Journal of the
Korean Mathematical Society JKMS

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 $B$ and any integer $i$ with $i>\dim{\alpha}$. In this paper, we consider the question of determining whether ${\Sigma}^k{\mathbb R}P^n$ is I-trivial or not, and to this question we give complete answers when $k\ne 1,3,8$ 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 ${\Sigma}^k{\mathbb R}P^n$ is a counterexample to the converse of this statement when $k=2,4$ or $8$ and $n\geq 2k$.

Keywords: sphere bundle, ${\mathbb Z}_2$-map, index

MSC numbers: Primary 55P91; Secondary 55R25