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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Linear isotopies in $E^{2}$
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by R. H. Bing and Michael Starbird PDF
Trans. Amer. Math. Soc. 237 (1978), 205-222 Request permission

Abstract:

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 Bd P and is linear with respect to T, then there is a continuous family of homeomorphisms ${h_t}:P \to P(t \in [0,1])$ such that ${h_0} = {\text {id}},{h_1} = f$, 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 $f|{\text {Bd}}\;P = g|{\text {Bd}}\;P$, 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.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 205-222
  • MSC: Primary 57A05
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0461510-7
  • MathSciNet review: 0461510