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Wild surfaces have some nice properties. (English) Zbl 0609.57005
If T is a triangulation of \({\mathbb{R}}^ 3\), \(S\subset {\mathbb{R}}^ 3\) is a 2-sphere (possibly wild) and \(\epsilon >0\) then there exists an \(\epsilon\)-isotopy \(H_ t\) of \({\mathbb{R}}^ 3\) fixed outside the \(\epsilon\)-neighbourhood of \(S\cap T^ 2\) such that: \(H_ 1(S)\) misses each vertex of T; for each 1-simplex \(\Delta^ 1\) of T, \(\Delta^ 1\cap H_ 1(S)\) is a finite set at each point of which \(\Delta^ 1\) pierces \(H_ 1(S)\); and for each 2-simplex \(\Delta^ 2\) of T each nondegenerate component of \(\Delta^ 2\cap H_ 1(S)\) is a polygonal arc, any arc in \(\Delta^ 2\) piercing \(\Delta^ 2\cap H_ 1(S)\) in \(\Delta^ 2\) also pierces \(H_ 1(S)\), for each \(\delta >0\) at most finitely many components of \(\Delta^ 2\cap H_ 1(S)\) have diameters greater than \(\delta\) and each degenerate component of \(\Delta^ 2\cap H_ 1(S)\) lies in int \(\Delta\) \({}^ 2\) and is a limit point of the union of the nondegenerate components. Proofs of this and related results are given ”so that the results would be believed even by mathematical agnostics”.
Reviewer: D.B.Gauld
57M30 Wild embeddings
57N37 Isotopy and pseudo-isotopy
57N45 Flatness and tameness of topological manifolds
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