The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
复制标题
具有水平结构的曲线模空间的高维上同调II：穿孔和边界
DOI：
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发表时间：
2020
影响因子：
1
通讯作者：
Andrew Putman
中科院分区：
文献类型：
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作者：
Tara E. Brendle;N. Broaddus;Andrew Putman
证明了具有穿孔/边界的曲面的映射类群的适当定义的同余子群在其虚拟上同调维有大量的有理上同调.特别地，我们给出了三个变量的超指数上界：穿孔数、边界分量数和亏格，推广了Fullarton-Putman的工作。在此过程中，我们简化了Heller的一个定理，该定理解释了如何通过一个类似于曲线复形的Bman精确序列的过程，将多个穿孔曲面的曲线复的同伦型与一次穿孔曲面的曲复的同伦型联系起来。
作为应用，我们证明了具有标记点的曲线模空间的凝聚上同调维的上下界。对于$g leq 5$，我们计算任意数目的标记点的这个凝聚上同调维度。与我们的上同调界相反，当曲面有$ngeq1$标记点时，这些界被证明是独立于$n$的，并且只依赖于亏格。
We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes.
As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For $g leq 5$, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has $n geq1$ marked points, these bounds turn out to be independent of $n$, and depend only on the genus.