Celebratio Mathematica

R. H. Bing and M. Starbird: “Super triangulations,” Pacific J. Math. 74 : 2 (1978), pp. 307–325. MR 0478170 Zbl 0384.57005 article

R. H. Bing and M. Starbird: “Linear isotopies in \( E^2 \),” Trans. Am. Math. Soc. 237 (March 1978), pp. 205–222. MR 0461510 Zbl 0397.57018 article

This paper deals with continuous families of linear embeddings (called linear isotopies) of finite complexes in the Euclidean plane \( E^2 \). Suppose \( f \) and \( g \) are two linear embeddings of a finite complex \( P \) with triangulation \( T \) into a simply connected open subset \( U \) of \( E^2 \) so that there is an orientation preserving homeomorphism \( H \) of \( E^2 \) to itself with \( H\circ f = g \). It is shown that there is a continuous family of embeddings \( h_t:P\to U \) (\( t\in [0,1] \)) so that \( h_0=f \), \( h_1=g \), and for each \( t \), \( h_t \) is linear with respect to \( T \).

It is also shown that if \( P \) is a PL star-like disk in \( E^2 \) with a triangulation \( T \) which has no spanning edges and \( f \) is a homeomorphism of \( P \) which is the identity on \( \operatorname{Bd} P \) and is linear with respect to \( T \), then there is an continuous family of homeomorphisms \( h_t:P\to P \) (\( t\in [0,1] \)) such that \( h_0=\mathrm{id} \), \( h_1=f \), and for each \( t \), \( h_t \) is linear with respect to \( T \). An example shows the necessity of the “star-like” requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theorem — namely, if \( f \) and \( g \) are two PL embeddings of a disk \( P \) into \( E^2 \) so that \( f|_{\operatorname{Bd} P} = g|_{\operatorname{Bd} P} \), then there is a linear isotopy with respect to some triangulation of \( P \) which starts at \( f \), ends at \( g \), and leaves the boundary fixed throughout.

R. H. Bing and M. Starbird: “A decomposition of \( S^3 \) with a null sequence of cellular arcs,” pp. 3–21 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537722 Zbl 0479.57004 incollection

Michael Peter Starbird	Related
James Cecil Cantrell	Related

Continua, decompositions, manifolds (University of Texas, Austin, TX, summer 1980). Edited by R. H. Bing, W. T. Eaton, and M. P. Starbird. University of Texas Press (Austin, TX), 1983. MR 711973 book

Michael Peter Starbird	Related
William T. Eaton	Related

M. Starbird: “R. H. Bing’s human and mathematical vitality,” pp. 453–466 in Handbook of the history of general topology (San Antonio, TX, 1993), vol. 2. Edited by C. E. Aull and R. Lowen. History of Topology 2. Kluwer Academic (Dordrecht), 1998. MR 1795160 Zbl 0932.01042 incollection

Michael Peter Starbird	Related
Robert Lowen	Related
Charles E. Aull	Related

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