| Division and k-th root theorems for Q-manifolds |
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| 引用本文： | Taras BANAKH,Duan REPOV. Division and k-th root theorems for Q-manifolds[J]. 中国科学A辑(英文版), 2007, 50(3) |
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| 作者姓名： | Taras BANAKH Duan REPOV |
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| 作者单位： | Department of Mathematics Lviv National University，Ukraine and Instytut Matematyki，Akademia wi■tokrzyska，Kielce，Poland，Institute for Mathematics，Physics and Mechanics，and Faculty of Education，University of Ljubljana，P．O．B．2964，Ljubljana，Slovenia |
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| 基金项目： | This work was supported by the Slovenian-Ukrainian(Grant No．SLO-UKR 04-06／07) |
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| 摘 要： | We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.
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Division and k-th root theorems for Q-manifolds |
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| Taras BANAKH,Duan REPOV. Division and k-th root theorems for Q-manifolds[J]. Science in China(Mathematics), 2007, 50(3) |
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| Authors: | Taras BANAKH Duan REPOV |
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| Affiliation: | 1. Department of Mathematics, Lviv National University, Ukraine and Instytut Matematyki, Akademia (S)wietokrzyska, Kielce, Poland
2. Institute for Mathematics, Physics and Mechanics, and Faculty of Education, University of Ljubljana, P.O.B.2964, Ljubljana, Slovenia |
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| Abstract: | We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countabledimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold.If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X × [0, 1] are Q-manifolds as well. We construct a countable family x of spaces with DDP and cd-AP such that no space X ∈ xis homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ x is homeomorphic to Q. We also show that no uncountable family x with such properties exists. |
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| Keywords: | Hilbert cube Cantor cube Tychonov cube ANR infinite-dimensional manifold Disjoint Disk Property cell-like map |
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