ABSTRACT. We define the notion of characteristic rank, charrank X (ξ), of a real vector bundle ξ over a connected finite CW -complex X. This is a bundledependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.
Let X k m,n = Σ k (RP m /RP n ). In this note we completely determine the values of k, m, n for which the total Stiefel-Whitney class w(ξ) = 1 for any vector bundle ξ over X k m,n .2010 Mathematics Subject Classification. 57R20 (55R40, 57R22).
The characteristic rank of a vector bundle $\xi$ over a finite connected
$CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq
\mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$,
$0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$. In
this note we compute the characteristic rank of vector bundles over the Stiefel
manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$
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