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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Strictly convex simplexwise linear embeddings of a $ 2$-disk

Author: Ethan D. Bloch
Journal: Trans. Amer. Math. Soc. 288 (1985), 723-737
MSC: Primary 57N05; Secondary 03H99, 57N35, 57Q99
MathSciNet review: 776400
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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) $ 2$-simplex $ \sigma $ of $ K$. Let $ E(K) = \{ {\text{orientation preserving SL embeddings}}\;K \to {{\mathbf{R}}^2}\} $, $ {E_{{\text{SC}}}}(K) = \{ f \in E(K)\vert f(K)\;{\text{is strictly convex}}\} $, and let $ \overline {E(K)} $ and $ \overline {{E_{{\text{SC}}}}(K)} $ denote their closures in the space of all $ {\text{SL}}$ maps $ K \to {{\mathbf{R}}^2}$. A characterization of certain elements of $ \overline {E(K)} $ is used to prove that $ {E_{{\text{SC}}}}(K) $ has the homotopy type of $ {S^1}$ and to characterize those elements of $ \overline {E(K)} $ which are in $ \overline {{E_{{\text{SC}}}}(K)} $, as well as to relate such maps to $ {\text{SL}}$ embeddings into the nonstandard plane.

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Additional Information

PII: S 0002-9947(1985)0776400-3
Keywords: Simplexwise linear, spaces of embeddings, nonstandard plane
Article copyright: © Copyright 1985 American Mathematical Society

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