The extension of Johnson's homomorphism from the Torelli group to the mapping class group
The extension of Johnson's homomorphism from the Torelli group to the mapping class group
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Johnson同态从Torelli群到映射类群的推广
DOI：
10.1007/bf01231286
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发表时间：
1993
影响因子：
3.1
通讯作者：
S. Morita
中科院分区：
文献类型：
--
作者：
S. Morita
设xG是亏格g的闭定向曲面，/Tg是它的映射类群。也就是说，它是拓扑群Diff+2‘的路分支群，它由2~的所有保向微分同态组成，具有C_a拓扑。在g>的情况下；2，diff+zg的单位分支Diffo xG是可压缩的[3]，使得分类空间BDiff+-yg具有Eilenberg-MacLane空间K(oj{/g，i)的同伦型，这样，原则上，关于BDiff+Zx的所有拓扑信息都包含在群#g的代数结构中。另一方面，映射类群~t/g适当间断地作用于Teichmtiller空间~g和商空间mg：=jg/，I/g是亏格为g的紧黎曼曲面的模空间，因此，本文还发现模空间mg的拓扑结构与代数结构有密切的关系。在这方面，映射类群在Riemann曲面上的共形场理论中起着重要的作用，目前正得到迅速的发展。通过Heegaard分解，我们还得到了三维流形理论与映射类群之间的经典关系。通过这种方式，映射类群r出现在数学的各个分支中，现在是从不同角度积极研究的对象。现在映射类群自然地作用于xG的基本群7h(Zg)，这个作用为我们提供了一种研究~d/g结构的代数方法，正如Dehn和Nielsen关于，，r的经典著作中已经看到的那样。在本文中，我们也将运用这一观点，并开始对映射类群的结构进行系统的研究。更确切地说，我们利用~(Zg)的下中心级数上的作用，得到JTG在某些李群的某些离散子群上的一系列表示，这些表示是辛群Sp(2g；R)与定义在Q上的某些幂零群的半直积。这些表示应该被认为是Siegel模群Sp(2g；Z)的经典表示的推广，它是由J/x在第一整同调群H~(xG；Z；Z)，并且在某种意义上这些表示的图像可以被认为是jhg的近似。建造
Let Xg be a closed oriented surface of genus g and let,/tg be its mapping class group. Namely it is the group of path components of the topological group Diff+ 2,'consisting of all the orientation preserving diffeomorphisms of 2,~ equipped with the C a topology. In the case of g> 2, the identity component Diffo Xg of Diff+ Zg is contractible [3] so that the classifying space BDiff+-Yg, which classifies oriented surface bundles with fibre 2g, has the homotopy type of an Eilenberg-MacLane space K (oJ {/g, I), Hence, in principle, all of the information about the topology of BDiff+ Zx is contained in the algebraic structure of the group,# g. On the other hand, the mapping class group~, t/g acts on the Teichmtiller space~ g properly discontinuously and the quotient space Mg:= Jg/, I/g is the moduli space of compact Riemann surfaces of genus g. Hence here also we find a close relationship between the topology of the moduli space Mg and the algebraic structure of,//~. In this connection, recently the mapping class group plays an important role in the conformal field theory on Riemann surfaces which is now making a rapid progress. We have also a classical relation between the theory of 3-dimensional manifolds and the mapping class group via the Heegaard splittings. In this way the mapping class group, r appears in diverse branches of mathematics and is now a target of vigorous researches from various points of view.Now the mapping class group acts naturally on the fundamental group 7h (Zg) of Xg and this action provides us with an algebraic method of studying the structure of~ d/g, as is already seen in the classical works on,, r by Dehn and Nielsen. In this paper we will also employ this point of view and begin a systematic study of the structure of the mapping class group. More precisely we use the action of, llg on the lower central series of~(Zg) to obtain a series of representations of Jtg onto some discrete subgroups of certain Lie groups, which are semi-direct products of the symplectic group Sp (2g; R) with certain nilpotent groups defined over Q. These representations should be considered as generalizations of the classical representation of,/t/g onto the Siegel modular group Sp (2g; Z), which is derived from the natural action of J/x on the first integral homology group H~(Xg; Z) of Zg, and in some sense the images of these representations can be thought of as approximations of JHg. To construct