A CW complex B is said to be I-trivial if there does not exist a

-map from

to S(

) for any vector bundle

over B a any integer i with i > dim

. In this paper, we consider the question of determining whether

is I-trivial or not, and to this question we give complete answers when k

1, 3, 8 and partial answers when k = 1, 3, 8. A CW complex B is I-trivial if it is "W-trivial", that is, if for every vector bundle over B, all the Stiefel-Whitney classes vanish. We find, as a result, that

is a counterexample to the converse of th statement when k = 2, 4 or 8 and n

2k.