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Preserving $$Z$$-sets by Dranishnikov’s resolution. (English) Zbl 1182.54040
A proper map $$h:Y\to Z$$ between separable complete $$\text{ANE}(n)$$ is a $$UV^{n-1}$$-divider of $$f:X\to Z$$ if there exists a topological embedding $$g:X\to Y$$ such that $$g(X)$$ is dense in $$Y$$, the inclusion of $$g(X)$$ in $$Y$$ is a $$UV^{n-1}$$-map and $$f=h\circ g$$. Let $$Q$$ be the Hilbert cube, $$\mu^k$$ the $$k$$-dimensional Menger compactum and $$\nu^k$$ the $$k$$-dimensional Nöbeling space. The authors prove that there exists a Dranishnikov resolution $$d_k:\mu^k\to Q$$ and a Chigogidze resolution $$c_k:\nu^k\to Q$$ such that $$d_k$$ is a $$UV^{k-1}$$-divider of $$c_k$$. This implies that $$d_k^{-1}(Z)$$ is a Z-set in $$\mu^k$$ for every Z-set $$Z$$ in $$Q$$.

##### MSC:
 54F65 Topological characterizations of particular spaces 58B05 Homotopy and topological questions for infinite-dimensional manifolds 54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) 57N20 Topology of infinite-dimensional manifolds
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##### References:
 [1] Ageev, S.M., Axiomatic method of partitions in the theory of Menger and Nöbeling spaces, (2000), preprint [2] Ageev, S.M., Axiomatic method of partitions in the theory of Nöbeling spaces. I. improvement of partition connectivity, Mat. sb., 198, 3, 3-50, (2007), (in Russian) · Zbl 1147.54019 [3] Ageev, S.M., Axiomatic method of partitions in the theory of Nöbeling spaces. II. unknotting theorem, Mat. sb., 198, 5, 3-32, (2007), (in Russian) · Zbl 1153.54018 [4] Ageev, S.M., Axiomatic method of partitions in the theory of Nöbeling spaces. III. consistency of the axiom system, Mat. sb., 198, 7, 3-30, (2007), (in Russian) · Zbl 1148.54018 [5] S.M. Ageev, Geometric Chigogidze’s resolution, Fund. Math., submitted for publication [6] Ageev, S.M.; Gruzdev, G.; Silaeva, Z., On characterization of 0-dimensional Chigogidze’s resolution, Vestnik BGU, 2, 45-48, (2006), (in Russian) [7] Ageev, S.M.; Repovš, D.; Shchepin, E.V., On the softness of the dranishnikov resolution, Proc. Steklov inst. math., 212, 1, 3-27, (1996) · Zbl 0877.54012 [8] Bestvina, M., Characterizing k-dimensional universal Menger compacta, Mem. amer. math. soc., 71, 380, (1988) · Zbl 0645.54029 [9] Bestvina, M.; Mogilski, J., Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan math. J., 33, 291-313, (1986) · Zbl 0629.54011 [10] Bowers, P.L., Dense embedding of sigma-compact, nowhere locally compact metric spaces, Proc. amer. math. soc., 95, 1, 123-130, (1985) · Zbl 0587.54021 [11] Chigogidze, A., Theory of n-shapes, Uspekhi mat. nauk, 44, 117-140, (1989), (in Russian) · Zbl 0696.54013 [12] Chigogidze, A.; Kawamura, K.; Tymchatyn, E.D., Nöbeling spaces and the pseudo-interiors of Menger compacta, Topology appl., 68, 33-65, (1996) · Zbl 0869.57024 [13] Chigogidze, A., $$\operatorname{UV}^n$$-equivalence and n-equivalence, Topology appl., 45, 283-291, (1992) · Zbl 0756.54009 [14] Chigogidze, A.; Zarichnyi, M., Universal Nöbeling spaces and pseudo-boundaries of Euclidean spaces, Mat. stud., 19, 2, 193-200, (2003) · Zbl 1023.54028 [15] Dobrowolski, T.; Mogilski, J., Problems on topological classification of incomplete metric spaces, (), 409-429 [16] Dranishnikov, A.N., Universal Menger compacta and universal mappings, Mat. sb., 129, 1, 121-139, (1986), (in Russian) · Zbl 0622.54026 [17] Fedorchuk, V.V.; Chigogidze, A.Ch., Absolute retracts and infinite-dimensional manifolds, (1992), Nauka Moscow, (in Russian) · Zbl 0762.54017 [18] Hu, S.T., Theory of retracts, (1965), Wayne State Univ. Press · Zbl 0137.01701 [19] Shchepin, E.V., On homotopically regular mappings of manifolds, (), 139-151 · Zbl 0644.57005 [20] Shchepin, E.V.; Brodskyi, N.B., Selections of filtered multivalued mapping, Proc. Steklov inst. math., 212, 1, 209-229, (1996) [21] Toruńczyk, H., Characterizing Hilbert space topology, Fund. math., 111, 247-262, (1981) · Zbl 0468.57015 [22] Zarichnyi, M.M., Absorbing sets for n-dimensional spaces in absolutely Borel and projective classes, Mat. sb., 188, 3, 435-447, (1997) · Zbl 0880.54019
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