Research on cohomological dimension of topological spaces and one of Coxeter groups

拓扑空间的上同调维数及Coxeter群之一的研究

• 批准号: 14540077
• 负责人: KOYAMA Akira
• 金额: $ 2.24万
• 依托单位: Osaka Kyoiku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540077/
• 关键词: topological spaces; dimension; cohomological dimension; Coxeter groups; ideal boundary; コホモロジ一次元; CAT(O)空間

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.

我们用归纳的方法在可分度量空间类中引入了一种新的上同调维，即大归纳维。我们用ND_GX表示可分度量空间X关于阿贝尔群G的强上同调维。下面的不等式明显成立：“Ind_GX[小于或等于]Dim_GX[小于或等于]Ind_GX+1，”这里，dim_G表示通常的上同调维度，与G有关。(1)如果可分度量空间X是ANR，则对每个阿贝尔群G，Ind_GX=dim_G X，(2)如果可分度量空间X是有限维的，且阿贝尔群G是可数的，Ind_GX=DIM_G X，(3)对任意无限维紧度量空间X，若DIM_Z X=2，Ind_ZX=3，(4)对于给定的素数p，存在紧度量空间X使得DIM_Z&lt；(P)&gt；X=2&lt；3=Ind_Z_&lt；(P)&gt；X，(5)对每个可分度量空间X和每个交换群G，Ind_G(X×i)=dim_GX+1，(6)对每个可分度量空间X，Ind_QX=dim_qX，这里Q是所有有理和的环。

项目成果

Boege, M., Dydak, J.R.Jimenez, Koyama, A., Shchepin, E.: "Borsuk-Sieklucki theorem in cohomological dimension"Fund.Math.. 171(3). 213-222 (2002)

Boege, M.、Dydak, J.R.Jimenez、Koyama, A.、Shchepin, E.：“上同调维数中的 Borsuk-Sieklucki 定理”Fund.Math.. 171(3)。

Koyama Akira, Yokoi Katsuya: "Cohomological dimension and acyclic resolutions"Topology and its Applications. 120(1-2). 175-204 (2002)

小山明、横井克也：“上同调维数和无环解析”拓扑及其应用。

Koyama, Akira, Yokoi, Katsuya: "Cohomological dimension and acyclic resolution"Topology and its Applications. 120(1-2). 175-204 (2002)

Koyama、Akira、Yokoi、Katsuya：“上同调维数和非循环解析”拓扑及其应用。

Koyama, A., Dydak, J.: "Cohomological dimension of locally connected compacta"Topology and its Appl.. 113(1-3). 39-50 (2001)

Koyama, A.，Dydak, J.：“局部连通紧致的上同调维数”拓扑及其应用。113(1-3)。

Koyama, A., Yokoi, K.: "On Dranishnikov's cell-like resolutions"Topology and its Appl.. 113(1-3). 87-106 (2001)

Koyama, A.，Yokoi, K.：“论 Dranishnikov 的类细胞分辨率”拓扑及其应用。113(1-3)。