Span of vector bundles \(m\xi_n\) over \({RP}^n\) and stable extendibility. III (Q2876000)

The authors study the problem to determine the span of some multiple of the canonical line bundle \(\xi_n\) over the real projective space \(\mathbb RP^n\), where the span of a vector bundle is the maximum number of linear independent cross sections of the bundle. In the main theorem the authors give conditions for \(\text{span}(v+n + r)\xi_n=v+s\), where \(v\) is some non-negative integer, \(r\geq 1\) and \(1 \leq s \leq 9\). The authors also consider the stable extendibility of a vector bundle over \(\mathbb RP^n\) which is stably equivalent to \((n + l)\xi_n\) with \(1\leq l\leq 10\).