Magnus展开式的上同调性质
Cohomological Aspects of Magnus Expansions
摘要：
我们推广了自由群的马格努斯展开的概念，以便将定义在具有一个边界分量的曲面的Torelli群的递减过滤上的每个Johnson同态扩展到自由群$\operatorname{Aut}(F_{n})$的整个自同构群。扩展的映射是{ \it not}同态，但满足无穷余边界关系序列，因此我们称它们为{ \it the Johnson映射}。在本文中，我们只限于研究第一和第二关系，这有上同调的后果，对群$\operatorname{Aut}(F_{n})$和映射类群的表面。第一个是指第一个Johnson映射是群$\operatorname{Aut}(F_{n})$的扭1-上圈。它的上同调类与曲面[Ka 1] [KM 1]的映射类群上所有Morita-Mumford类的“唯一基本粒子”相一致。第二个限制在映射类群上的关系等价于Garoufalten和Nakamura [GN]提出的、Morita和作者[KM 2]建立的扭曲Morita-Mumford类之间的一个基本关系。这意味着我们对基本关系给出了一个简单而连贯的证明。第一个Johnson映射给出了自由群的诱导自同构群$IA_n$的阿贝尔化.
英文摘要：
We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended ones are {\it not} homomorphisms, but satisfy an infinite sequence of coboundary relations, so that we call them {\it the Johnson maps}. In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group $\operatorname{Aut}(F_{n})$ and the mapping class groups for surfaces. The first one means that the first Johnson map is a twisted 1-cocycle of the group $\operatorname{Aut}(F_{n})$. Its cohomology class coincides with ``the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface [Ka1] [KM1]. The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes proposed by Garoufalidis and Nakamura [GN] and established by Morita and the author [KM2]. This means we give a simple and coherent proof of the fundamental relation. The first Johnson map gives the abelianization of the induced automorphism group $IA_n$ of a free group in an explicit way.
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