Thickenings of CW complexes of the form 𝑆^{𝑚}∪_{𝛼}𝑒ⁿ

Cooke, George

doi:10.1090/S0002-9947-1979-0517691-0

Thickenings of CW complexes of the form $S^{m}\cup _{\alpha }e^{n}$
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by George Cooke

Trans. Amer. Math. Soc. 247 (1979), 177-209

DOI: https://doi.org/10.1090/S0002-9947-1979-0517691-0

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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.

References

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 97060, DOI 10.1090/S0002-9947-1958-0097060-7
A. L. Blakers and W. S. Massey, The homotopy groups of a triad. II, Ann. of Math. (2) 55 (1952), 192–201. MR 44836, DOI 10.2307/1969428
A. L. Blakers and W. S. Massey, The homotopy groups of a triad. III, Ann. of Math. (2) 58 (1953), 409–417. MR 58971, DOI 10.2307/1969744
A. L. Blakers and W. S. Massey, Products in homotopy theory, Ann. of Math. (2) 58 (1953), 295–324. MR 60820, DOI 10.2307/1969790
George Cooke, Embedding certain complexes up to homotopy type in euclidean space, Ann. of Math. (2) 90 (1969), 144–156. MR 242152, DOI 10.2307/1970685
Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492
Brayton Gray, On the homotopy groups of mapping cones, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970) Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 104–142. MR 0350728
P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154
P. J. Hilton and E. H. Spanier, On the imbeddability of certain complexes in euclidean spaces, Proc. Amer. Math. Soc. 11 (1960), 523–526. MR 124902, DOI 10.1090/S0002-9939-1960-0124902-3
I. M. James, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374–383. MR 91467, DOI 10.1090/S0002-9939-1957-0091467-4
I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107
Norman Levitt, Normal fibrations for complexes, Illinois J. Math. 14 (1970), 385–408. MR 275442
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258, DOI 10.1515/9781400883875
Hirosi Toda, On iterated suspensions. I, II, J. Math. Kyoto Univ. 5 (1966), 87–142, 209–250. MR 210130, DOI 10.1215/kjm/1250524559

p-primary components of homotopy groups

32

Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR 192509, DOI 10.1016/0040-9383(66)90005-X
J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MR 4123, DOI 10.2307/1968907
N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075, DOI 10.1307/mmj/1028999711
Tudor Ganea, Monomorphisms and relative Whitehead products, Topology 10 (1971), 391–403. MR 283800, DOI 10.1016/0040-9383(71)90030-9
Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, DOI 10.1016/0040-9383(65)90039-X