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

SIMULTANEOUS Z/P-ACYCLIC RESOLUTIONS OF EXPANDING SEQUENCES

Leonard Rubin and Vera Tonić

Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel
e-mail: vera.tonic@gmail.com

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, dim_Z/p X_k≤ l_k. 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 Z_k≤ l_k,
(b) π(Z_k)=X_k, and
(c) π|_Zk:Z_k→ X_k is a Z/p-acyclic map.
Moreover, there is a sequence A₁⊂ A₂⊂⋅⋅⋅ of closed subspaces of Z such that for each k, dim A_k≤ l_k, π|_Ak:A_k → X is surjective, and for kN, Z_k⊂ A_k and π|_Ak:A_k→ X is a UV^l_k-1-map.

It is not required that X=∪^∞_k=1 X_k or that Z=∪^∞_k=1 Z_k. 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, UV^k-map.

DOI: 10.3336/gm.48.2.15

References:

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.
   MathSciNet    

3. A. N. Dranishnikov, Cohomological dimension theory of compact metric spaces, Topology Atlas Invited Contributions, http://at.yorku.ca/t/a/i/c/43.pdf.

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.
   MathSciNet    

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

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.
    MathSciNet    

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

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