#### 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 *l*_{1}≤ l_{2}≤ ⋅⋅⋅ be a sequence in **N**, and let *X*_{1} ⊂
X_{2}⊂ ⋅⋅⋅ be a sequence of nonempty closed subspaces of *X* such
that for each *k***N**, dim_{Z/p} X_{k}≤ l_{k}. Then there
exists a compact metrizable space *Z*, having closed subspaces
*Z*_{1}⊂ Z_{2}⊂ ⋅⋅⋅, and a (surjective) cell-like map
*π:***Z** → X, such that for each *k***N**,

(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 *A*_{1}⊂ A_{2}⊂⋅⋅⋅ of
closed subspaces of **Z** such that for each *k*, dim *A*_{k}≤ l_{k},
*π|*_{Ak}:A_{k} → X is surjective, and for *k***N**,
*Z*_{k}⊂ A_{k} and *π|*_{Ak}:A_{k}→ X is a *UV*^{lk-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.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.48.2.15

**References:**

- 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

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

MathSciNet

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

- 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

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

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

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

MathSciNet

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

MathSciNet

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

MathSciNet
CrossRef

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

MathSciNet
CrossRef

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

MathSciNet

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

MathSciNet

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

MathSciNet

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