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Transactions of the American Mathematical Society

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



Linear isotopies in $ E\sp{2}$

Authors: R. H. Bing and Michael Starbird
Journal: Trans. Amer. Math. Soc. 237 (1978), 205-222
MSC: Primary 57A05
MathSciNet review: 0461510
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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\vert{\text{Bd}}\;P = g\vert{\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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Keywords: Linear isotopy, push, linear embedding
Article copyright: © Copyright 1978 American Mathematical Society

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