Composition properties of projective homotopy classes

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

Vol. 68 (1977), No. 1, 47–61

DOI: 10.2140/pjm.1977.68.47

Abstract

1. Introduction. A homotopy class x ∈ π_q(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 p^q, as shown in diagram (I), where π is the double covering map.

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 S^m. Since S^m 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 P^q to the truncated projective space P_t^q = P^q∕P^t−1. 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

Milestones

Received: 1 October 1975

Published: 1 January 1977