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

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

Part II:
Trans. Amer. Math. Soc. (1973), 235-243

MathSciNet review:
0322891

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Abstract: Let be a proper rectilinear triangulation of a -simplex in the plane and be the space of all homeomorphisms of which are linear on each simplex of and are fixed on Bd. The author shows in this paper that with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space is pathwise connected. Both results will be used in Part II of this paper to show that where is a space of p.l. homeomorphisms of an -simplex, a space introduced by . Thom in his study of the smoothings of combinatorial manifolds.

**[1]**S. S. Cairns,*Isotopic deformations of geodesic complexes on the -sphere and plane*, Ann. of Math. (2)**45**(1944), 207-217. MR**5**, 273. MR**0010271 (5:273d)****[2]**-,*Deformations of plane rectilinear complexes*, Amer. Math. Monthly**51**(1944), 247-252. MR**5**, 273. MR**0010273 (5:273f)****[3]**C.-W. Ho,*On a space of piecewise linear homeomorphisms of a -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.**[5]**-,*On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms*. II, Trans. Amer. Math. Soc.**181**(1973), 235-243.**[6]**N. H. Kuiper,*On the smoothings of triangulated and combinatorial manifolds*, Differential and Combinatorial Topology (A Sympos. in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 3-22. MR**33**#4941. MR**0196755 (33:4941)****[7]**R. Thom,*Des variétés triangulées aux variétés différentiables*, Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, New York, 1960, pp. 248-255. MR**22**#12536. MR**0121806 (22:12536)**

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Additional Information

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

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