Resolutions of generalized 3-manifolds whose singular sets have general position dimension one.
A Euclidean neighborhood retract is a generalized 3-manifold if for each point \(p \in X\) \(H_{k}(X, X \setminus \{p\})\) is isomorphic to \(H_{k}({\mathbb R}^{3}, {\mathbb R}^{3} \setminus \{(0,0,0)\})\) for all \(k \in {\mathbb N}\). The manifold set of \(X\) is the set of points that have neighborhoods homeomorphic to \({\mathbb R}^{3}\) and the singular set is the complement of the manifold set of \(X\). The singular subset is of general position dimension one if any map \(f: B^{2} \rightarrow X\) of a 2-ball into \(X\) can be approximated by one that meets the singular set in a \(0\)-dimensional set. In an earlier paper [Proc. Lond.Math. Soc., III. Ser. 55, 378--432 (1987; Zbl 0629.57011)] the author showed that modulo the classical Poincaré conjecture a generalized 3-manifold with singular set of general position dimension zero is resolvable; meaning that it is the cell like image of a closed 3-manifold. In this paper the result is improved to cover the case where the singular set is of general position dimension one. The paper begins with some errata and clarification of the original paper. This is followed by a reworking of the original material to cover the extension. The author, wisely, advises that the paper should be read with the previous paper at hand as there are numerous references to specific pages in the original.
scientific article; zbMATH DE number 2056819
scientific article