A Borsuk-Ulam type theorem for sphere bundles over stunted projective spaces
A \(CW\)-complex \(B\) is 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 the author studies whether or not the stunted projective space \(FP_m^{n}= \frac{FP^n}{FP^{m-1}}\), \(1\leq m \leq n\) is I-trivial. So, \(FP_{m}^{n} \) is not I-trivial in the case \(m=1\) while it is I-trivial in the case where \(m=n\) and, for \(d=\dim_{\mathbb R}F\), \(dn \neq 1,2,4,8\) . A space \(B\) is \(W\)-trivial if \(W(\alpha)=1\) holds for every vector bundle \(\alpha\) over \(B\) (where \(W(\alpha)\) denotes the total Stiefel-Whitney class of \(\alpha\)).
scientific article; zbMATH DE number 5539517