Studies on the Cohomology of Mapping Class Groups, Coxeter Groups and Artin Groups

离散群的上同调，如闭曲面的映射类群、Coxeter群和Artin群，是拓扑学和几何学中的重要研究对象之一。(1)与有限子群上同调的关系(2)离散群在流形/复形和组合结构上的作用(3)关于(1)的代数方法(如组合学和自由分解)，Akita和Kawazumi证明了映射类群的循环子群的积分Riemann-Roch公式，它们是积分上同调的Grothedieck-Riemann-Roch定理的变体。关于(2)，Izeki和Hosaka得到了关于群作用和几何结构的各种结果。最后，Akita关于(3)证明了偶数维三角流形的Euler特征的替代公式。证明这些公式的关键因素是Klee得到的推广的Dehn-Sommerville方程。此外，Akita还利用经典数论中的Kummer同余证明了mod p Riemann-Roch公式在各种情况下都成立。

The cohomology of discrete groups, such as mapping class groups of closed surfaces, Coxeter groups and Artin groups, is one of the important objects in topology as well as geometry In this research project, we studied the cohomology of discrete groups. The following three subjects were emphasized in the project:(1) Relations with the cohomology of finite subgroups(2) Actions of discrete groups on manifolds/complexes and combinatorial structures(3) Algebraic methods (such as combinatorics and free resolutions)Concerning of (1), Akita and Kawazumi proved integral Riemann-Roch formulae for cyclic subgroups of mapping class groups, which are variants of Grothedieck-Riemann-Roch theorem for integral cohomology. Concerning of (2), Izeki and Hosaka obtained various results concerning of group actions and geometric structures Finally, concering of (3), Akita proved alternative formulae for the Euler characteristics of even dimensional triangulated manifolds. The key ingredient to prove the formulae is generalized Dehn-Sommerville equations obtained by Klee. In addition, Akita showed that mod p Riemann-Roch formulae hold for various cases, by using Kummer's congruences in classical number theory.