论文标题
具有水平结构的模量空间的高维共同体II：穿刺和边界
The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
论文作者
论文摘要
我们给出了两个证据，表明具有穿刺/边界的表面映射类组的适当定义的一致性亚组具有大量的合理共同体学在其虚拟的共同体学维度中。特别是，我们给出了三个变量中每个变量中每个界限的界限：穿刺数，边界组件的数量和属，概括了Fullarton-Putman的工作。一路走来，我们简化了一个harer定理的解释，该定理解释了如何通过一个过程通过一个过程将多重函数表面的曲线复合物与曾经启动的表面的曲线复合物联系起来，该过程可以将其视为曲线精确序列的Birman精确序列的类似物。 作为一种应用，我们证明了具有标记点的曲线模量空间的相干共同体学维度上的上限和下限。对于$ g \ \ leq 5 $，我们为任何数量的标记点计算此连贯的共同体学维度。与我们在同胞学方面的界限相反，当表面具有$ n \ geq1 $标记点时，这些界限被证明独立于$ n $，并且仅取决于属。
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.
