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, University of Oklahoma, 601 Elm Ave, room 423, Norman, Oklahoma 73019, USA
e-mail: lrubin@ou.edu

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, 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


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