Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The homotopy type of the space of diffeomorphisms. II


Authors: Dan Burghelea and Richard Lashof
Journal: Trans. Amer. Math. Soc. 196 (1974), 37-50
MSC: Primary 57E05; Secondary 58D05
DOI: https://doi.org/10.1090/S0002-9947-74-99951-6
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The result (proved in Part I) that $ {\operatorname{Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text{PL}_n}/{O_n})$ is used to compute some new homotopy of $ {\operatorname{Diff}}({D^n},\partial {D^n})$. The relation between smooth and PL pseudo-isotopy is explored. Known and new results on the homotopy of $ {\text{PL}_n}$ are summarized.


References [Enhancements On Off] (What's this?)

  • [26a] A. Haefliger and C. T. C. Wall, Piecewise linear bundles in the stable range, Topology 4 (1965), 209-214. MR 32 #1716. MR 0184243 (32:1716)
  • [27] G. Bredon, A $ {\pi _ \ast }$-module structure for $ {\theta _ \ast }$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434-448. MR 36 #4570. MR 0221518 (36:4570)
  • [28] J. G. Miller, Homotopy groups of diffeomorphism groups (preprint). MR 0362355 (50:14797)
  • [29] W. Browder, The Kervaire invariant of framed manifolds and its generalizations, Ann. of Math. (2) 90 (1969), 157-186. MR 40 #4963. MR 0251736 (40:4963)
  • [30] M. G. Barratt and M. Mahowald, The metastable homotopy of $ O(n)$, Bull. Amer. Math. Soc. 70 (1964), 758-760. MR 31 #6229. MR 0182004 (31:6229)
  • [31] I. James, On the iterated suspension, Quart. J. Math. Oxford Ser. (2) 5 (1954), 1-10. MR 15, 891. MR 0061836 (15:891g)
  • [32] Mark Mahowald, Some Whitehead products in $ {S^n}$, Topology 4 (1965), 17-26. MR 31 #2724. MR 0178467 (31:2724)
  • [33] G. F. Paechter, The groups $ {\pi _r}({V_{n,m}})$. I, II, III, IV, Quart. J. Math. Oxford Ser. (2) 7 (1956), 249-268; ibid. 9 (1958), 8-27; 10 (1959), 17-37; 241-260. MR 24 #A1725; 25 #4540; #4541; #4542.
  • [34] C. Rourke and B. Sanderson, Block bundles. I, Ann. of Math. (2) 87 (1968), 1-28. MR 37 #2234a. MR 0226645 (37:2234a)
  • [35] I. A. Volodin, Algebraic K-theory, Uspehi Mat. Nauk 27 (1972), no. 4 (166), 207-208. (Russian) MR 0393191 (52:14001)
  • 1. -, Generalized Whitehead groups and pseudo-isotopy, Uspehi Mat. Nauk (to appear).
  • [36] R. J. Milgram, On the Haefliger knot groups, Bull. Amer. Math. Soc. 78 (1972), 861-865. MR 0315728 (47:4277)
  • [37] A. Chenciner, Pseudo-isotopies différentiables et pseudo-isotopies linéaires par morceaux, C. R. Acad. Sci. Paris Sér. A--B 270 (1970), A1312-A1315. MR 42 #2489. MR 0267587 (42:2489)
  • [38] M. Hirsch, On tubular neighborhoods of piecewise linear and topological manifolds, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber and Schmidt, Boston, Mass., 1968, pp. 63-80. MR 38 #5226. MR 0236933 (38:5226)
  • [39] N. Kuiper and R. Lashof, Microbundles and bundles. I, II, Invent. Math. 1 (1966), 1-17, 243-259. MR 35 #7339; #7340. MR 0216507 (35:7340)
  • [40] G. Brumfiel, On the homotopy groups of BPL and PL$ /O$, Ann. of Math. (2) 88 (1968), 291-311. MR 38 #2775. MR 0234458 (38:2775)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57E05, 58D05

Retrieve articles in all journals with MSC: 57E05, 58D05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-74-99951-6
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society