Linear isotopies in
Authors:
R. H. Bing and Michael Starbird
Journal:
Trans. Amer. Math. Soc. 237 (1978), 205222
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 . Suppose f and g are two linear embeddings of a finite complex P with triangulation T into a simply connected open subset U of so that there is an orientation preserving homeomorphism H of to itself with . It is shown that there is a continuous family of embeddings so that , and for each t, is linear with respect to T. It is also shown that if P is a PL starlike disk in 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 such that , and for each t, is linear with respect to T. An example shows the necessity of the ``starlike'' requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theoremnamely, if f and g are two PL embeddings of a disk P into so that , 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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 , Deforming P. L. homeomorphisms on a convex 2disk, Bull. Amer. Math. Soc. 81 (1975), 726728. MR 51 # 11529. MR 0375334 (51:11529)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197804615107
PII:
S 00029947(1978)04615107
Keywords:
Linear isotopy,
push,
linear embedding
Article copyright:
© Copyright 1978
American Mathematical Society
