Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
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. II


Author: Chung Wu Ho
Journal: Trans. Amer. Math. Soc. 181 (1973), 235-243
MSC: Primary 57E05; Secondary 57C05
Part I: Trans. Amer. Math. Soc. (1973), 213-233
MathSciNet review: 0322891
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In his study of the smoothings of p. l. manifolds, R. Thom considered the homotopy groups of a certain space $ {L_n}$ of p.l. homeomorphisms on an $ n$-simplex. N. H. Kuiper showed in 1965 that the higher homotopy groups of $ {L_n}$ were in general nontrivial. The main result in this paper is that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$.

The proof of this result is based on a theorem of S. S. Cairns in 1944 on the deformation of rectilinear complexes in $ {R^2}$ and a theorem established in Part I of this paper.


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] V. K. A. M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres. I, II, Proc. London Math. Soc. (3) 3 (1953), 29–53, 129–152. MR 0058204 (15,336d)
  • [4] P. J. Hilton and S. Wylie, Homology theory, 2nd ed., Cambridge Univ. Press, Cambridge, 1962.
  • [5] Chung Wu Ho, On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I, II, Trans. Amer. Math. Soc. 181 (1973), 213–233; ibid. 181\ (1973), 235–243. MR 0322891 (48 #1252)
  • [6] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844 (40 #2094)
  • [7] 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)
  • [8] James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR 0198479 (33 #6637)
  • [9] 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)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57E05, 57C05

Retrieve articles in all journals with MSC: 57E05, 57C05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-99929-3
PII: S 0002-9947(1973)99929-3
Keywords: Simplicial subdivision, proper subdivision, rectilinear cell complex, inductive limit, the space of linear isomorphisms on a complex
Article copyright: © Copyright 1973 American Mathematical Society