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

通过归纳的方法，在可分离度量空间中引入了一个新的上同维，即大归纳维数。我们用nd_GX表示可分度量空间X相对于阿贝群g的强上同调维数。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，”这里的dim_G表示相对于g的通常上同调维数。(1)如果一个分离度量空间X是一个ANR, Ind_GX = dim_G X每阿贝尔群G,(2)如果一个分离度量空间X是有限维空间和一个阿贝尔群G是可数,Ind_GX = dim_G X,(3)为每一个无限维空间紧凑度规空间X dim_Z X = 2, Ind_ZX = 3,(4)对于一个给定的素数p,存在一个紧度量空间X这样dim_Z_ < (p) > X = 2 < 3 = Ind_Z_ < (p) > X,(5)为每一个可分离的度量空间X和每一个交换群G,对于每一个可分度量空间X， Ind_QX = dim_qx，其中Q是有理数环。

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.