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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebra of dimension theory


Author: Jerzy Dydak
Journal: Trans. Amer. Math. Soc. 358 (2006), 1537-1561
MSC (2000): Primary 54F45, 55M10, 55N99, 55Q40, 55P20
Published electronically: April 22, 2005
MathSciNet review: 2186985
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03690-1
PII: S 0002-9947(05)03690-1
Keywords: Cohomological dimension, dimension, Eilenberg-Mac\, Lane complexes, graded groups, infinite symmetric products, Moore spaces
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.