On quadratic forms of 3-manifolds
On quadratic forms of 3-manifolds
复制标题
关于 3-流形的二次形式
DOI：
10.1007/bf01390003
复制
发表时间：
1977
影响因子：
3.1
通讯作者：
A. Kawauchi
中科院分区：
文献类型：
--
作者：
A. Kawauchi
JW Milnor[16]定义了一个经典的驯服纽结KC S 3的纽结外部(即闭纽结)E(K)的二次型，它由纽结k和包含3-球的S a的方向所指定的生成元H1(E(K)；Z)相联系。纽结外部的二次型必然是非奇异的，它被应用于Fox-Milnor纽结共边群[5]。本文定义了具有可定向手柄S 1×S 2的积分同调群的闭三维流形的类似非奇异二次型，并将这种二次型应用于同调可定向手柄的/~-余边群~(S 1×S 2)。本文的主要目的是定义具有非零第一Betti数的任意紧致连通定向三维流形的二次型，并推导其基本性质。为了定义二次型，我们将需要Milnor对偶定理[16]的推广，该对偶定理也可以被视为流形的无限循环覆盖上的Blanchfield对偶定理[1]的一个版本。在许多情况下，我们的二次型将是奇异的，并且，如在链接论[7，17，18]等中，我们将定义(单变量)Alexander多项式、细川多项式、Murasugi签名和零性。亚历山大多项式和细川多项式将被定义为总是非零多项式。因此，我们对Alexander(或Hosokawa)多项式的定义将与原始定义严格不同，但如果它不是零，则原始Alexander(或Hosokawa)多项式将等于我们的Alexander(或Hosokawa)多项式。村木签名和形式的无效性的概念将有别于[17]的原始概念。二次形式和这些不变量将被视为与三维流形的各种(受限)余边问题密切相关。本文具体包括三个方面的应用。第一章给出了一个有限表示群是3-流形群的多项式条件。例如，我们将证明鲍姆斯拉格-索利塔群G~vq~=
JW Milnor [16] defined a quadratic form of the knot exterior (ie, the closed knot complement) E (k) of a classical tame knot kc S 3 associated with a generator of H 1 (E (k); Z) that is specified by the orientations of the knot k and the containing 3-sphere S a. The quadratic form of the knot exterior is necessarily nonsingular and it was applied for the Fox-Milnor knot cobordism group [5]. The author defined analogously non-singular quadratic forms for closed 3-manifolds having the integral homology groups of an orientable handle S 1 x S 2 and this quadratic form was applied for the/~-cobordism group~(S 1 x S 2) of the homology orientable handles.(See [11].)The main purpose of this paper is to define quadratic forms for arbitrary, compact, connected and oriented 3-manifolds with non-zero first Betti numbers and to deduce basic properties of the forms. In order to define the quadratic form, we will need a generalization of the Milnor duality theorem [16], which may be also regarded as a version of the Blanchfield duality theorem [1] on the infinite cyclic covers of manifolds. Our quadratic form will be, in many cases, singular and, as in the link theory [7, 17, 18] etc., we will define the (one variable) Alexander polynomial, the Hosokawa polynomial, the Murasugi signature and the nullity. The Alexander polynomial and the Hosokawa polynomial will be defined so as to be always non-zero polynomials. Accordingly, our definition of the Alexander (or Hosokawa) polynomial will be strictly distinct from the original definition, but if it is not zero, then the original Alexander (or Hosokawa) polynomial will be equal to our Alexander (or Hosokawa) polynomial. The concepts of the Murasugi signature and the nullity of the form will be distinct from the original concepts of [17]. The quadratic forms and hence these invariants will be seen to be closely related to the various (restricted) cobordism problems of 3-manifolds. The paper contains concrete three applications. The first will give a polynomial condition for a certain finitely presented group to be a 3-manifold group. For example, we shall show that the Baumslag-Solitar group G~ vq~=