We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

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关于 Johnson 子群映射曲面类群的商

我们研究映射类组的商${\伽玛_{g,1}}$由子组组成的具有一个边界分量的定向表面${{\cal I}_{g,1}}(k)$在约翰逊过滤中，我们证明了稳定的分类空间${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$加号构造之后是无限循环空间，适合在无限循环空间之间插值的无限循环空间图塔${\mathbb {Z}} \times B\Gamma _\infty ^ + $和${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B {\rm{Sp}}{({\mathbb {Z}})^ + }$. 我们还表明，对于每个级别ķ在约翰逊过滤中，这些商与合适的扭曲系数系统的同源性随着曲面的属趋于无穷大而稳定。