Vol. 68, No. 1, 1977

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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