Wild surfaces have some nice properties. (English) Zbl 0609.57005

Bing, R. H.

Mich. Math. J. 33, 403-415 (1986).

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

MSC:

57M30	Wild embeddings
57N37	Isotopy and pseudo-isotopy
57N45	Flatness and tameness of topological manifolds

Keywords:

wild 2-spheres in \({bbfR}^ 3\)