## 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, - 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**

*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.

## 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