# Near-Homeomorphisms of Nobeling Manifolds

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## Abstract

NEAR-HOMEOMORPHISMS OF NO¨BELING MANIFOLDS A. CHIGOGIDZE AND A. NAGO´RKO Abstract. We characterize maps between n-dimensional No¨beling manifolds that can be approximated by homeomorphisms. 1. Introduction A long standing problem (see, for example, [8, TC 10], [3, Conjecture 5.0.5]) of characterizing topologically universal n-dimensional No¨beling space, as well as manifolds modeled on it, was solved recently by M. Levin [5] and A. Nago´rko [7]. Theory of No¨beling manifolds, developed in [5], [6], [7] based on completely differ- ent approaches, among other things contains various versions of Z-set unknotting theorem, open embedding theorem, n-homotopy classification theorem, etc. In this note we complete the picture by proving that for n-dimensional No¨beling manifolds classes of near-homeomorphisms, approximately n-soft maps, fine n-homotopy equivalences and UVn−1-mappings coincide. Recall that an n- dimensional No¨beling manifold is a Polish space locally homeomorphic to νn, the subset of R2n+1 consisting of all points with at most n rational coordinates. Definition 1.1. For each map f from a space X into a space Y , for each open cover U of Y and for each integer n, we define the following conditions. (NHU ) There exists a homeomorphism of X and Y that is U-close to f . (AnSU ) For each at most n-dimensional metric space B, its closed subset A and maps ϕ and ψ such that the following diagram commutes X f // Y A � i / ϕ OO B ψ OO k ` there exists a map k : B → X such that k|A = ϕ and f ◦ k is U-close to ψ. (FnHEU ) There exists a map g from Y to X such that f ◦ g is U-n-homotopic1 to the identity on Y and g ◦ f is f−1(U)-n-homotopic to the identity on X (with f−1(U) denoting {f−1(U)}U∈U). (UVn−1U ) The star of the image of f in U is equal to Y and there exists an open cover W of Y such that for each W in W there exists U in U such that the inclusion f−1(W ) ⊂ f−1(U) induces trivial (zero) homomorphisms on 1991 Mathematics Subject Classi

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