A CW complex B is said to be I-trivial if there does not exist a $mathbb{Z}_2$-map from $S^{i-1}$ to S(${alpha}$) for any vector bundle ${alpha}$ over B a any integer i with i > dim ${alpha}$. In this paper, we consider the question of determining whether $Sigma^kmathbb{R}P^n$ is I-trivial or not, and to this question we give complete answers when k $ eq$ 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 $Sigma^kmathbb{R}P^n$ is a counterexample to the converse of th statement when k = 2, 4 or 8 and n $geq$ 2k.