Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-07-01T07:30:32.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Two Applications of Homology Decompositions

Published online by Cambridge University Press:  20 November 2018

Graham Hilton Toomer
Graham Hilton Toomer*
Affiliation:
Cornell University, Ithaca, New York; Ohio State University, Columbus, Ohio
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that a map of rational spaces (see Definition 1) induces a map of homology sections at each stage, and that the k'-invariants are mapped naturally. This is used to characterize rational spaces in which all (matric) Massey products vanish as wedges of rational spheres, and yields the precise Eckmann-Hilton dual of a result of M. Dyer [7]. Berstein's result on co-H spaces [3] is also deduced. These results form a part of the author's doctoral dissertation at Cornell University written under Professor I. Berstein, to whom I express my sincere thanks for his patient help and encouragement. Extensions and counterexamples will appear in a future paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 323 - 329
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Adams, J. F. and Hilton, P. J., On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305–330.Google Scholar
2. Allday, C., Rational Whitehead products and a spectral sequence of Quillen, Pacific J. Math. 46 (1973), 313–323.Google Scholar
3. Berstein, I., Homotopy mod C of spaces of category 2, Comment. Math. Helv. 35 (1961), 9–14.Google Scholar
4. Berstein, I. and Hilton, P. J., Category and generalized Hopf invariants, Illinois J. Math. 4 (1960), 437–451.Google Scholar
5. Brown, E. H. and Copeland, A. H., An homology analogue of Postnikov systems, Michigan Math. J. 6 (1959), 313–330.Google Scholar
6. Curjel, C. R., A note on spaces of category 2, Math. Z. 80 (1963), 293–299.Google Scholar
7. Dyer, M., Rational homotopy and Whitehead products, Pacific J. Math. 40 (1972), 59–71.Google Scholar
8. Hilton, P. J., Homotopy theory and duality (Gordon and Breach, New York, 1965).Google Scholar
9. Kahn, D. W., Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963), 432–450.Google Scholar
10. Massey, W. and Uehara, H., The Jacobi identity for Whitehead products, Algebraic Geometry and Topology Symposium in Honour of S. Lefshetz (Princeton U. Press, Princeton, 1957).Google Scholar
11. May, J. P. and Guggenheim, V. K., On the theory and applications of differential torsion products (to appear).Google Scholar
12. Milnor, J., On spaces having the homotopy type of a CW complex, Trans. Amer. Math. Soc. 90 (1959), 272–280.Google Scholar
13. Porter, G. J., Higher products, Trans. Amer. Math. Soc. 148 (1970), 315–345.Google Scholar
14. Spanier, E. H., Algebraic topology (McGraw Hill, New York, 1966).Google Scholar
15. Sternstein, M., Necessary and sufficient conditions for homotopy classifications by cohomology and homotopy homomorphisms, Proc. Amer. Math. Soc. 34 (1972), 250–256.Google Scholar
16. Sullivan, D., Geometric topology, M.I.T. mimeographed notes, June, 1970.Google Scholar
17. Toomer, G. H., On Liusternik-Schnirelmann category and the Moore spectral sequence, Ph.D. thesis, Cornell University, 1971.Google Scholar
18. Toomer, G. H., On the kernel of the rationalized Freudenthal suspension homomorphism (to appear in J. Pure Appl. Algebra).Google Scholar