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Homotopy Pull-Backs and Applications to Duality

Published online by Cambridge University Press:  20 November 2018

Marshall Walker
Marshall Walker*
Affiliation:
York University, Downsview Ontario
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Abstract

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The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 1 , 01 February 1977 , pp. 45 - 64
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Barratt, M. G. and Whitehead, J. H. C., On the second non-vanishing homotopy groups of pairs and triads, Proc. London Math. Soc. (3) 5 (1955), 392–406.Google Scholar
2. Blakers, A. L. and Massey, W. S., The homotopy groups of a triad, i”, Annals of Math. 53 (1951), 161–205.Google Scholar
3. Blakers, A. L. and Massey, W. S. The homotopy groups of a triad, II, Annals of Math. 55 (1952), 192–201.Google Scholar
4. Blakers, A. L. and Massey, W. S. The homotopy groups of a triad, III, Annals of Math. 58 (1953), 409–417.Google Scholar
5. Boardman, J. M. and Vogt, R. M., Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347 (Springer, Berlin-Heidelberg-New York, 1973).Google Scholar
6. Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Math. 304 (Springer, Berlin-Heidelberg-New York, 1972).Google Scholar
7. Fantham, P. H. H., Lectures in homotopy theory (University of Toronto, 1973).Google Scholar
8. Ganea, T., A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322.Google Scholar
9. Hall, I. M., The generalized Whitney Sum, Quart. J. Math. Oxford 16 (1965), 360–384.Google Scholar
10. Hilton, P., Homotopy theory and duality (Gordon and Breach Science Publishers, New York, 1965).Google Scholar
11. Mather, M., Pull-backs in homotopy theory, Can. J. Math. 28 (1976), 225–263.Google Scholar
12. Nomura, Y., On extensions of triads, Nagoya Math. J. 22 (1963), 169–188.Google Scholar
13. Nomura, Y. The Whitney join and its dual, Osaka. J. Math. 7 (1970), 353–373.Google Scholar
14. Sugawara, M., On a condition that a space is an H-space, Math. J. Okayama University 6 (1957), 109–129.Google Scholar
15. Svarc, S., The genus of a fibre space, AMS Translations (2) 55, 49–140.Google Scholar
16. Vogt, R. M., Homotopy limits and colimits, Math. Zeit. 134 (1973), 11–52.Google Scholar