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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cohomological dimension
and metrizable spaces. II


Author: Jerzy Dydak
Journal: Trans. Amer. Math. Soc. 348 (1996), 1647-1661
MSC (1991): Primary 55M11, 54F45
DOI: https://doi.org/10.1090/S0002-9947-96-01536-X
MathSciNet review: 1333390
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Abstract | References | Similar Articles | Additional Information

Abstract: The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$.

Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$.

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension:

Theorem. Suppose $A,B$ are subsets of a metrizable space. Then

\begin{equation*}\dim _{{\mathbf R} }(A\cup B)\le \dim _{{\mathbf R} }A+\dim _{{\mathbf R} }B+1 \end{equation*}

for any ring ${\mathbf R} $ with unity and

\begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*}

for any abelian group $G$.

The second part of the paper is devoted to the question of existence of universal spaces:

Theorem. Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then

a.
Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$.
b.
There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$.
c.
There is a completely metrizable and separable space $Z$ such that $ K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z'$ with $ K_{i}\in AE(Z')$ for all $i\ge 1$ embeds in $Z$ as a closed subset.


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  • [Bo] B. F. Bockstein, Homological invariants of topological spaces, I, (English translation in Amer. Math. Soc. Transl. 11 (1959)), Trudy Moskov. Mat. Obshch. 5 (1956), 3--80 (Russian). MR 18:813
  • [Ch-1] A. Chigogidze, Uncountable powers of the real line and the natural series and n-soft maps, Soviet Math. Dokl. 90 (1984), 342--345. MR 86m:54049
  • [Ch-2] A. Chigogidze, A note on cohomological dimension, preprint.
  • [Do] A. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin, 1972. MR 54:3685
  • [Dr-1] A. N. Dranishnikov, Homological dimension theory, Russian Maht. Surveys 43:4 (1988), 11--62. MR 90e:55003
  • [Dr-2] A. N. Dranishnikov, Extension of maps into CW complexes, Math. USSR Sbornik 74 (1993), 47--56. MR 93a:55002
  • [Dr-3] A. N. Dranishnikov, Cohomological dimension is not preserved by Stone-\v{C}ech compactification, Comptes Rendus Bulgarian Acad. of Sci. 41 (1988), 9--10 (Russian). MR 90e:55002
  • [Dr-4] A. N. Dranishnikov, Alternative construction of compacta with different dimensions, Proceedings of the Graduate Workshop in Mathematics and its Applications in Social Sciences, Ljubljana University 1991, Ljubljana (Slovenia), pp. 33--36.
  • [Dr-5] A. N. Dranishnikov, On the mapping intersection problem, preprint.
  • [D-R] A. Dranishnikov and D. Repov\v{s}, The Urysohn-Menger Sum Formula: An improvement of the Dydak-Walsh theorem to dimension one, J.Austral. Math. Soc., Ser.A (to appear).
  • [D-R-S] A. Dranishnikov, D. Repov\v{s} and E. \v{S}\v{c}epin, On the failure of the Urysohn-Menger sum formula for cohomological dimension, Proc. Amer. Math. Soc. 120 (1994), 1267--1270. MR 94f:55001
  • [D-S] J. Dydak and J. Segal, Shape theory: An introduction, Lecture Notes in Math., vol. 688, Springer Verlag, 1978, pp. 1-150. MR 80h:54020
  • [D-T] A. Dold and R. Thom, Quasifaserungen und Unendliche Symmetrische Produkte, Annals of Math. 67 (1958), 239--281 (German). MR 21:3855
  • [D-M] J. Dydak and J. Mogilski, Universal cell-like maps, Proceedings of AMS 122 (1994), 943--948. MR 95a:55003
  • [D-W-1] J. Dydak and J. J. Walsh, Aspects of cohomological dimension for principal ideal domains, preprint.
  • [D-W-2] J. Dydak and J. J. Walsh, Spaces without cohomological dimension preserving compactifications, Proceedings of the Amer. Math. Soc. 113 (1991), 1155--1162. MR 92c:54039
  • [Dy-1] J. Dydak, Cohomological dimension and metrizable spaces, Transactions of the Amer. Math. Soc. 337 (1993), 219--234. MR 93g:55001
  • [Dy-2] J. Dydak, pp. 219--234; Compactifications and cohomological dimension, Topology and its Appl. 50 (1993), 1--10. MR 94c:55002
  • [Dy-3] J. Dydak, Union theorem for cohomological dimension: A simple counterexample, Proceedings of AMS 121 (1994), 295--297. MR 94g:55001
  • [Dy-4] J. Dydak, Realizing dimension functions via homology, Topology and its Appl. 64 (1995), 1--7.
  • [Fu] L. Fuchs, Infinite abelian groups, Academic Press, New York and London, 1970. MR 41:333
  • [Hu] S. T. Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 31:6202
  • [Ka] A. I. Karinski, On cohomological dimension of the Stone-\v{C}ech compactification, Vestnik Mosk. Univ., Ser. 1 Mat. 1991, no. 4, 8--11 (Russian).
  • [Ku] W. I. Kuzminov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1--45. MR 39:2158
  • [M-S] S. Mardesic and J. Segal, Shape theory, North-Holland, Amsterdam, 1982. MR 84b:55020
  • [Ol-1] W. Olszewski, Completion theorem for cohomological dimensions, Proc. Amer. Math. Soc. 123 (1995), 2261--2264. CMP 95:10
  • [Ol-2] W. Olszewski, Universal separable metrizable spaces for given cohomological dimension, preprint.
  • [Ru] L. R. Rubin, Characterizing cohomological dimension: The cohomological dimension of $A\cup B$, Topology and its Appl. 40 (1991), 233--263. MR 92g:55002
  • [R-S] L. R. Rubin and P. J. Schapiro, Cell-like maps onto non-compact spaces of finite cohomological dimension, Topology and its Appl. 27 (1987), 221--244. MR 89b:55002
  • [Sp] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35:1007
  • [Su] D. Sullivan, Geometric Topology, Part I: Localization, Periodicity, and Galois Symmetry, M.I.T. Press, 1970. MR 58:13006a
  • [Wa] J. J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Lecture Notes in Math. 870, 1981, pp. 105--118. MR 83a:57021
  • [Was] A. Wa\'{s}ko, Spaces universal under closed embeddings for finite dimensional complete metric spaces, Bull. London. Math. Soc. 18 (1986), 293--298. MR 87e:54084
  • [We] J.West, Open problems in infinite dimensional topology, in Open Problems in Topology, North-Holland, 1990. MR 90e:57070
  • [Wh] George W.Whitehead, Elements of homotopy theory, Springer-Verlag, 1978. MR 80b:55001

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Additional Information

Jerzy Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: dydak@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01536-X
Keywords: Dimension, cohomological dimension, Menger-Urysohn Theorem, absolute extensors, Eilenberg--Mac Lane spaces, universal spaces, compactifications
Received by editor(s): December 11, 1992
Received by editor(s) in revised form: May 3, 1995
Additional Notes: Supported in part by a grant from the National Science Foundation
Article copyright: © Copyright 1996 American Mathematical Society

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