Torsion invariants of $Spin^c$-structures on 3-manifolds
Torsion invariants of $Spin^c$-structures on 3-manifolds
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3 流形上 $Spin^c$ 结构的扭转不变量
DOI：
10.4310/mrl.1997.v4.n5.a6
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发表时间：
1997
影响因子：
1
通讯作者：
V. Turaev
中科院分区：
文献类型：
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作者：
V. Turaev
最近，人们对三维流形的Seiberg-Witten不变量的兴趣激增，参见[3]，[4]，[7]。闭定向三维流形M的Seiberg-Witten不变量是M上自旋结构集到Z的函数SW。这个函数是在假设b1（M）≥ 1下定义的，其中b1（M）是M的第一个贝蒂数;在b1（M）= 1的情况下，函数SW取决于H（M ; Z）= Z的生成元的选择。SW的定义与四维流形的SW-不变量的定义平行：计算塞伯格-威滕方程解的规范等价类。Meng和Taubes [4]观察到函数SW（M）与M的Reidemeister型挠率密切相关。所讨论的挠是由Milnor [5]引入的; Meng和Taubes使用的改进版本是由于作者[12]。考虑到符号，这个挠率等价于M的基本群的亚历山大多项式，见[5]，[8]。本文的目的是讨论自旋结构和挠率之间的关系。我们使用作者在[9]、[12]、[13]中引入的挠率来定义封闭定向3-流形上自旋结构的数值不变量。假设在b1 ≥ 1的情况下，这个不变量等价于塞伯格-威滕理论中的不变量。在[11]中，结合纽结理论中的一个分类问题，研究了在三维流形上求自旋结构的拓扑不变量的相关问题。在[11]中观察到，在3球面S中的链的取向在S的相应的2-片分支覆盖上诱导自旋结构。为了区分3-流形上的自旋结构，可以使用挠率，参见[13]。作为一个特殊的应用，注意3维透镜空间上自旋结构的同胚分类：一个具有偶数p的透镜空间L（p，q）允许一个保持方向的自同胚置换L（p，q）上的两个结构当且仅当q = p + 1（mod 2 p），见[13]，定理C.3.1。这意味着（困难的部分）的分类有两个桥梁在S首先建立的舒伯特在一个不同的方式。
Recently there has been a surge of interest in the Seiberg-Witten invariants of 3-manifolds, see [3], [4], [7]. The Seiberg-Witten invariant of a closed oriented 3-manifold M is a function SW from the set of Spin-structures on M to Z. This function is defined under the assumption b1(M) ≥ 1 where b1(M) is the first Betti number of M ; in the case b1(M) = 1 the function SW depends on the choice of a generator of H(M ; Z) = Z. The definition of SW runs parallel to the definition of the SW-invariant of 4-manifolds: one counts the gauge equivalence classes of solutions to the Seiberg-Witten equations. It was observed by Meng and Taubes [4] that the function SW (M) is closely related to a Reidemeister-type torsion of M . The torsion in question was introduced by Milnor [5]; the refined version used by Meng and Taubes is due to the author [12]. Considered up to sign, this torsion is equivalent to the Alexander polynomial of the fundamental group of M , see [5], [8]. The aim of this paper is to discuss relationships between Spin-structures and torsions. We use the torsions introduced by the author in [9], [12], [13] to define a numerical invariant of Spin-structures on closed oriented 3-manifolds. Presumably, in the case b1 ≥ 1, this invariant is equivalent to the one arising in the Seiberg-Witten theory. A related question of finding topological invariants of Spin-structures on 3manifolds was studied in [11] in connection with a classification problem in the knot theory. It was observed in [11] that an orientation of a link in the 3sphere S induces a Spin-structure on the corresponding 2-sheeted branched covering of S. To distinguish Spin-structures on 3-manifolds one can use torsions, see [13]. As a specific application, note the homeomorphism classification of Spin-structures on 3-dimensional lens spaces: a lens space L(p, q) with even p admits an orientation-preserving self-homeomorphism permuting the two Spinstructures on L(p, q) if and only if q = p + 1(mod 2p), see [13], Theorem C.3.1. This implies (the hard part of) the classification of oriented links with two bridges in S first established by Schubert in a different way.