## Algebra of dimension theory

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**358**(2006), 1537-1561 Request permission

## 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 $X$.

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## Additional Information

**Jerzy Dydak**- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
- Email: dydak@math.utk.edu
- Received by editor(s): August 14, 2001
- Received by editor(s) in revised form: April 12, 2004
- Published electronically: April 22, 2005
- Additional Notes: This research was supported in part by grant DMS-0072356 from the National Science Foundation
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 1537-1561 - MSC (2000): Primary 54F45, 55M10, 55N99, 55Q40, 55P20
- DOI: https://doi.org/10.1090/S0002-9947-05-03690-1
- MathSciNet review: 2186985