Flat connections, the Alexander invariant, and Casson’s invariant
Flat connections, the Alexander invariant, and Casson’s invariant
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平坦连接、亚历山大不变量和卡森不变量
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发表时间：
1997
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通讯作者：
Christopher Herald
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作者：
Christopher Herald
本文研究了同调纽结补的平坦模空间与纽结的其它拓扑不变量之间的关系。我们定义了一个不变量，通过计算的平坦轨道与meridinal holonomy的迹固定，并给出了一个公式的不变量的Tristam-Levine等变结签名和卡森不变量的同源3-球的纽结生活。卡森定义了一个拓扑不变量，现在被称为卡森不变量，为封闭定向同调3-球，粗略地说，计数（一半）的数量不可约表示从基本群的3-流形到SU（2）模共轭（见[AM]）。此后不久，Taubes提供了一个解析定义相同的不变量，计数而不是平坦SU（2）连接模规范等价（见[T]）。这些文件，沿着与相关的步行者[瓦]和弗洛尔[F]，激发了一长串的文件都从拓扑的观点和规范理论之一。在1992年，Lin通过计算纽结群到SU（2）模共轭中的无迹不可约表示的数目，定义了一个5纽结不变量（参见[Li]）。这里无迹意味着所有结子午线都被取为无迹矩阵。然后，他表明了一个聪明的拓扑参数，这个不变量等于一半的结签名。鲁伯曼提出了一个推广的结果，涉及等变结签名，删除了无迹条件。对于2桥结的情况，Heusener在[He]中给出了类似的公式。本文的目的是建立同调三维球面中任意纽结的一般公式。在S中纽结的特殊情况下，这证明了鲁伯曼提出的公式。我们的主要结果是：如果AS：5 -> X是定向同调3-球面中的光滑纽结，则（在一定的横截性假设下）以迹2cos a的矩阵为经线的非交换表示p：7 ri（Xn）-SU（2）的共轭类数等于X的Casson不变量的负4倍负1
This paper explores the relationship between the flat moduli space of a homology knot complement and other topological invariants of the knot. We define an invariant by counting the flat orbits with trace of the meridinal holonomy fixed and give a formula for the invariant in terms of the TristamLevine equivariant knot signature and the Casson invariant of the homology 3-sphere in which the knot lives. Casson defined a topological invariant, now known as the Casson invariant, for closed oriented homology 3-spheres which, roughly speaking, counts (one half) the number of irreducible representations from the fundamental group of the 3-manifold into SU(2) modulo conjugation (see [AM]). Shortly thereafter, Taubes provided an analytic definition of the same invariant, counting instead flat SU(2) connections modulo gauge equivalence (see [T]). These papers, along with related ones by Walker [Wa] and Floer [F], have inspired a long list of papers both from the topological viewpoint and from the gauge theoretic one. In 1992, Lin defined an 5 knot invariant by counting the number of trace-free irreducible representations of the knot group into SU(2) modulo conjugation (see [Li]). Here trace-free means that all knot meridians are taken to trace-free matrices. He then showed by a clever topological argument that this invariant equals one half of the knot signature. Ruberman suggested a generalization of this result involving equivariant knot signature which removed the trace-free condition. For the case of 2-bridge knots, a similar formula was conjectured by Heusener in [He]. The aim of this article is to establish a general formula for arbitrary knots in homology 3-spheres. In the special case of knots in S, this proves the formula suggested by Ruberman. Our main result states that if AS : 5 -> X is a smooth knot in an oriented homology 3-sphere then (under certain transversality assumptions) the number of conjugacy classes of nonabelian representations p : 7ri(X n) —► SU(2) taking the meridians to matrices of trace 2 cos a equals minus four times the Casson invariant of X minus one