A Borsuk–Ulam type theorem for sphere bundles over stunted projective spaces

A CW complex B is described 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 . For n > 1 , we determine all positive integers m for which the stunted projective space FP m n is I-trivial, where F = R , C or H .