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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.
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0322891-3
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