Simplexwise linear near-embeddings of a $2$-disk into $\textbf {R}^ 2$

Abstract:

Let $K \subset {{\mathbf {R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf {R}}^2}$ is called simplexwise linear $(SL)$ if $f|\sigma$ is affine linear for each (closed) simplex $\sigma$ of $K$. Interest in ${\text {SL}}$ maps originated with work of S. S. Cairns and subsequent work of R. Thom and N. H. Kuiper. Let $E(K) = \{ {\text {orientation preserving SL embeddings}}\;K \to {{\mathbf {R}}^2}\}$, $L(K) = \{ {\text {SL homeomorphism}}\;K \to K\;{\text {fixing}}\;\partial K\;{\text {pointwise}}\}$, and $\overline {E(K)} ,\overline {L(K)}$ denote their respective closures in the space of all ${\text {SL}}$ maps $K \to {{\mathbf {R}}^2}$ and the space of all ${\text {SL}}$ maps $K \to K$ fixing $\partial K$. The main result of this paper is useful characterizations of maps in $\overline {L(K)}$ and some maps in $\overline {E(K)}$, including the relation of such maps to ${\text {SL}}$ embeddings into the nonstandard plane.