Algebra of dimension theory
Author:
Jerzy Dydak
Journal:
Trans. Amer. Math. Soc. 358 (2006), 15371561
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 of graded groups . There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes and , if and only if the infinite symmetric products and are of the same extension type (i.e., iff for all compact ). 2) For pointed compact spaces and , if and only if and are of the same dimension type (i.e., for all Abelian groups ). Dranishnikov's version of the Hurewicz Theorem in extension theory becomes for all simply connected . The concept of cohomological dimension of a pointed compact space with respect to a graded group is introduced. It turns out iff for all . If and are two positive graded groups, then if and only if for all compact .
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 A. Dranishnikov and J. Dydak, Extension theory of separable metrizable spaces with applications to dimension theory, Transactions of the American Math. Soc. 353 (2000), 133156. MR 1694287 (2001f:55002)
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Additional Information
Jerzy Dydak
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 379961300
Email:
dydak@math.utk.edu
DOI:
http://dx.doi.org/10.1090/S0002994705036901
PII:
S 00029947(05)036901
Keywords:
Cohomological dimension,
dimension,
EilenbergMac\,
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 DMS0072356 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.
