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Transactions of the American Mathematical Society

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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 $X$.

References [Enhancements On Off] (What's this?)

  • [Git] M. Aguilar, S. Gitler, and C. Prieto, Algebraic Topology from a Homotopical Viewpoint, Springer-Verlag (2002). MR 1908260 (2003c:55001)
  • [BK] A. K. Bousfield and D. M. Kan, Homotopy limits, Completions, and Localizations, Lecture Notes In Math., Springer-Verlag (1972). MR 0365573 (51:1825)
  • [CD] M. Cencelj and A. N. Dranishnikov, Extension of maps to nilpotent spaces II, Topology Appl. 124 (2002), no. 1, 77-83. MR 1926136 (2003f:55002)
  • [D$_3$] A. N. Dranishnikov, On intersection of compacta in euclidean space II, Proceedings of AMS 113 (1991), 1149-1154. MR 1060721 (92c:54015)
  • [D$_4$] A. N. Dranishnikov, On the mapping intersection problem, Pacific Journal of Mathematics 173 No.2 (1996), 403-412. MR 1394397 (97e:54030)
  • [D$_5$] A. N. Dranishnikov, Extension of maps into CW complexes, Math. USSR Sbornik 74 (1993), 47-56. MR 1133570 (93a:55002)
  • [D$_7$] A. N. Dranishnikov, On dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space, Transactions of the American Math.Soc. 352 (2000), 5599-5618.MR 1781276 (2001j:55002)
  • [D$_8$] A. N. Dranishnikov, Basic elements of the cohomological dimension theory of compact metric spaces (1999).
  • [D-D$_1$] A. Dranishnikov and J. Dydak, Extension dimension and extension types, Proceedings of the Steklov Institute of Mathematics 212 (1996), 55-88. MR 1635023 (99h:54049)
  • [D-D$_2$] A. Dranishnikov and J. Dydak, Extension theory of separable metrizable spaces with applications to dimension theory, Transactions of the American Math. Soc. 353 (2000), 133-156. MR 1694287 (2001f:55002)
  • [Dy$_3$] J. Dydak, Cohomological dimension and metrizable spaces II, Trans. Amer. Math. Soc. 348 (1996), 1647-1661. MR 1333390 (96h:55001)
  • [Dy$_4$] J. Dydak, Realizing dimension functions via homology, Topology and its Appl. 64 (1995), 1-7. MR 1354376 (96k:54058)
  • [Dy$_6$] J. Dydak, Extension theory of infinite symmetric products, Fundamenta Mathematicae, to appear.
  • [HMR] P. Hilton, G. Mislin, and J. Roitberg, Localizations of Nilpotent Groups and Spaces, North-Holland (1975).MR 0478146 (57:17635)
  • [Krull1931] W. Krull, Allgemeine Bewertungstheorie, J.Reine Angew. Math 167 (1931), 160-196.
  • [K] V. I. Kuzminov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1-45. MR 0240813 (39:2158)
  • [N] J. A. Neisendorfer, Primary homotopy theory, Memoirs of AMS 232 (1980), 1-67. MR 0567801 (81b:55035)
  • [O$_1$] W. Olszewski, Completion theorem for cohomological dimensions, Proceedings of AMS 123 (1995), 2261-2264. MR 1307554 (95k:54064)
  • [S] E. V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (1998), 975-1069. MR 1691185 (2002a:55002)
  • [Sp] E. Spanier, Algebraic Topology, McGraw-Hill (1966), New York, NY. MR 0210112 (35:1007)
  • [Su] D. Sullivan, Geometric Topology, Part I: Localization, Periodicity, and Galois Symmetry, M.I.T. Press (1970). MR 0494074 (58:13006a)

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Additional Information

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

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.

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