Vol. 68, No. 1, 1977

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Composition properties of projective homotopy classes

Samuel S. Feder, Samuel Carlos Gitler and K. Y. Lam

Vol. 68 (1977), No. 1, 47–61

1. Introduction. A homotopy class x πq(X) is said to be projective on X, or projectively carried by X, if it can be represented by a map that factors through the projective space pq, as shown in diagram (I), where π is the double covering map.

         Sq  −x→   X              Sq  −x→   Sm
(I)    ˆπ  ↘        ↗    (II)   π  ↘         ↗     (t ≦ m).
P q                      Pqt

When x is a stable homotopy class of spheres, it is of interest to ask for the values of m such that x be projective on Sm. Since Sm is (t 1)-connected for t m, this amounts to the factorisation problem posed in diagram (II) above, where π is π followed by the collapsing map from Pq to the truncated projective space Ptq = Pq∕Pt1. We give an answer to this problem when x is a generator of the image of the J-homomorphism.

Mathematical Subject Classification
Primary: 55E45, 55E45
Received: 1 October 1975
Published: 1 January 1977
Samuel S. Feder
Samuel Carlos Gitler
K. Y. Lam