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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
##### MSC:
 57M30 Wild embeddings 57N37 Isotopy and pseudo-isotopy 57N45 Flatness and tameness of topological manifolds
##### Keywords:
wild 2-spheres in $${bbfR}^ 3$$
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