Linear isotopies in

Authors:
R. H. Bing and Michael Starbird

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

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Abstract | References | Similar Articles | Additional Information

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 star-like 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 ``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 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0461510-7

Keywords:
Linear isotopy,
push,
linear embedding

Article copyright:
© Copyright 1978
American Mathematical Society