2007 Fiscal Year Final Research Report Summary
Studies on the Cohomology of Mapping Class Groups, Coxeter Groups and Artin Groups
| Project/Area Number |
17540056
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Hokkaido University |
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Principal Investigator |
AKITA Toshiyuki Hokkaido University, Faculty of Science, Associated Professor (30279252)
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| Co-Investigator(Kenkyū-buntansha) |
IZEKI Hiroyasu Tohoku University, Graduate School of Science, Associated Professor (90244409)
HIROSE Susumu Saga University, Faculty of Science and Engineering, Associated Professor (10264144)
HOSAKA Tetsuya Utsunomiya University, Faculty of Education, Associated Professor (50344908)
KAWAZUMI Nariya The University of Tokyo, Grad. School of Mathematical Sciences, Associated Prof. (30214646)
OHMOTO Toru Hokkaido University, Faculty of Science, Associated Professor (20264400)
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| Project Period (FY) |
2005 – 2007
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| Keywords | discrete groups / cohomology / topology / geometry / mapping class groups / Euler characteristic / complex |
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| Research Abstract |
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.
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Research Products
(8 results)
