论文标题
自由群体的IA-Automormormormit群体的稳定合理同源性
Stable rational homology of the IA-automorphism groups of free groups
论文作者
论文摘要
自由组$ f_n $的IA-Automormormorming群体的合理同源性$ \ permatatorname {ia} _n $仍然是神秘的。我们研究了$ \ perialatorname {ia} _n $的合理同源性的商，该图作为Abelianization Map引起的地图图像，我们称之为$ \ operatatorName {ia} _n $的Albanese同源性。我们获得表示稳定的$ \ operatorName {gl}（n，\ mathbb {q}）$ - $ \ operatatorName {ia} _n $的阿尔巴尼亚同源性的亚Quortient，它与$ \ \ operatornamame {ia} _n $的整个Albanese同源物相吻合。特别是，我们获得了$ \ operatorname {ia} _n $的Albanese同源性的尺寸的下限，每个同源度均在稳定范围内。此外，我们确定了$ \ operatorname {ia} _n $ for $ n \ ge 9 $的整个第三个Albanese同源性。我们还研究了$ \ operatorname {ia} _n $类似物的Albanese同源性，$ f_n $和torelli表面的Albanese同源性。此外，我们研究了$ \ operatorname {ia} _n $的Albanese同源性与$ \ operatatorName {aut}（aut}（f_n）$的共同体，具有扭曲的系数。
The rational homology of the IA-automorphism group $\operatorname{IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname{IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname{IA}_n$. We obtain a representation-stable $\operatorname{GL}(n,\mathbb{Q})$-subquotient of the Albanese homology of $\operatorname{IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname{IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname{IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname{IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname{IA}_n$ to the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname{IA}_n$ and the cohomology of $\operatorname{Aut}(F_n)$ with twisted coefficients.
