳森）。在b1 = 1的情况下，定义了一个拓扑不变量，它只依赖于与扰动项有关的上同调信息。我们证明了一个“穿墙”公式，告诉我们如何不变量的变化与不同的选择，这种扰动。在整个过程中，我们都非常注意一般性陈述以及所有关系中的方向和符号问题。在积分同调球的情况下，这个不变量与卡森不变量的等价性在Lim，1999中得到了处理（另见Nicolescu的作品，预印本）。在b1 > 0的情况下与Reidemeister挠的等价性是Meng & Taubes，1996的结果。一些相关材料载于Marcolli（1996年）、Froyshov（1996年）和唐纳森（1996年）的调查。Taubes，1990包含了本文在平坦SU（2）-联络的背景下的最初构造。
In this note we give a detailed exposition of the SeibergWitten invariants for closed oriented 3-manifolds paying particular attention to the case of b1 = 0 and b1 = 1. These are extracted from the moduli space of solutions to the SeibergWitten equations which depend on choices of a Riemannian metric on the underlying manifold as well as certain perturbation terms in the equations. In favourable circumstances this moduli space is finite and naturally oriented and we may form the algebraic sum of the points. Given any two sets of choices of metric and perturbation which are connected by a 1-parameter family, we analyse in detail the singularities which may develop in the interpolating moduli space. This leads then to an understanding of how the algebraic sum changes. In the case b1 = 0 a topological invariant can be extracted with the addition of a suitable counter-term, which we identify (this idea is attributed to Donaldson). In the case b1 = 1 a topological invariant is defined which depends only on cohomological information related to the perturbation term. We prove a ‘wall-crossing’ formula which tells us how the invariant changes with different choices of this perturbation. Throughout we pay careful attention to genericity statements and the issue of orientations and signs in all the relations. The equivalence of this invariant in the case of an integral homology sphere with the Casson invariant is treated in Lim, 1999 (see also works of Nicolescu, preprint). The equivalence with Reidemeister Torsion in the case b1 > 0 is a result of Meng & Taubes, 1996. Some related material is in Marcolli, 1996, Froyshov, 1996 and in the survey Donaldson, 1996. Taubes, 1990 contains the originating construction in this article in the context of flat SU(2)-connections.