Nilpotent homotopy types of closed 3-manifolds
Nilpotent homotopy types of closed 3-manifolds
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闭合 3 流形的幂零同伦型
DOI：
10.1007/bfb0099951
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
V. Turaev
中科院分区：
文献类型：
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作者：
V. Turaev
本文的出发点是以下问题：什么是上同调环和更一般的梅西产品结构的3流形？我们将把我们自己限制在闭连通定向的情况下。表示这类流形为·流形Me：的系数在zl 11（= Z/l1 t，0）中的上同调环完全由形式确定。Massey产品的特征化问题要微妙得多。其中有些是有定义的，有些没有，总的来说很难跟踪它们。然而，很明显，1维上同调的Massey乘积。我的班级：e由：JC= $ iCM）和由1 Ml表示的Hs（：Ji）中的类确定。(Here以及以下，除非另有说明，所有同调和上同调均采用无扭曲整数系数）。此外，Massey乘积由其下中心级数的项和相应的三维同调类的因子群决定。例如，形式（1）可以容易地从Hi（M）和由[M]表示的类导出。问题出现了：什么样的幂零群和它们的同调类对应于3-流形？为了精确地表述这个问题，我们需要一些符号。对于组$和整数，将F 1（：JO=
The starting point of this paper was the following question: What are the cohomology rings and more generally the Massey product structures of 3-manifolds? We will confine ourselves to the case of closed connected oriented Denote this class of manifolds by• The cohomology ring with coefficients in zl11 (= Z/l1t, 0) of a manifold Me: is completely determined by the formForms obtained in this way were characterized algebraically by Postnikov [5J for l1t= 2, by Sullivan [8] for 11== 0 and by the author [14J for all n. The characterization problem for the Massey products is much more subtle. Some of them are defined, some are not, and on the whole it is difficult to keep track of them. However, it is quite clear that the Massey products of 1-dimensional cohomology. classes of Me:.-e are determined by: JC= $ iCM) and the class in Hs (: Ji) represented by 1Ml.(Here and below, unless otherwise specified, all homology and cohomology is taken with untwisted integer coefficients). Moreover, the Massey products are determined by the factorgroups of by the terms of its lower central series and the associated 3-dimensional homology classes. For example, the form (1) can be easily derived from Hi (M) and the class in represented by [M]. the problem arises: What nilpotent groups and their homology classes correspond to 3-manifolds? To formulate this problem in precise terms we need some notation. For a group $ and an integer put F 1 (: JO=