Original scientific paper

https://doi.org/10.3336/gm.48.2.15

Simultaneous Z/p-acyclic resolutions of expanding sequences

Leonard Rubin orcid.org/0000-0002-1108-0267 ; Department of Mathematics, University of Oklahoma, 601 Elm Ave, room 423, Norman, Oklahoma 73019, USA
Vera Tonić ; 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.

Keywords

Cell-like map; cohomological dimension; CW-complex; dimension; Edwards-Walsh resolution; Eilenberg-MacLane complex; G-acyclic map; inverse sequence; simplicial complex; UVk-map

Hrčak ID:

112219

URI

https://hrcak.srce.hr/112219

Publication date:

16.12.2013.

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