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Transactions of the American Mathematical Society

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Thickenings of CW complexes of the form $ S\sp{m}\cup \sb{\alpha }e\sp{n}$


Author: George Cooke
Journal: Trans. Amer. Math. Soc. 247 (1979), 177-209
MSC: Primary 55P99; Secondary 57P10, 57Q35
DOI: https://doi.org/10.1090/S0002-9947-1979-0517691-0
MathSciNet review: 517691
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Abstract: Necessary conditions are given for the existence of a thickening of $ {S^m}\,{ \cup _\alpha }\,{e^n}$ in codimension k. I give examples of such complexes requiring arbitrarily large codimension in order to thicken. Sufficient conditions are given for the existence of a tractable thickening in codimension $ k\, + \,1$. The methods used include the study of the reduced product space of a pair of CW complexes.


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  • [1] W. D. Barcus and M. G. Barratt, On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc. 88 (1958), 57-74. MR 0097060 (20:3540)
  • [2] A. L. Blakers and W. S. Massey, Homotopy groups of a triad. II, Ann. of Math. (2) 55 (1952), 192-201. MR 0044836 (13:485f)
  • [3] -, Homotopy groups of a triad. III, Ann. of Math. (2) 58 (1953), 409-417. MR 0058971 (15:458b)
  • [4] -, Products in homotopy theory, Ann. of Math. (2) 58 (1953), 295-324. MR 0060820 (15:731f)
  • [5] George Cooke, Embedding certain complexes up to homotopy type in Euclidean space, Ann. of Math. (2) 90 (1969), 144-156. MR 0242152 (39:3486)
  • [6] S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N. J., 1952. MR 0050886 (14:398b)
  • [7] Brayton Gray, On the homotopy groups of mapping cones, Proc. Advanced Study Inst. Algebraic Topics (1970), Aarhus, p. 104. MR 0350728 (50:3220)
  • [8] P. J. Hilton, Homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154-172. MR 0068218 (16:847d)
  • [9] P. J. Hilton and E. H. Spanier, On the embeddability of certain complexes in Euclidean spaces, Proc. Amer. Math. Soc. 11 (1960), 523-526. MR 0124902 (23:A2211)
  • [10] I. M. James, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374-383. MR 0091467 (19:974a)
  • [11] -, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197. MR 0073181 (17:396b)
  • [12] Norman Levitt, Normal fibrations for complexes, Illinois J. Math. 14 (1970), 385-408. MR 0275442 (43:1198)
  • [13] N. Steenrod, Topology of fibre bundles, Princeton, Univ. Press, Princeton, N. J., 1951. MR 0039258 (12:522b)
  • [14] H. Toda, On iterated suspensions. I, J. Math. Kyoto Univ. 5 (1965). MR 0210130 (35:1024)
  • [15] -, p-primary components of homotopy groups. IV, Mem. Coll. Sci. Univ. Kyoto Ser. A 32 (1959).
  • [16] -, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, Princeton Univ. Press, Princeton, N. J., 1962. MR 0143217 (26:777)
  • [17] C. T. C. Wall, Classification problems in differential topology. IV, Thickenings, Topology 5 (1966), 73-94. MR 0192509 (33:734)
  • [18] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. 42 (1941), 409-428. MR 0004123 (2:323c)
  • [19] N. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 0210075 (35:970)
  • [20] Tudor Ganea, Monomorphisms and relative Whitehead products, Topology 10 (1971), 391-403. MR 0283800 (44:1030)
  • [21] Gerald J. Porter, Higher order Whitehead products, Topology 3 (1965), 123-135. MR 0174054 (30:4261)

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DOI: https://doi.org/10.1090/S0002-9947-1979-0517691-0
Article copyright: © Copyright 1979 American Mathematical Society

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