论文标题
大型映射课程组具有无数的积分同源性
Big mapping class groups with uncountable integral homology
论文作者
论文摘要
我们证明，对于任何无限型Surface $ s $，关闭紧凑型绘制的映射类$ \ OVERLINE {\ MATHRM {PMAP} _C（s s）} $以及Torelli Group $ \ Mathcal $ \ Mathcal {t}（S）$的积分同源。通过我们在Arxiv：2211.07470和其他已知计算中的结果，对于所有无限型型表面$ s $的完整映射类组$ \ mathrm {map}（s）$而言，此类语句是正确的。但是，我们仍然能够证明$ \ mathrm {map}（s）$的积分同源性在所有正学位上都是无数级的无限型表面$ s $。这类表面的关键属性大致是表面$ s $的末端空间包含拓扑区分点的限制点。我们的结果尤其包括所有有限型表面，具有可数端空间，其最大坎托 - 宾多克森等级$α$的独特点，其中$α$是后继的序列。我们还观察到订单的订单-10 $元素在任何表面的纯映射类组的第一个同源性$ 2 $中，回答了G. domat的最新问题。
We prove that, for any infinite-type surface $S$, the integral homology of the closure of the compactly-supported mapping class group $\overline{\mathrm{PMap}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive degree. By our results in arXiv:2211.07470 and other known computations, such a statement cannot be true for the full mapping class group $\mathrm{Map}(S)$ for all infinite-type surfaces $S$. However, we are still able to prove that the integral homology of $\mathrm{Map}(S)$ is uncountable in all positive degrees for a large class of infinite-type surfaces $S$. The key property of this class of surfaces is, roughly, that the space of ends of the surface $S$ contains a limit point of topologically distinguished points. Our result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor-Bendixson rank $α$, where $α$ is a successor ordinal. We also observe an order-$10$ element in the first homology of the pure mapping class group of any surface of genus $2$, answering a recent question of G. Domat.
