Simplexwise linear nearembeddings of a disk into
Author:
Ethan D. Bloch
Journal:
Trans. Amer. Math. Soc. 288 (1985), 701722
MSC:
Primary 57N05; Secondary 03H99, 57N35, 57Q99
MathSciNet review:
776399
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Abstract 
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Additional Information
Abstract: Let be a finitely triangulated disk; a map is called simplexwise linear if is affine linear for each (closed) simplex of . Interest in maps originated with work of S. S. Cairns and subsequent work of R. Thom and N. H. Kuiper. Let , , and denote their respective closures in the space of all maps and the space of all maps fixing . The main result of this paper is useful characterizations of maps in and some maps in , including the relation of such maps to embeddings into the nonstandard plane.
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Ethan
D. Bloch, Robert
Connelly, and David
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 R. H. Bing and M. Starbird, Linear isotopies in , Trans. Amer. Math. Soc. 237 (1978), 205222. MR 0461510 (57:1495)
 [BS2]
 , Super triangulations, Pacific J. Math. 74 (1978), 307325. MR 0478170 (57:17659)
 [BCH]
 E. D. Bloch, R. Connelly and D. W. Henderson, The space of simplexwise linear homeomorphisms of a convex disk, Topology (to appear). MR 744848 (85m:57005)
 [C]
 S. S. Cairns, Isotopic deformations of geodesic complexes on the sphere and plane, Ann. of Math. (2) 45 (1944), 207217. 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)
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 M. Davis, Applied nonstandard analysis, Wiley, New York, 1977. MR 0505473 (58:21590)
 [H]
 D. W. Henderson, The space of simplexwisegeodesic homeomorphisms of the sphere (to appear).
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 C.W. Ho, On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I, Trans. Amer. Math. Soc. 181 (1973), 213233. MR 0322891 (48:1252)
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 , On the space of the linear homeomorphisms of polyhedral cell with two interior vertices, Math. Ann. 243 (1979), 227236. MR 548803 (80k:57004)
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 , On the extendability of a linear embedding of the boundary of a triangulated cell to an embedding of the cell, Amer. J. Math. 103 (1981),
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 [T]
 R. Thom, Des variétés triangulées aux variétés différentiables, Proc. Internat. Congr. Math., Edinburgh, 1958, pp. 248255. MR 0121806 (22:12536)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719850776399X
PII:
S 00029947(1985)0776399X
Keywords:
Simplexwise linear,
spaces of embeddings,
nonstandard plane
Article copyright:
© Copyright 1985
American Mathematical Society
