Investigation of unstable homotopy theory

Research Project

Project/Area Number	60540045
Research Category	Grant-in-Aid for General Scientific Research (C)
Allocation Type	Single-year Grants
Research Field	代数学・幾何学
Research Institution	Hyogo University of Teacher Education
Principal Investigator	NOMURA Yasutoshi Hyogo University of Teacher Education, 学校教育学部, 教授 (20029630)
Co-Investigator(Kenkyū-buntansha)	MATSUYAMA Hiroshi Hyogo University of Teacher Education, 学校教育学部, 助教授 (80028266) KOIKE Satoshi Hyogo University of Teacher Education, 学校教育学部, 助教授 (60161832) WATANABE Kinji Hyogo University of Teacher Education, 学校教育学部, 助教授 (20004468) ITAGAKI Yoshio Hyogo University of Teacher Education, 学校教育学部, 教授 (30006431) YANAGIHARA Hiroshi Hyogo University of Teacher Education, 学校教育学部, 教授 (00033803)
Project Period (FY)	1985 – 1986
Project Status	Completed (Fiscal Year 1986)
Budget Amount *help	¥1,400,000 (Direct Cost: ¥1,400,000) Fiscal Year 1986: ¥700,000 (Direct Cost: ¥700,000) Fiscal Year 1985: ¥700,000 (Direct Cost: ¥700,000)
Keywords	Principal fibration / Category / <M_k> -map / <N_k> -space / Homotopy fibre / Pushout / Phantom map / 双対性 / 左ホモトピー逆 / ファントム写像
Research Abstract	We have defined the notion of <M_k> -map as a generalization of a homotopy monomorphism introduced by P. J. Hilton and studied by T. Ganea. Let a principal fibration call an <M_(infinte)> -map if it is an <M_k> -map for every positive integer k. No principal fibration has not been known which is an <M_(infinte)> -map but admits no left homotopy inverse. In this research we have succeeded to construct such examples, using phantom maps in complex K-theory. The notion of homotopy pullbacks and pushouts is important in developing unstable homotopy theory. In this investigation we introduce the notion of a triad and cotriad of squares and interprete a theorem of Mather-Walker concerning commutativity of homotopy limits and colimits, which allows us to extend their result. Further we can obtain some approximation theorems to the duals of preceding theorems.