Detecting codimension one manifold factors with 0-stitched disks
This paper introduces a new property that can be used to show that a space \(X\) is a codimension one manifold factor, meaning that \(X \times {\mathbb R}\) is a manifold. It is known that all finite dimensional resolvable generalized manifolds are codimension 2 manifold factors [\textit{R. J. Daverman}, Pac. J. Math. 93, 277--298 (1981; Zbl 0415.57007)]. Two maps \(f,g: D^{2} \rightarrow X\) are said to be \(0\)-stitched provided that there are 0-dimensional \(F_{\sigma}\) sets \(A\) and \(B\) contained in the interior of \(D^{2}\) such that \(f(D^{2} - A) \cap g(D^{2} - B) = \emptyset \). If, further, \(Y\) and \(Z\) are sets in \(D^{2}\) missing \(A\) and \(B\) respectively, then \(f\) and \(g\) are \(0\)-stitched away from \(Y\) and \(Z\). A space \(X\) has the \(0\)-stitched disks property if any two maps \(f,g: D^{2} \rightarrow X\) can be approximated by maps \(f',g'\) such that there are 0-dimensional \(F_{\sigma}\) sets \(A\) and \(B\) contained in the interior of \(D^{2}\) that are \(0\)-stitched away from infinite \(1\)-skeleta \((K_{j}^{\infty})^(1), j = 1,2\) of \(D^{2}\) and such that \(f'\mid (K_{1}^{\infty})^{(1)} \cup g'\mid (K_{2}^{\infty})^{(1)}\) is \((1-1)\). The main result of the paper is that a space \(X\) with the \(0\)-stitched disks property has the disjoint homotopy property. From this it follows that if \(X\) is also a resolvable generalized manifold of dimension \(\geq 4\) then it is a codimension \(1\) manifold factor. In a final section there is a brief discussion of cases in which the \(0\)-stitched disks property can be easily verified.
scientific article; zbMATH DE number 5159485