2003 Fiscal Year Final Research Report Summary
Research on cohomological dimension of topological spaces and one of Coxeter groups
| Project/Area Number |
14540077
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka Kyoiku University |
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Principal Investigator |
KOYAMA Akira Osaka Kyoiku Univ., Fac.of Education, Professor, 教育学部, 教授 (40116158)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Tadashi Yamaguchi Univ., Fac.of Education, Professor, 教育学部, 教授 (10107724)
MACHIGASHIRA Yoshiro Osaka Kyoiku Univ., Fac.of Education, Associate Professor, 教育学部, 助教授 (00253584)
SUGAHARA Kunio Osaka Kyoiku Univ., Fac.of Education, Professor, 教育学部, 教授 (20093255)
YOKOI Katsuya Shimane Univ., Fac.of Sci., Associate Professor, 総合理工学部, 助教授 (90240184)
YAGASAKI Tatsuhiko Kyoto Inst.Tech., Fac.of Tech., Associate Professor, 工芸学部, 助教授 (40191077)
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| Project Period (FY) |
2002 – 2003
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| Keywords | topological spaces / dimension / cohomological dimension / Coxeter groups / ideal boundary |
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| Research Abstract |
We introduced a new cohomological dimension to the class of separable metric spaces by an inductive way as large inductive dimension. By nd_GX we denote our strong cohomological dimension of a separable metric space X with respect to an abelian group G. The following inequalities are clearly hold : "Ind_GX 【less than or equal】 dim_GX 【less than or equal】 Ind_GX + 1," here dim_G means the usual cohomological dimension with resprect to G. Relate to the fundamental properties we showed the following.
(1) If a separable metric space X is an ANR, Ind_GX = dim_G X for every abelian group G,
(2) If a separable metric space X is finite-dimensional and an abelian group G is countable, Ind_GX = dim_G X,
(3) For every infinite-dimensional compact metric space X with dim_Z X = 2, Ind_ZX = 3,
(4) For a given prime number p, there exists a compact metric space X such that dim_Z_<(p)> X = 2 < 3 = Ind_Z_<(p)> X,
(5) For every separable metric space X and every abelian group G, Ind_G(X × I) = dim_G X + 1,
(6) For every separable metric space X, Ind_QX = dim_Q X, here Q is the ring of all rational sumbers.
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