| Co-Investigator(Kenkyū-buntansha) |
MATSUZAWA Jun-ichi Kyoto University, Faculty of Engineering, Lecturer, 工学研究科, 講師 (00212217)
FUKAYA Kenji Kyoto University, Faculty of Science, Professor, 理学研究科, 教授 (30165261)
KONO Akira Kyoto University, Faculty of Science, Professor, 理学研究科, 教授 (00093237)
HARADA Masana Kyoto University, Faculty of Science, Instructor, 理学研究科, 助手 (80181022)
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| Research Abstract |
In this project, we developed so-called Novikov Conjecture, a conjecture in differential topology. First, we treated combing groups, as a class of discrete groups. This class, being a big class, includes cases where a segment may be far from being a geodesic. Therefore, we introduced the geometric notion of properness and its class turned out to be very easy to treat. Secondly, as a Fredholm representation corresponding to the E-theory introduced by Connes and Higson, we introduced the notion of asymptotic Lipschitz maps of spaces. Under these preparations, we proved the Novikov Conjecture for torsion-free, proper combing groups.
For a map of discrete metric spaces, we can consider conditions on the metric, such as the Lipschitz condition Putting such conditions on the metric, we defined a map being a fiber structure. By studying fiber structures in the cases of discrete groups, we obtained the following : Let Γ be a fundamental group of almost non positively curved manifold. Then any class in H^* (Γ ; R) is a proper Lipschitz class. In particular, Γ satisfies Novikov conjecture.
We studied versal deformations of reflexive modules on rational double points. We constructed a natural stratification of the deformation space and a desingularization of the closure of a stratum as a moduli space, representing a functor defined over the deformation space as a base. In particular, the closure relation of the classes of reflexive modules coincides with the usual order of dominant weights of the corresponding root system. Moreover, we described the singularities arising from adjacent strata.
Finally, we generalized Ito-Nakamura type results on McKay correspondence to the cases of general quotient surface singularities, as conjectured by Riemenschneider.
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