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On the Mapping Theorem for Lusternik-Schnirelmann Category II

Published online by Cambridge University Press:  20 November 2018

Yves Félix  and
Jean-Michel Lemaire
Yves Félix
Affiliation:
Université Catholique de Louvain, Louvain, Belgique
Jean-Michel Lemaire
Affiliation:
Université de Nice, Nice, France
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Let X and Y be 1-connected spaces having the homotopy type of cw-complexes.

Definition 0.1. A continuous map f:X → Y is Ω-split if Ωf:ΩX → ΩY admits a retraction up to homotopy.

In [6] we prove the following “mapping theorem”:

THEOREM 0.1. (a) If f is Ω-split, then cat(X) ≦ cat(Y);

(b) If π*(f) is split injective and ΩY has the homotopy type of a product of Eilenberg-MacLane spaces, then f is Ω-split.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1389 - 1398
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Baues, H., Obstruction theory, Springer Lect. Notes Math. 628 (1977).CrossRefGoogle Scholar
2. Boullay, P., Kiefer, F., Majewski, M., Stelzer, M., Scheerer, H., Unsold, M. and Vogt, E., Tame homotopy theory via differential forms, Freie U. Berlin preprint 223 (1986).Google Scholar
3. Dwyer, W., Tame homotopy theory, Topology 18 (1979), 321–338.Google Scholar
4. Eilenberg, S. and Ganéa, T., On the Lusternik-Schnirelmann category of abstract groups, Ann. Math. 65 (1957), 517–518.Google Scholar
5. Félix, Y. and Halperin, S., Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1–37.Google Scholar
6. Félix, Y. and Lemaire, J.-M., On the mapping theorem for Lusternik-Schnirelmann category, Topology 24(1985), 41–43.Google Scholar
7. Henn, H. W., On almost rational co-H-spaces, Proc. Amer. Math. Soc. 87 (1983), 164–168.Google Scholar
8. Lemaire, J.-M., Lusternik-Schnirelmann category: an introduction, in Algebra, algebraictopology and their interactions, Springer Lect. Notes Math. 1183 (1986), 259–276.Google Scholar
9. Scheerer, H., One more facet of a mapping theorem for Lusternik-Schnirelmann category, Bonn preprint (1985).Google Scholar
10. Strøm, A., Note on cofibrations II, Math. Scand. 22 (1968), 130–142.Google Scholar