| Co-Investigator(Kenkyū-buntansha) |
YAGITA Nobuaki Ibaraki University Faculty of Education, Professor, 教育学部, 教授 (20130768)
KOMATSU kazushi Kochi University, Faculty of Science, Associate Professor, 理学部, 助教授 (00253336)
HEMM Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
NAKAI Hirofumi Oshima National College of Maritime Technology, Lecturer, 講師 (80343739)
OHKAWA Tetsusuke Hiroshima Institute of Technology, Associate Professor, 工学部, 助教授 (60116548)
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| Research Abstract |
In this research, we aimed to understand the stable homotopy category L_n, localized with respect to the Johnson-Wilson spectrum E(n)(in a sense of Bousfield). As a way of understanding it, we study a geometric properties of spaces in the unstable homotopy category, which is a base of the stable homotopy category, and study properties of cohomologies of groups. We also have a little more direct way of understanding L_n, in other words, the determination of the homotopy groups π_*(L_nS^O) of the sphere L_nS^O of the category L_n4 and the determination of the Picard group Pic(L_n). For the viewpoint of unstable homotopy theory, we have Hemmi's results, which showed how much we can say about the associativity of a finite complex with H-structure, by observing the cohomology classes after embedding it to relevant loop spaces. Komatsu studied a condition on dimensions of immersions of a. real projective space by observing the bundle structure on the space. Ohkawa studied the stable homotopy category from the viewpoint of the Bousfield classes. For the viewpoint of cohomologies of groups, Yagita investigated the realization map from a motivic cohomology to an ordinary cohomology on a compact group by using the Milnor operations. On stable homotopy groups, Nakai determined the E_2-term, which will be a base of a future computation, of the Adams-Novikov spectral sequence converging to homotopy groups of a spectrum relating to the spheres. For the homotopy groups of spheres, they are known for n<2 and n=2 and p> 3.Shimomura determined the homotopy groups and the Adams-Novikov E_2-term in the cases where n= 2 and p=3 and where n= and p=2, respectively. On the Picard group, we showed a relation between the E_r-terms of the E(n)-based Adams spectral sequence converging to π_*(L_nS^O) and the Picard group Pic(L_n), and gave an example of an element of Pic(L_n), that is not the sphere, in the case where n=2 and p=3.
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