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Simplexwise linear near-embeddings of a $ 2$-disk into $ {\bf R}\sp 2$


Author: Ethan D. Bloch
Journal: Trans. Amer. Math. Soc. 288 (1985), 701-722
MSC: Primary 57N05; Secondary 03H99, 57N35, 57Q99
DOI: https://doi.org/10.1090/S0002-9947-1985-0776399-X
MathSciNet review: 776399
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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\vert\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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776399-X
Keywords: Simplexwise linear, spaces of embeddings, nonstandard plane
Article copyright: © Copyright 1985 American Mathematical Society

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