Project/Area Number  09640138 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Fukuoka University 
Principal Investigator 
ISHIGURO Kenshi Fukuoka Univ., Fac.of Science, Assoc.Prof., 理学部, 助教授 (00268971)

CoInvestigator(Kenkyūbuntansha) 
AKITA Toshiyuki Fukuoka Univ., Fac.of Science, Assistant, 理学部, 助手 (30279252)
KUROSE Takashi Fukuoka Univ., Fac.of Science, Asso.Prof., 理学部, 助教授 (30215107)
ODA Nobuyuki Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (80112283)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,300,000 (Direct Cost : ¥2,300,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Classifying spaces / Homotopy / pcompact groups / Compact Lie groups / Pairing / Admissible maps / 分類空間 / ホモトピー論 / コホモロジー論 / 有限ループ空間 / pコンパクト群 / コンパクトLie群 / Whitehead積 / Coxter群 
Research Abstract 
The research on the classifying spaces of compact Lie groups has been one of the major area in Homotoy Theory. Our results obtained during 1997 through 1998 are basically concerned with maps between classifying spaces and their applications. DwyerWilkerson defined apcompact group and studied its properties. The purely homotopy theoritic object appears to be a good generalization of a compact Lie group. A pcompact group has rich structure, such as a maximal torus, a Weyl group, etc. A note wrtten by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of the classifying spaces of pcompact groups are being made. We state here our main results. First, we consider the maps of pcompact groups of the form BX * BY*BZ.The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the pcompact group Z.Next, for G =S^3 * .. * S^3, let X be a genus of BG.We investigate the monoid of rational equivalences of X, denoted by epsilon(X). It is shown that a submonoid of epsilon_0(X), denoted by delta_00(X), determines the decomposability of the space X.We also show converses to some known results for the classifying spaces of ptoral groups or pcompact toral group. Suppose G is a compact Lie group. The following results are obtained. If there is a positive integer k such that the nth homotopy groups of the pcompletion of BG are zero for all n <greater than or equal> k then the loop space of this space is a pcompact toral group. If the canonical map Rep(G, K)*[BG, BK] is bijective for any compact connected Lie group K, then G is a ptoral group. in addition, our work containesa research on the conditions of a compact Lie group that its loop space of the pcompleted classifying space be a pcompact group.
