|Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
The head investigator introduced the concept of "quasi KO_*-equivalence" in 1990, in order to give a certain classification of CW-complexes or manifolds, which is weaker than the classification based on the K_*-localization. Since then, he has continued to research mainly the several subjects concerned with the quasi KO_*-equivalence and the K_*-localization. He has already obtained some satisfactory results, and moreover obtained a new result mentioned below during the period of Scientific Research Project in 2003-2005.
The subject is to classify CW-complexes (or CW-spectra) X by the K_*-localization when once X has been classified by the quasi KO_*-equivalence. For our purpose it is necessary to investigate the behavior of the Adams operation on KO_*X. However it is never easy to determine their K_*-local types because the behavior of the Adams operation on KO_*X is very complicated. In 2001-2002 he determined the K_*-local type of X when KU_*X is isomorphic to Z Z the free abelian gr
oup of rank 2.
During the period of Scientific Research Project in 2003-2005 he moreover established to determine the K_*-local type of X when KU_* X is isomorphic to Z Z/2^m the direct sum of the free abelian group of rank 1 and a 2-cyclic group.
Our result is mentioned below :
In order to determine the K_*-local type of X with KU_*X Z Z/2^m we may treat CW-spectra X under the restriction that KU_0X Z Z/2^m and KU_1X 0 by virtue of the duality theorem. Such a CW-spectrum X is quasi K0_*-equivalent to one of the following 7 kinds of spectra Σ^<2i>∨SZ/2^m,Σ^<2i>∨V_m, Σ^<2i>∨W_m,M_m,N_m,Q_m,R_m (0【less than or equal】i【less than or equal】3). When a CW-spectrum X is quasi KO_*-equivalent to each of these small spectra, it has one or two kinds of the Adams complex operation on KU_*X following its quasi KO_*-equivalent type. The Adams operation on its KO-homology groups KO_*X are completely determined by the behavior of the Adams complex operation on KU_*X. By these observation of the Adams operation and construction of several small cells complexes we can completely determine the K_*-local type of CW-spectra X with KU_0X Z Z/2^m and KU_1X 0. Consequently, a CW-spectrum X has essentially 31,33,1,4,2,3,12 kinds of K_*-local types if it is quasi KO_*- equivalent to Σ^<2i>∨SZ/2^m, Σ^<2i>∨V_m,Σ^<2i>∨W_m,M_m,N_m,Q_m,R_m respectively. Less