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


On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I

Author: Chung Wu Ho
Journal: Trans. Amer. Math. Soc. 181 (1973), 213-233
MSC: Primary 57E05; Secondary 57C05
Part II: Trans. Amer. Math. Soc. (1973), 235-243
MathSciNet review: 0322891
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Abstract: Let $ K$ be a proper rectilinear triangulation of a $ 2$-simplex $ S$ in the plane and $ L(K)$ be the space of all homeomorphisms of $ S$ which are linear on each simplex of $ K$ and are fixed on Bd$ (S)$. The author shows in this paper that $ L(K)$ with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space $ L(K)$ is pathwise connected. Both results will be used in Part II of this paper to show that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$ where $ {L_n}$ is a space of p.l. homeomorphisms of an $ n$-simplex, a space introduced by $ {\mathbf{R}}$. Thom in his study of the smoothings of combinatorial manifolds.

References [Enhancements On Off] (What's this?)

  • [1] Stewart S. Cairns, Isotopic deformations of geodesic complexes on the 2-sphere and on the plane, Ann. of Math. (2) 45 (1944), 207–217. MR 0010271 (5,273d)
  • [2] S. S. Cairns, Deformations of plane rectilinear complexes, Amer. Math. Monthly 51 (1944), 247–252. MR 0010273 (5,273f)
  • [3] C.-W. Ho, On a space of piecewise linear homeomorphisms of a $ 2$-simplex, Ph.D. Dissertation, M. I. T., Cambridge, Mass., 1970.
  • [4] -, On the existence of certain linear homeomorphisms of a convex polyhedral disk, Notices Amer. Math. Soc. 19 (1972), A-406. Abstract #693-Gl.
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  • [6] Nicolaas H. Kuiper, On the smoothings of trangulated and combinatorial manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 3–22. MR 0196755 (33 #4941)
  • [7] R. Thom, Des variétés triangulées aux variétés différentiables, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 248–255 (French). MR 0121806 (22 #12536)

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Additional Information

PII: S 0002-9947(1973)0322891-3
Keywords: Simplicial subdivision, star, link, the join of two complexes, decomposable subdivision, pivot triangle, binding pair, the accessibility of a vertex, incidence number, Seifert and Van Kampen theorem
Article copyright: © Copyright 1973 American Mathematical Society