Note on the Characteristic rank of Vector bundles

We define the notion of characteristic rank, ${\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}}_X(\xi )$, of a real vector bundle $\xi$ over a connected finite $CW$-complex $X$. This is a bundle-dependent version of the notion of characteristic rank introduced by J'{u}lius Korba\v{s} in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korba\v{s} and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.