Representation theory of groupoids and its applications to complex cobordism theory
| Project/Area Number |
11640092
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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 | Osaka Prefecture University |
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Principal Investigator |
YAMAGUCHI Atsushi Osaka Prefecture University, Collage of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (80182426)
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| Project Period (FY) |
1999 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2002: ¥100,000 (Direct Cost: ¥100,000)
Fiscal Year 2001: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 2000: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | groupoid / fibered category / representation theory / Steenrod algebra / unstable module / K-thoery / formal group / complex projective space / 群スキームの表現 / 双対Steenrod代数 / 双対Stennrod代数 |
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| Research Abstract |
First, we considered the relations between the notions of fibered categories and 2-categories and showed that the 2-category consisting of fibered categories over a category ε is equivalent to the 2-category consisting of 2-functors from the opposite category of ε to the 2-category cat consisting of categories. Then, we gave a general definition of the representation of groupoid using the notion of fibered category. Moreover, we collected basic facts by constructing various examples of representations of groupoids and defining the trivial representations and regular representations.
We denote by F_p the prime field of characteristic p and by TopVect^* the category of graded topological vector spaces over F_p which have fundamental systems of the neighborhood of 0 consisting of sub vector spaces. We observed that the ordinary mod p cohomology theory is regaded as a functor from the category of topological spaces to TopVect^* by giving a suitable topology to each mod p cohomology group of
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a space. After defining the space Hom(V^*, W^*) of linear maps suitably, we showed that a functor Z^* → Hom(W^*, Z^*) is a substitute for the right adjoint of the functor V^* → V^*【cross product】^^^W^* given by the completed tensor product and investigated properties of these functors. With the preparations above, we showed that the mod p cohomology theory of spaces are regarded as representations of the affine group scheme represented by the dual Steenrod algebra and tried to reconstruct the Lanne's theory of unstable modules over the Steenrod algebras and submit several axioms which characterize the Steenrod algebras.
On the other hand, in order to investigate the relations between the formal group law of the complex K-theory and the real K-theory which has the associated Hopf algebroid though it is not complex oriented, we determined the KO^*-algebra structures of the real K-cohomology of complex projective spaces CP^l and their product CP^l × CP^m. Moreover, we determined the KO_*-algebra structure of the real K-cohomology of infinite dimensional complex projective space. Less
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Report
(5 results)
Research Products
(6 results)
