Celebratio Mathematica — Bing

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R. H. Bing and J. M. Kister: “Taming complexes in hyperplanes,” Duke Math. J. 31 : 3 (1964), pp. 491–511. MR 0164329 Zbl 0124.16701 article

In this paper we investigate conditions that suffice for an imbedding of a complex in a Euclidean space to be tame. We also consider the more general question of when two imbeddings are equivalent. We show, for example, with a dimension restriction that if the image under an imbedding of a \( k \)-complex is contained in a hyperplane of codimension \( k \), then the imbedding is tame. More precisely we prove:

Let \( K \) be a finite \( k \)-dimensional complex and \( h \) an imbedding of \( K \) into an \( n \)-plane \( E^n \) in \( E^{n+k} \), where \( k+2\leq n \). Let \( \varepsilon \) be any positive number. Then there exists an isotopy \( G_t \) (\( t\in I \)) of \( E^{n+k} \) onto itself such that

\( g_0 \) is the identity,
\( g_1 h \) is piecewise linear,
\( g_t \) is the identity outside an \( \varepsilon \)-neighborhood of \( h(K) \) for each \( t \) in \( I \),
each point of \( E^{n+k} \) moves along a polygonal path under \( g_t \) (\( t\in I \)) having length less than \( \varepsilon \).