Characteristic rank of vector bundles over Stiefel manifolds (Q1934245)

Let \(X\) be a connected finite CW-complex and \(\xi\) a real vector bundle over \(X\). The characteristic rank of \(\xi\) over \(X\), denoted by \({charrank}_X (\xi)\), is the largest integer \(k, 0\leq k\leq dim(X)\), 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 \(w_i(\xi)\). The upper characteristic rank of \(X\), denoted by \({upcharrank} (X)\), is the maximum of \({charrank}_X (\xi)\) as \(\xi\) varies over all vector bundles over \(X\). The upper characteristic rank is a homotopy invariant so it is also a topological invariant. When \(X\) is a connected closed smooth manifold and \(TX\) the tangent bundle of \(X\), then \({charrank}_X (TX)\), denoted by \(charrank(X)\), is called the characteristic rank of the manifold \(X\). In previous work the authors computed the characteristic rank of vector bundles over a product of spheres, the real and complex projective spaces, the Dold manifolds \(P(m,n)\), the Moore spaces \(M(\mathbb Z_2, n)\) and stunted projective space \(\mathbb R {\mathbb P}^n / \mathbb R {\mathbb P}^m\). Let \(\mathbb F\) denote either the field \(\mathbb R\) of reals, the field \(\mathbb C\) of complex numbers, or the skew-field \(\mathbb H\) of quaternions. Let \(V_k ({\mathbb F}^n)\) denote the Stiefel manifold of orthonormal \(k\)-frames in \({\mathbb F}^n\). In this paper the authors compute the characteristic rank of vector bundles over \(V_k ({\mathbb F}^n)\). The paper is interesting and it is well written.