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 S i − 1 to S ( α ) for any vector bundle α over B and any integer i with i > dim α . We prove that the m-fold suspension of projective plane F P 2 is I-trivial if and only if m ≠ 0 , 2 , 4 for F = C , m ≠ 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 m > 0 .