Algebra of dimension theory

Author:
Jerzy Dydak

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

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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Additional Information

**Jerzy Dydak**

Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300

Email:
dydak@math.utk.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03690-1

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.