Subspace and addition theorems for extension and cohomological dimensions. A problem of Kuzminov

Abstract

Let $K$ be either a CW or a metricsimplicial complex. We find sufficient conditions for the subspace inequality$$A\subset X, \quad Kın \text{\rm AE}(X)\Rightarrow Kın \text{\rm AE}(A).$$For the Lebesgue dimension ($K=S^n$) our result is a slightgeneralization of Engelking's theorem for a strongly hereditarilynormal space $X$. As a corollary we get the inequality$$A\subset X\Rightarrow\dim_GA\leq\dim_GB.$$for a certain class of paracompact spaces $X$ and an arbitrary abelian group $G$.As for the addition theorems$$\gatherKın \text{\rm AE}(A), \;\; Lın\text{\rm AE}(B)\Rightarrow K\ast Lın\text{\rm AE}(A\cup B), \dim_G(A\cup B)\leq\dim_GA+\dim_GB+1,\endgather$$we extend Dydak's theorems for metrizable spaces ($G$ is a ring with unity) to some classes of paracompact spaces.