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Transactions of the American Mathematical Society

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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
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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  • [B] E. D. Bloch, Strictly convex simplexwise linear embeddings of a $ 2$-disk, Trans. Amer. Math. Soc. Soc. 288 (1985), 723-727. MR 776400 (86h:57010b)
  • [BS1] R. H. Bing and M. Starbird, Linear isotopies in $ {E^2}$, Trans. Amer. Math. Soc. 237 (1978), 205-222. MR 0461510 (57:1495)
  • [BS2] -, Super triangulations, Pacific J. Math. 74 (1978), 307-325. MR 0478170 (57:17659)
  • [BCH] E. D. Bloch, R. Connelly and D. W. Henderson, The space of simplexwise linear homeomorphisms of a convex $ 2$-disk, Topology (to appear). MR 744848 (85m:57005)
  • [C] S. S. Cairns, Isotopic deformations of geodesic complexes on the $ 2$-sphere and plane, Ann. of Math. (2) 45 (1944), 207-217. MR 0010271 (5:273d)
  • [CHHS] R. Connelly, D. W. Henderson, C.-W. Ho and M. Starbird, On the problems related to linear homeomorphism, embeddings and isotopies, Topology Symposium 1980, Univ. of Texas Press, Austin, 1983. MR 711994 (84i:57014)
  • [D] M. Davis, Applied nonstandard analysis, Wiley, New York, 1977. MR 0505473 (58:21590)
  • [H] D. W. Henderson, The space of simplexwise-geodesic homeomorphisms of the $ 2$-sphere (to appear).
  • [Ho1] C.-W. Ho, On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I, Trans. Amer. Math. Soc. 181 (1973), 213-233. MR 0322891 (48:1252)
  • [Ho2] -, On the space of the linear homeomorphisms of polyhedral $ n$-cell with two interior vertices, Math. Ann. 243 (1979), 227-236. MR 548803 (80k:57004)
  • [Ho3] -, On the extendability of a linear embedding of the boundary of a triangulated $ N$-cell to an embedding of the $ n$-cell, Amer. J. Math. 103 (1981),
  • [K] N. H. Kuiper, On the smoothings of triangulated and combinatorial manifolds, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N. J., 1965, pp. 3-22. MR 0196755 (33:4941)
  • [T] R. Thom, Des variétés triangulées aux variétés différentiables, Proc. Internat. Congr. Math., Edinburgh, 1958, pp. 248-255. MR 0121806 (22:12536)

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Keywords: Simplexwise linear, spaces of embeddings, nonstandard plane
Article copyright: © Copyright 1985 American Mathematical Society

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