2012 Fiscal Year Final Research Report
Multiplicative structures of Hopf spaces and its applications
| Project/Area Number |
22540079
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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 |
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| Project Period (FY) |
2010 – 2012
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| Keywords | 幾何学 / トポロジー |
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| Research Abstract |
Multiplicative structures of Hopf spaces were studied through gauge groups. A gauge group is the topological group of automorphisms of a given principal bundle, which reflects the multiplicative structure of the structure group of the bundle. It is known that the classifying space of gauge groups can be identified with certain mapping spaces. In this programme, I gave the mod p decomposition of gauge groups which is the gauge group analogue of the mod p decomposition of Lie groups, and as its applications, the p-local homotopy types of gauge groups were determined and the relation between the decomposition of the adjoint bundles and the mod p decomposition of gauge groups were elucidated. This yields a basis of the study of the p-local homotopy theory of gauge groups. The foundation of fiberwise higher homotopy commutativity is developed. The existence of a section of the natural homomorphism from the gauge group to the structure group of the principal bundle as An spaces is related to the triviality of the adjoint bundle as a fiberwise An-space. From this, it is shown that the the existence of the An-section of the homomorphism is equivalent to the triviality of the adjoint bundle as a fiberwise An-space. Moreover, these equivalent condition turns out to have a connection with the higher homotopy commutativity of the structure group of the principal bundle.
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Research Products
(10 results)
