A fiberwise analogue of the Borsuk-Ulam theorem for sphere bundles over a 2-cell complex II

We describe a finite complex B as I-trivial if there does not exist a Z(2)-map from Si-1 to S(alpha) for any vector bundle alpha over B and any integer i with i > dim alpha. We prove that the in-fold suspension of projective plane F P-2 is I-trivial if and only if m not equal 0, 2, 4 for F = C, m not equal 0.4 for F = H. In the case where F is the Cayley algebra, the m-fold suspension is shown to be I-trivial for every in > 0. (c) 2007 Elsevier B.V. All rights reserved.