Algebra of dimension theory

Abstract

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \linebreak 2. For pointed compact spaces $X$ and $Y$, $\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))$ if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of Hurewicz Theorem in extension theory becomes $\dim(π_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\dim_A(X)$ of a pointed compact space $X$ with respect to a graded group $A$ is introduced. It turns out $\dim_A(X)\leq 0$ iff $\dim_{A(n)}(X)\leq n$ for all $n\in\Z$. If $A$ and $B$ are two positive graded groups, then $\dim(A)=\dim(B)$ if and only if $\dim_A(X)=\dim_B(X)$ for all compact $X$.