Classification theorem for Menger manifolds

Abstract

Abstract.We introduce the notion of the n-homotopy kernel of a Menger manifold and prove the following theorem: Menger manifolds are w-homotopy equivalent if and only if the «-homotopy kernels are homeomorphic This paper is devoted to some problems induced by the internal development of Menger manifold theory. It is closely related to previous papers [1,8,[3][4][5] and can be considered as their natural continuation. We gave a detailed discussion of certain phenomena in [5] that are inherent in this theory (in comparison with Hubert cube manifold theory) and that are clearly manifested even in the formulations of such important components of the theory as stability and triangulation theorems [8,5] as well as the open embedding theorem [5]. Let us consider the last proposition more formally (see §1 for definitions). It states that each Menger (ai + l)-manifold N contains a Z-set Z satisfying the following conditions: (1) Z is homeomorphic to N; (2) the complement N-Z admits an open embedding into Mn+l (throughout the paper Mk denotes the Ac-dimensional universal Menger compactum). It should be observed (it was remarked in [5], too) that there is a certain arbitrariness in the above formulation connected with the choice of homeomorphism from condition (1). It will be shown that under minimal restrictions the mentioned arbitrariness can be eliminated. At the same time this specifies the second condition, because in this way the complement N-Z will acquire completely correct sense and will depend only on the Menger (ai + l)-manifold N. Moreover, this complement (which will be called an Ai-homotopy kernel of N and will be denoted by Ker(/V)) becomes a very important object associated with N. Indeed, if we bear in mind the strong parallel between Hubert cube manifold theory and Menger (ai + 1)-manifold theory then we shall be able to conclude confidently that Ker(/V) plays the role of the product N x [0, 1 ) (compare our modified open embedding theorem below and Chapman's open embedding theorem for Hubert cube manifolds [2]). Moreover, the consideration of Ai-homotopy kernels gives us the possibility to solve the problem of classification of Menger (ai + l)-manifolds by «-homotopy type. Our main result states in this connection that Menger (ai + l)-manifolds are Ai-homotopy equivalent if and only if their Ai-homotopy kernels are homeomorphic (compare with Chapman's theorem that states that

Menger (Nöbeling) Manifolds versus Hilbert Cube (Space) Manifolds - A Categorical Comparison

“…In particular, we may assume [7,Proposition 4.1.6] that f ′ and f are n-homotopic (in N). Considerations similar to the argument used in the proof of [8,Theorem2.2], [7,Theorem 4.4.7] guarantee that the n-homotopy class of the restriction…”

“…we use concept of the n-homotopy kernel Ker n (M) of an (n + 1)-dimensional Menger manifold M (see, [8], [7,Section 4.4.1]). It is important to note that Ker n (M) plays the role of the product M × [0, 1) in the category of (n + 1)dimensional Manger manifolds.…”