Resolving wild embeddings of codimension-one manifolds in manifolds of dimensions greater than 3
This paper systematically surveys the topic suggested by the title and simultaneously generalizes to dimension 4 certain results previously known for embeddings in manifolds of dimension \(n\geq 5\). The central item is the codimension one tame approximation theorem: every embedding of an (n-1)-manifold in an n-manifold can be approximated by a locally flat embedding. \textit{R. H. Bing} [Ann. Math., II. Ser. 65, 484-500 (1957; Zbl 0079.388)] first proved this for \(n=3\); later the author and \textit{J. W. Cannon} [ibid. 109, 61-86 (1979; Zbl 0405.57007)] did the same for \(n\geq 5\). Here the author derives the theorem when \(n=4\), with an argument markedly different from that of the earlier cases modelled on one set forth by \textit{J. W. Cannon}, \textit{J. L. Bryant} and \textit{R. C. Lacher} [Geometric topology, Proc. Conf., Athens/Ga. 1977, 261-300 (1979; Zbl 0476.57006)]. At the heart of the discussion is the following tool, called the \(\delta\)-resolution theorem: given an embedding \(e: M\to N\) of an (n-1)- manifold M in an n-manifold N, \(n\geq 4\), and given a map \(\delta\) : \(N\to (0,\infty)\), there exist a locally flat embedding \(f: M\to N\) and a cell- like map \(G: N\to N\) such that G moves each \(x\in N\) less than \(\delta\) (x) and \(G\circ f=e\). The 4-dimensional version follows from recent results of \textit{F. Quinn} [J. Differ. Geom. 17, 503-521 (1982; Zbl 0533.57009)], while the (previously known) higher-dimensional ones depend on work of \textit{R. D. Edwards} [Proc. int. Congr. Math., Helsinki 1978, Vol. I, 111-127 (1980; Zbl 0428.57004)] and other work of \textit{F. Quinn} [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0394.57022)]. In the final two sections the author studies extensions of the results mentioned above in the situation where M is allowed to be a generalized (n-1)-manifold.
codimension one tame approximation theorem
embedding of an (n-1)- manifold in an n-manifold
scientific article; zbMATH DE number 3982080