论文标题
Torelli集团同源性的Abelian周期
Abelian Cycles in the Homology of the Torelli group
论文作者
论文摘要
在1980年代初，约翰逊定义了同构$ \ nathcal {i} _ {g}^1 \ to \ bigwedge^3 H_1（s_ {g}，\ m athbb {z}）$边界组件和$ s_g $是没有边界组件的相应表面。这被称为约翰逊同态。 我们研究了约翰逊同构对理性同源组引起的地图，并将其应用于通过脱节界对确定的Abelian循环，以计算稳定范围内$ H_N的大商量（\ Mathcal {i} _ {i} _ {i} _ {g} _ {g}^1，\ Mathb {q}）$。这也意味着Torelli组$ \ MATHCAL {I} _ {g，1} $具有标记点而不是边界组件的表面的稳定合理同源性的类似结果。此外，我们研究了此地图的图像中的数量是由此类周期的图像生成的，并用它来证明，在尖锐的情况下，它们生成了$ H_N（\ Mathcal {i} _ {g，1}）$的适当子代理，对于$ n \ ge 2 $和$ g $足够大。
In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is the Torelli group of a closed, connected and oriented surface of genus $g$ with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding pair maps, in order to compute a large quotient of $H_n(\mathcal{I}_{g}^1,\mathbb{Q})$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal{I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n(\mathcal{I}_{g,1})$ for $n\ge 2$ and $g$ large enough.
