The structure of the Torelli group—III: The abelianization of I
The structure of the Torelli group—III: The abelianization of I
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Torelli群的结构—III：I的阿贝尔化
DOI：
10.1016/0040-9383(85)90050-3
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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：
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作者：
D. L. Johnson
本文是关于Torelli群J的三篇论文中的第三篇。Torelli群J是在同调上起平凡作用的映射类群~t的子群。在第一篇论文[-3]中，我们讨论了有限生成J的问题；这篇论文的思想和符号将在这里重现，我们将假定它们是熟悉的。本文将计算J的两个阿贝尔商，即泛阿贝尔商J/J‘和泛Ze-向量空间商J/J2，其中Je是J中所有平方生成的子群，如果G是具有一个边界分支的亏格g>；3的曲面，JG，1是它的Torelli群，则作者先前构造了两个满射同态a：JG，1/Jd21-’B3，1，其中目标群是由“三次多项式”构成的某个ZZ-向量空间(见[5])，z：0，g.1/o。1~A3Ht(mg.I，Z)，其中目标是MG的同调的第三次外幂。1(见[6])。主要结果如下：(A)a是同构，因此j2=[~t，~2)，j]=if，其中~/(T2)是平凡作用于H1(mg，1，z2)上的~‘的子群，c~是所有Bman-Craggs同态的公共核(关于这些同态的定义见[1]和[5]).(B)o，1=Ker tr c~Ker3，因此JG，1/J~，1是由Ba，1和A3HI(mg.L，Z)。这些结果分别验证了[7]和[6]的猜想。本文的结构如下：在接下来的两节中，我们简要回顾了同态a和z的定义和性质，以及文献[7]中的一些必要结果，并建立了一些关系，说明了如何从J/J2计算~‘/o’。在第四节中，我们首先证明了~t‘~2)对J/J~2的平凡作用，从而证明了o~2=[~t’t~2~，J]和j/j2具有~i/Jt‘t2J上的模的结构。在下一节中，我们首先建立亏格3曲面的a的同构性质，然后利用这一性质和j/j2上的模结构来证明一般结果。在最后一节中，我们对亏格g>；3的闭曲面mg，o进行了相应的计算。本文将始终如一地使用如下记号：mg，L是亏格#>；3的曲面，有一个边界分支.~t‘g，1=no(diff+mg.1，Reid)是它的保定向微分同胚映射类群./，~2)和~？g10，1，如上所述.3“0，1是JG，1的子群，它是由有界简单闭曲线上的扭曲生成的。简单闭曲线y的扭转映射记为T~。AG，1=JG，1/J~1是J0，1的泛阿贝尔商。Tg，1是~g，1在A0，1中的像。UO，1=JO，1/J21是JG，1的泛Z-向量空间商。群运算将在上述三个群中加写。Akl=akhi(mg，I，Z)
THIS is the third of three papers on the Torelli group J, that is, the subgroup of the mapping class group~¢ t which acts trivially on homology. In the first paper [-3] we treated the problem of finite generation of J; the ideas and notation of that paper will reoccur here, and we shall assume familiarity with them. In this paper we will calculate two abelian quotients of J, namely the universal abelian quotient J/J'and the universal Ze-vector space quotient J/J2, where je is the subgroup generated by all squares in J. IfMg, l is a surface of genus g> 3 with one boundary component and Jg, 1 is its Torelli group, then the author has previously constructed two surjective homomorphisms a: Jg, 1/Jd21--'B3, 1, where the target group is a certain Zz-vector space of" cubic polynomials (see [5]), and z: o¢ g. 1/o¢~. 1~ A 3 Ht (Mg. I, Z) where the target is the 3rd exterior power of the homology of Mg. 1 (see [6]). The principal results are as follows:(a) a is an isomorphism, and hence j2=[~ t,~ 2), j]= if, where~/(t2) is the subgroup of~'which acts trivially on H1 (Mg, 1, Z2) and c~ is the common kernel of all the Birman-Craggs homomorphisms (see [1] and [5] for definition of these homomorphisms).(b) o¢~, 1= Ker tr c~ Ker 3, and hence Jg, 1/J~, 1 is a certain pullback constructed from B a, 1 and A3HI (Mg. l, Z). These results verify conjectures of [7] and [6] respectively. The paper is constructed as follows: in the next two sections we review briefly the definitions and properties of the homomorphisms a and z, as well as some necessary results from [7], and then construct some relations which show how to calculate~'/o¢'from J/J 2. In Section 4 we begin the calculation ofJ/J 2 by showing that~ t'~ 2) acts trivially on it and hence that o¢ 2=[~ t't2~, J] and j/j2 has the structure of a module over~¢¢/Jt't2J. In the following section we begin by establishing the isomorphic nature of a for genus 3 surfaces, and then use this and the module structure on j/j2 to prove the general result. In the final section we do the corresponding calculations for a closed surface Mg, o of genus g> 3. The following notation will be used consistently throughout the paper: Mg, l is a surface of genus#> 3 and one boundary component.~ t'g, 1= no (Diff+ Mg. 1, reid) is its mapping class group of orientation preserving diffeomorphisms../¢~ 2) and~¢ g 1 0, 1, are as defined above. 3" 0, 1 is the subgroup of Jg, 1 generated by twists on bounding simple closed curves. The twist map of a simple closed curve y is denoted T~. Ag, 1= Jg, 1/J~, 1 is the universal abelian quotient of J0, 1. Tg, 1 is the image of~ g, 1 in A0, 1. Uo, 1= Jo, 1/j21 is the universal Z 2-vector space quotient of Jg, 1. The group operation will be written additively in the above three groups. Akl= AkHI (Mg, I, Z)