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

通过归纳的方法，在可分度量空间类中引入了一个新的上同调维数作为大归纳维数。用nd_GX表示可分度量空间X关于交换群G的强上同调维数。下面的不等式显然成立：“Ind_GX [小于或等于] dim_GX [小于或等于] Ind_GX + 1”，这里dim_G表示与G有关的通常的上同调维数。与我们展示的基本性质有关。(1)若可分度量空间X是ANR，则对每个阿贝尔群G，有Ind_GX = dim_GX，（2）若可分度量空间X是有限维的，且阿贝尔群G是可数的，则有Ind_GX = dim_GX，（3）对每个无穷维紧致度量空间X，且dim_ZX = 2，有Ind_ZX = 3，（4）对给定素数p，存在紧度量空间X，使得dim_Z_（p）> X = 2 < 3 = Ind_Z_（p）> X，（5）对每个可分度量空间X和每个阿贝尔群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是所有有理数和的环。

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.