The algebra and topology of the mapping class group

所提出的研究涉及的映射类组的表面和相关的群体。它包含四个项目家族。第一个是关于有限指数子群的上同调性质。这里的具体目标包括证明一个类似的博雷尔稳定性定理的映射类组和证明一种“等变同调稳定性”定理的同余子群的特殊线性群。第二个家庭的项目涉及皮卡德集团有限覆盖的模空间的曲线。这里的目标是了解这些有限覆盖上的某些自然线丛的整除性。第三族工程涉及映射类群的Torelli子群，它是映射类群在曲面的第一同调群上的作用核。这里的目标是澄清这个群及其子群的基本上同调和组合性质。最后一个家庭的项目涉及类似的Torelli子群的自同构群的自由群。这里的目标是调整已经成功研究映射类组的工具，以设置自由组的自同构组。特别是，类似物的曲线complex将进行研究。建议的项目涉及映射类组，这在许多数学领域中发挥着关键作用，从代数几何和低维拓扑数学物理。 问题涉及映射类群的上同调群寻求衡量这些群体的最基本的不变量之一？粗略地说，k维上同调群计算了群的几何模型中的k维“洞”。 这些在应用中起着重要的作用。 另一组问题涉及这些群的组合学。 这应该允许在它们内部进行实际的具体计算，促进对与它们相互作用的各种物体的研究。

The proposed research concerns the mapping class group of a surface and related groups. It contains four families of projects. The first concerns the cohomological properties of finite-index subgroups. Specific goals here include proving an analogue of the Borel stability theorem for the mapping class group and proving a sort of "equivariant homological stability" theorem for congruence subgroups of the special linear group. The second family of projects concerns the Picard groups of finite covers of the moduli space of curves. The goal here is to understand the divisibility properties of certain natural line bundles on these finite covers. The third family of projects concerns the Torelli subgroup of the mapping class group, which is the kernel of action of the mapping class group on the first homology group of the surface. The goal here is to clarify the basic cohomological and combinatorial properties of this group and its subgroups. The final family of projects concerns the analogue of the Torelli subgroup in the automorphism group of a free group. The goal here is to adapt tools that have been successful in studying the mapping class group to the setting of the automorphism group of a free group. In particular, analogues of the curve complex will be studied.The proposed projects concern mapping class groups, which play a key role in many fields of mathematics, ranging from algebraic geometry and low dimensional topology to mathematical physics. The problems which involve the cohomology groups of the mapping class group seek to measure one of the most basic invariants of these groups ? roughly, the k-dimensional cohomology groups count the k-dimensional "holes" in geometric models for the groups. These play an important role in the applications. Another set of problems concern the combinatorics of these groups. This should allow actual concrete calculations within them, facilitating the investigation of the diverse objects with which they interact.