Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Fibered stable compacta have finite homotopy type


Author: Ross Geoghegan
Journal: Proc. Amer. Math. Soc. 71 (1978), 123-129
MSC: Primary 55D15; Secondary 57A65
Erratum: Proc. Amer. Math. Soc. 74 (1979), 391.
MathSciNet review: 0515418
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that a fibered compact metric space having the shape of a CW complex has the homotopy type of that complex, and that its Wall obstruction to finiteness is zero.


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

  • [1] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. MR 0115157 (22:5959)
  • [2] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976. MR 0423357 (54:11336)
  • [3] T. tom Dieck, Partitions of unity in homotopy theory, Compositio Math. 23 (1971), 159-167. MR 0293625 (45:2702)
  • [4] D. A. Edwards and R. Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975), 521-535; correction 104 (1976), p. 389. MR 0375330 (51:11525)
  • [5] -, The stability problem in shape and a Whitehead theorem in pro-homotopy, Trans. Amer. Math. Soc. 214 (1975), 261-277. MR 0413095 (54:1216)
  • [6] D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math., vol. 542, Springer-Verlag, Berlin and New York, 1976. MR 0428322 (55:1347)
  • [7] R. Geoghegan, Compacta with the homotopy type of finite complexes, (Proc. Georgia Conf. on Geometric Topology, 1977), Academic Press, New York (to appear). MR 537753 (81a:55018)
  • [8] R. Geoghegan and R. C. Lacher, Compacta with the shape of finite complexes, Fund. Math. 92 (1976), 25-27. MR 0418029 (54:6073)
  • [9] H. M. Hastings, Fibrations of compactly generated spaces, Michigan Math. J. 21 (1974), 243-251. MR 0367985 (51:4227)
  • [10] S. Mardešić and J. Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), 41-59. MR 0298634 (45:7686)
  • [11] M. Mather, Counting homotopy types of manifolds, Topology 4 (1965), 93-94. MR 0176470 (31:742)
  • [12] J. Milnor, On spaces having the homotopy type of CW complexes, Trans. Amer. Math. Soc. 90 (1959), 272-280. MR 0100267 (20:6700)
  • [13] K. Morita, On shapes of topological spaces, Fund. Math. 86 (1975), 251-259. MR 0388385 (52:9222)
  • [14] D. G. Quillen, Homotopical algebra, Lecture Notes in Math., vol. 43, Springer-Verlag, Berlin and New York, 1967. MR 0223432 (36:6480)
  • [15] L. C. Siebenmann, L. Guillou and H. Hähl, Les voisinages réguliers ouverts, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 253-293. MR 0331399 (48:9732)
  • [16] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [17] A. Stróm, Note on cofibrations, Math. Scand. 19 (1966), 11-14. MR 0211403 (35:2284)
  • [18] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. (2) 81 (1965), 55-69. MR 0171284 (30:1515)
  • [19] J. E. West, Mapping Hilbert cube manifolds to ANRs, Ann. of Math. (2) 106 (1977) (to appear). MR 0451247 (56:9534)
  • [20] J. H. C. Whitehead, A certain exact sequence, Ann. of Math. (2) 52 (1950), 51-110. MR 0035997 (12:43c)
  • [21] R. Geoghegan, The inverse limit of homotopy equivalences between towers of fibrations is a homotopy equivalence-a simple proof (submitted).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55D15, 57A65

Retrieve articles in all journals with MSC: 55D15, 57A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0515418-4
Keywords: Homotopy type of a CW complex, shape, Wall obstruction, sequence of fibrations
Article copyright: © Copyright 1978 American Mathematical Society