The computation of the topological invariants of mapping spaces between Lie gorups
Project/Area Number  05640134 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
Geometry

Research Institution  Kanagawa University 
Principal Investigator 
KOZIMA Kazumoto Kanagawa University, Faculty of Science, Professor, 理学部, 教授 (00127078)

CoInvestigator(Kenkyūbuntansha) 
KAWAHIGASHI Ken Kanagawa Univercity, Faculty of Science, Assistant, 理学部, 助手 (70231272)
MATSUI Shogo Kanagawa Univercity, Faculty of Science, Lecturer, 理学部, 講師 (00221581)

Project Fiscal Year 
1993 – 1994

Project Status 
Completed(Fiscal Year 1994)

Budget Amount *help 
¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1994 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1993 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Keywords  Lie group / Loop space / Loop group / Cohomology group / Function space / リー群 / ループ空間 / ループ群 / コホモロジー群 / 写像空間 / ホモロジー群 
Research Abstract 
Let X be a connected fiber of a Lie group (or finite Hspace) G.By evaluating of the adjoint map G*OMEGA^nX*OMEGA^nX on cohomology, one can get information of the cohomology of the space of type Map (S^n, G). We investigate this problem by using a computer aided symbolic computation and the theoritical computation based Hopf algebra structures and cohomology operations. We got some results by the computer aided computation and could get more general results by the theoritical method. Especialy, for the case G is the DwyerWilkerson's Hspace, Spin (n) and Sp (n) (and the connected fibers of these spaces), we could evaluate the adjoint maps on the cohomology. These result show that the homotopy type of the space like Map (S^n, G) is ristricted strongly by these adjoint map. We got also the result about the commutator map and its lift to the connected fiber space. As an application of this, we can prove the unknown result about the homotopy commutativity of the rotation group. The symbolic caliculation like effective homology causes a hevy intermidiate explosion of computation and it makes very hevy garbage collection on the symbolic computaion system. About this problem, S.Matsui applied the method of parallel and conditional garbage collection and reduced GC time. K.Kawahigashi investigated the application to phisical phenomina which some properties of the low dimensional Lie group affects.

Report
(4results)
Research Output
(12results)