The structure of the Torelli group—II: A characterization of the group generated by twists on bounding curves
The structure of the Torelli group—II: A characterization of the group generated by twists on bounding curves
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托雷利群的结构—II：由边界曲线扭曲产生的群的表征
DOI：
10.1016/0040-9383(85)90049-7
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
D. L. Johnson
中科院分区：
文献类型：
--
作者：
D. L. Johnson
这是关于Torelli群•的三篇论文中的第二篇，Torelli群•是映射类群~的子群，它对同调起着平凡的作用。在第一篇论文中，我们讨论了J的有限生成问题；那篇论文的思想和符号在这里将经常出现，我们假定对它们很熟悉。本文研究了由有界简单闭合曲线上的扭转所产生的群~ c J。设Mg， t为具有一个边界分量的g属曲面，Jg， i为它的Torelli群；在[2]中构造了一个从j0,1到三次外幂A3HI （Mg. 1, z）的满射同态z，并证明了3”Ker z，由此推测~= Ker z；这个猜想的证明是本文的目的。我们还假定您熟悉参考[2]的思想和术语。整个纸张的所有表面都是紧凑的，可定向的和定向的。除非另有说明，所有地图将是平滑的；所有同调群都使用Z系数。对于表面Mg， 1， n~是自由的，其基表示为SCC的~ i，~ i (i= 1，…， g)除基点处外均互不相交，排列如图la所示；这样一组曲线被称为“规范基”。（在图中，基准点用~ M表示，但下面的情况可能并不总是如此。）图lb显示了我们将在本文中使用的曲面形式上的~ k，/3k曲线。对于封闭曲面，情况基本上是一样的
THIS is the second of three papers on the Torelli group•, that is, the subgroup of the mapping class group~ which acts trivially on homology. In the first paper [1] we treated the problem of finite generation of J; the ideas and notation of that paper will occur frequently here, and we shall assume familiarity with them. In this paper we study the group~ c J generated by twists on bounding simple closed curves. Let Mg, t be a surface of genus g> 2 with one boundary component and Jg, i be its Torelli group; in [2] the author constructed a surjective homomorphism z from J0, 1 to the third exterior power A3HI (Mg. 1, Z) and showed that 3" Ker z. It was conjectured there that in fact~= Ker z; the proof of this conjecture is the purpose of the present paper. We will also assume familiarity with the ideas and terminology of the reference [2].Throughout the paper all surfaces are compact, orientable and oriented. All maps will be smooth unless otherwise stated; all homology groups use Z coefficients. For a surface Mg, 1, n~ is free and has a basis represented by SCC's~ i,~ i (i= 1,..., g) which are mutually disjoint except at the base point and are arranged there as in Fig. la; such a set of curves is known as a" canonical basis".(In the figure, the base point is shown in~ M, but this may not always be the case in what follows.) Figure lb shows the curves~ k,/3k on the form of the surface we will use in this paper. The situation is essentially the same for a closed surface