Algebra of dimension theory

Abstract:

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this 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 these equivalence classes: 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 $X$). 2) For pointed compact spaces $X$ and $Y$, $\dim (\mathcal {H}^{-\ast }(X))=\dim (\mathcal {H}^{-\ast }(Y))$ 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 the Hurewicz Theorem in extension theory becomes $\dim (\pi _\ast (K))=\dim (H_\ast (K))$ 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 $n\in \mathbf {Z}$. 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 $X$.