First cohomologies and the Johnson homomorphisms of the automorphism group of a free group
First cohomologies and the Johnson homomorphisms of the automorphism group of a free group
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自由群自同构群的第一上同态和约翰逊同态
DOI：
10.1016/j.jpaa.2012.06.017
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发表时间：
2013
影响因子：
0.8
通讯作者：
Takao Satoh
中科院分区：
文献类型：
--
作者：
Dai Tamaki;Dai Tamaki;玉木 大;Dai Tamaki;Dai Tamaki;Dai Tamaki;Dai Tamaki;Dai Tamaki;Dai Tamaki;Dai Tamaki;Dai Tamaki;M. Fujii;Dai Tamaki;Michihiko Fujii;Dai Tamaki;M. Fujii and Takao Satoh;Dai Tamaki;Dai Tamaki;Dai Tamaki;Takao Satoh
我们计算了一个自由群的自同构群的积分扭第一上同群，其系数在自由群的ia自同构群的阿贝尔化V中。特别地，我们证明了它是由分别由Morita和Kawazumi的Magnus表示和Magnus展开构造的两个交叉同态生成的。作为推论，我们看到，如果自由群的秩大于4，则第一Johnson同态不能扩展到自由群的自同态群作为交叉同态。
We compute an integral twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization V of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism if the rank of the free group is greater than 4.