Glasnik Matematicki, Vol. 48, No. 2 (2013), 443-466.


Leonard Rubin and Vera Tonić

Department of Mathematics, University of Oklahoma, 601 Elm Ave, room 423, Norman, Oklahoma 73019, USA

Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel

Abstract.   We prove the following theorem.
Theorem. Let X be a nonempty compact metrizable space, let l1≤ l2≤ ⋅⋅⋅ be a sequence in N, and let X1 ⊂ X2⊂ ⋅⋅⋅ be a sequence of nonempty closed subspaces of X such that for each kN, dimZ/p Xk≤ lk. Then there exists a compact metrizable space Z, having closed subspaces Z1⊂ Z2⊂ ⋅⋅⋅, and a (surjective) cell-like map π:Z → X, such that for each kN,
(a) dim Zk≤ lk,
(b) π(Zk)=Xk, and
(c) π|Zk:Zk→ Xk is a Z/p-acyclic map.
Moreover, there is a sequence A1⊂ A2⊂⋅⋅⋅ of closed subspaces of Z such that for each k, dim Ak≤ lk, π|Ak:Ak → X is surjective, and for kN, Zk⊂ Ak and π|Ak:Ak→ X is a UVlk-1-map.

It is not required that X=∪k=1 Xk or that Z=∪k=1 Zk. This result generalizes the Z/p-resolution theorem of A. Dranishnikov and runs parallel to a similar theorem of S. Ageev, R. Jiménez, and the first author, who studied the situation where the group was Z.

2010 Mathematics Subject Classification.   55M10, 54F45, 55P20.

Key words and phrases.   Cell-like map, cohomological dimension, CW-complex, dimension, Edwards-Walsh resolution, Eilenberg-MacLane complex, G-acyclic map, inverse sequence, simplicial complex, UVk-map.

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DOI: 10.3336/gm.48.2.15


  1. S. Ageev, R. Jiménez and L. Rubin, Cell-like resolutions in the strongly countable Z-dimensional case, Topology Appl. 140 (2004), 5-14.
    MathSciNet     CrossRef

  2. R. Daverman, Decompositions of Manifolds, Academic Press, Orlando, 1986.

  3. A. N. Dranishnikov, Cohomological dimension theory of compact metric spaces, Topology Atlas Invited Contributions,

  4. J. Dydak and J. Walsh, Complexes that arise in cohomological dimension theory: a unified approach, J. London Math. Soc. (2) (48) (1993), 329-347.
    MathSciNet     CrossRef

  5. R. D. Edwards, A theorem and a question related to cohomological dimension and cell-like maps, Notices Amer. Math. Soc. 25 (1978), A58-A59.

  6. L. C. Glaser, Geometrical combinatorial topology, Vol. I, Van Nostrand Math. Studies, 1970.

  7. W. Hurewicz and H. Wallman, Dimension theory, Princeton University Press, Princeton, 1941.

  8. V. I. Kuz'minov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1-45.

  9. A. Koyama and K. Yokoi, Cohomological dimension and acyclic resolutions, Topology Appl. 120 (2002), 175-204.
    MathSciNet     CrossRef

  10. M. Levin, Acyclic resolutions for arbitrary groups, Isr. J. Math. 135 (2003), 193-203.
    MathSciNet     CrossRef

  11. S. Mardešić and J. Segal, Shape theory, North-Holland, Amsterdam, 1982.

  12. E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966.

  13. J. Walsh, Dimension, cohomological dimension, and cell-like mappings, in Shape theory and geometric topology, Springer, Berlin, 1981, 105-118.

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