Morita's trace maps on the group of homology cobordisms
Morita's trace maps on the group of homology cobordisms
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森田在同调配边群上的迹图
DOI：
10.1142/s179352531950064x
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发表时间：
2019
影响因子：
0.8
通讯作者：
Takuya Sakasai
中科院分区：
文献类型：
--
作者：
Gwenael Massuyeau;Takuya Sakasai
Morita 在 2008 年引入了关于无限维向量空间中具有值的曲面的同调余弦群的 a-cocycle。 His-cocycle 包含约翰逊同态的所有“痕迹”，这是他 15 年前在映射类群的研究中引入的。在本文中，我们提出了基于简单而明确的构造的新版本 Morita’s-cocycle。我们的余循环被证明满足几个基本性质，包括与 Magnus 表示和 LMO 同态的联系。作为一个应用，我们证明了同调配边群的有理阿贝尔化是不平凡的。此外，我们应用一些代数方法来比较有限生成自由群的自同构群上的两个自然过滤。
Morita introduced in 2008 a-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His-cocycle contains all the “traces” of Johnson homomorphisms which he introduced 15 years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita’s-cocycle based on a simple and explicit construction. Our-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.