Linear isotopies in $E^{2}$

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.