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Detecting codimension one manifold factors with the disjoint homotopies property. (English) Zbl 0992.57024

A generalized \(n\)-manifold \(X\) is a finite-dimensional, locally contractible metric space such that \(H_n(X,X \smallsetminus \{x\};\mathbb{Z}) \cong \mathbb{Z}\) and otherwise \(H_i(X,X \smallsetminus\{x\}; \mathbb{Z})\cong 0\); such a space is said to be resolvable if there exist an \(n\)-manifold \(M\) and a cell-like mapping \(f\) of \(M\) onto \(X\) \((f\) is cell-like provided for each \(x\in X\) and neighborhood \(U\) of \(f^{-1}(x)\), the inclusion \(f^{-1}(x) \hookrightarrow U\) is null-homotopic). An approximately 40 year old problem traceable to R. H. Bing asks whether, for every resolvable generalized \(n\)-manifold \(X\), \(X\times \mathbb{R}\) is a manifold. If so, then \(X\) is said to be a codimension one manifold factor. It is known that in this setting \(X\times\mathbb{R}^2\) always is an \((n+2)\)-manifold. The reviewer proved [Pac. J. Math. 93, 277-298 (1981; Zbl 0415.57007)] that a generalized \(n\)-manifold \(X\) \((n>3)\) is a codimension one manifold factor if each pair of maps \(f:I\to X\), \(g:B^2\to X\) can be approximated by maps \(F:I\to X\), \(G:B^2\to X\) with disjoint images. In what may be the first result on the topic since then, here the author introduces a Disjoint Homotopies Property and shows that all generalized \(n\)-manifolds \(X\) \((n>3)\) having it are codimension one manifold factors; \(X\) has the Disjoint Homotopies Property if any two homotopies \(f_t,g_t: I\to X\) can be approximated by homotopies \(F_t,G_t:I\to X\) such that \(F_t(I)\cap G_t(I) =\emptyset\) for all \(t\).
The author introduces another property, the Plentiful 2-manifolds Property, which (for metric spaces \(X)\) is characterized by the fact that each path in \(X\) can be approximated by another path which lies in a 2-manifold \(N \subset X\). Then in another key result the author establishes that all resolvable generalized \(n\)-manifolds having the Plentiful 2-manifolds Property are codimension one manifold factors. She concludes by constructing a new class of codimension one manifold factors, namely, the \(k\)-ghastly examples \((2<k<n)\), which contain no embedded \(k\)-cells but do contain embedded \((k-1)\)-cells; these are constructed so as to have the Plentiful 2-manifolds Property.

MSC:

57P05 Local properties of generalized manifolds
54B10 Product spaces in general topology
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)

Citations:

Zbl 0415.57007
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References:

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