Fibered stable compacta have finite homotopy type
Author:
Ross Geoghegan
Journal:
Proc. Amer. Math. Soc. 71 (1978), 123129
MSC:
Primary 55D15; Secondary 57A65
Erratum:
Proc. Amer. Math. Soc. 74 (1979), 391.
MathSciNet review:
0515418
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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.
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R. Geoghegan, The inverse limit of homotopy equivalences between towers of fibrations is a homotopy equivalencea simple proof (submitted).
 [1]
 M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478483. 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), 159167. 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), 521535; correction 104 (1976), p. 389. MR 0375330 (51:11525)
 [5]
 , The stability problem in shape and a Whitehead theorem in prohomotopy, Trans. Amer. Math. Soc. 214 (1975), 261277. 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, SpringerVerlag, 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), 2527. MR 0418029 (54:6073)
 [9]
 H. M. Hastings, Fibrations of compactly generated spaces, Michigan Math. J. 21 (1974), 243251. MR 0367985 (51:4227)
 [10]
 S. Mardešić and J. Segal, Shapes of compacta and ANRsystems, Fund. Math. 72 (1971), 4159. MR 0298634 (45:7686)
 [11]
 M. Mather, Counting homotopy types of manifolds, Topology 4 (1965), 9394. MR 0176470 (31:742)
 [12]
 J. Milnor, On spaces having the homotopy type of CW complexes, Trans. Amer. Math. Soc. 90 (1959), 272280. MR 0100267 (20:6700)
 [13]
 K. Morita, On shapes of topological spaces, Fund. Math. 86 (1975), 251259. MR 0388385 (52:9222)
 [14]
 D. G. Quillen, Homotopical algebra, Lecture Notes in Math., vol. 43, SpringerVerlag, 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), 253293. MR 0331399 (48:9732)
 [16]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
 [17]
 A. Stróm, Note on cofibrations, Math. Scand. 19 (1966), 1114. MR 0211403 (35:2284)
 [18]
 C. T. C. Wall, Finiteness conditions for CWcomplexes, Ann. of Math. (2) 81 (1965), 5569. 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), 51110. MR 0035997 (12:43c)
 [21]
 R. Geoghegan, The inverse limit of homotopy equivalences between towers of fibrations is a homotopy equivalencea simple proof (submitted).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197805154184
PII:
S 00029939(1978)05154184
Keywords:
Homotopy type of a CW complex,
shape,
Wall obstruction,
sequence of fibrations
Article copyright:
© Copyright 1978 American Mathematical Society
