论文标题
在光谱序列上，用于3属Torelli组对循环复合物的作用
On spectral sequence for the action of genus 3 Torelli group on the complex of cycles
论文作者
论文摘要
torelli属于$ g $ g $的表面$ s_g $的torelli组是映射类$ \ mathrm {modrm {mod}（s_g）$的子组$ \ mathcal {i} _g $，由所有映射类组成，这些映射类以$ s_g $ $ s_g $的同源为单独行动。关于Torelli组的最有趣的开放问题之一是该组是否有限地提出了该组$ \ Mathcal {i} _3 $。解决此问题的一种方法取决于使用频谱序列$ e^r_ {p，q} $的第二个同源组{i} _3 $的研究，用于$ \ Mathcal {i} _3 $的动作。在本文中，我们为猜想获得了部分结果，即$ h_2（\ mathcal {i} _3; \ mathbb {z}）$不是有限生成的，因此不是有限地生成的。也就是说，我们证明了光谱序列的$ e^3_ {0,2} $是无限生成的，也就是说，$ e^1_ {0,2} $在通过差异图像$ d^1 $ d^1 $和$ d^2 $的图像后，保持无限生成。如果一个人在用$ d^3 $的图像摄取商后也保持了无限生成的证明，他将完成证明$ \ Mathcal {i} _3 $没有有限提出的事实。
The Torelli group of a genus $g$ oriented surface $S_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group $\mathrm{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$ is finitely presented or not. A possible approach to this problem relies upon the study of the second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain a partial result towards the conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence $\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is infinitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by images of the differentials $d^1$ and $d^2$. If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of $d^3$, he would complete the proof of the fact that $\mathcal{I}_3$ is not finitely presented.
