2006 Fiscal Year Final Research Report Summary
Homotopy Theoretic Study of Higher Dimensional Categories
| Project/Area Number |
16540061
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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 | Kyoto University |
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Principal Investigator |
NISHIDA Goro Kyoto University, Graduate school of sciences, Professor, 大学院理学研究科, 教授 (00027377)
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| Co-Investigator(Kenkyū-buntansha) |
FUKAYA Kenji Kyoto University, Graduate school of sciences, Professor, 大学院理学研究科, 教授 (30165261)
KONO Akira Kyoto University, Graduate school of sciences, Professor, 大学院理学研究科, 教授 (00093237)
NAKAJIMA Hiraku Kyoto University, Graduate school of sciences, Professor, 大学院理学研究科, 教授 (00201666)
MINAMI Norihiko Nagoya institute of technology, Graduate school of Engineering, Professor, 大学院工学研究科, 教授 (80166090)
SHIMOMURA Katumi Kochi University, Faculty of sciences, Professor, 理学部, 教授 (30206247)
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| Project Period (FY) |
2004 – 2006
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| Keywords | formal group / simplicial set / symmetric group / cohomology / linear group / K-theory / homotopy type |
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| Research Abstract |
I tried to make a new construction of algebraic K-theory based on formal group laws of general heights, e.g., Honda group law. Formal group laws other than additive or multiplicative group, do not give algebraic groups, but can be regarded as a gamma set defined by Segal. Since a gamma set is a simplicial set, we have a homotopy type by taking the geometric realization. We consider the full matrix algebra over a ring of integers of a local field. Then the resulting homotopy type is a candidate for a space of algebraic K-theory of higher height. First I studied the cohomology group of the space. In the case of multiplicative group, the space is the classifying space of the general linear group. The cohomology is symmetric polynomials of the cohomology of the maximal torus. If the height of the formal group law is h, then the dimension of the maximal torus is h-times of the ordinary case. Then we obtain a multi-symmetric polynomials as cohomology of the space for the candidate. What we have to do next is to define or construct the representation theory based on a given formal group law of higher height. In particular I hope to get a group theoretic interpretation of Hokins-Kuhn-Ravenel character not using the Morava K-theory.
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