In [7] Varadarajan denned the notion of a cyclic map f : A → X. The collection of all homotopy classes of such cyclic maps forms the Gottlieb subset G(A, X) of [A, X]. If A = S 1 this reduces to the group G(X, X0 ) of Gottlieb [5]. We show that a cyclic map f maps ΩA into the centre of ΩX in the sense of Ganea [4]. If A and X are both suspensions, we then show that if f : A → X maps ΩA into the centre of ΩX, then f is cyclic. Thus for maps from suspensions to suspensions, Varadarajan's cyclic maps are just those maps considered by Ganea. We also define G (Σ4, ΣX) in terms of the generalized Whitehead product [1], This gives the computations for G(Sn+k, Sn ) in terms of Whitehead products in π2n+k(Sn).We work in the category of spaces with base points and having the homotopy type of countable CW-complexes. All maps and homotopies are with respect to base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class.
Hoo, C. S. Cyclic Maps From Suspensions to Suspensions. Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 789-791. doi: 10.4153/CJM-1972-075-1
@article{10_4153_CJM_1972_075_1,
author = {Hoo, C. S.},
title = {Cyclic {Maps} {From} {Suspensions} to {Suspensions}},
journal = {Canadian journal of mathematics},
pages = {789--791},
year = {1972},
volume = {24},
number = {5},
doi = {10.4153/CJM-1972-075-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-075-1/}
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