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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I
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by Chung Wu Ho PDF
Trans. Amer. Math. Soc. 181 (1973), 213-233 Request permission

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 $\text {Bd}(S)$. 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
  • 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 10271, DOI 10.2307/1969263
  • S. S. Cairns, Deformations of plane rectilinear complexes, Amer. Math. Monthly 51 (1944), 247–252. MR 10273, DOI 10.2307/2304300
  • C.-W. Ho, On a space of piecewise linear homeomorphisms of a $2$-simplex, Ph.D. Dissertation, M. I. T., Cambridge, Mass., 1970. —, On the existence of certain linear homeomorphisms of a convex polyhedral disk, Notices Amer. Math. Soc. 19 (1972), A-406. Abstract #693-Gl. —, On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. II, Trans. Amer. Math. Soc. 181 (1973), 235-243.
  • 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
  • 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
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 181 (1973), 213-233
  • MSC: Primary 57E05; Secondary 57C05
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0322891-3
  • MathSciNet review: 0322891