Characteristic rank of vector bundles over Stiefel manifolds

Abstract

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 $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$

On Stiefel-Whitney classes of vector bundles over real Stiefel Manifolds

“…We have complete answer when 2 ϕ(n−k−1) = n − k. We observed in the above proof that 2 ϕ(n−k−1) = n − k if and only if n − k = 1, 2, 4, 8. In the case when n − k = 1, 2, the existence of a vector bundle ξ such that [6]). In the following example, when n − k = 4, 8 we construct a vector bundle…”

“…In [6], it was observed that for a vector bundle ξ over V k (R n ), n > k, the Stiefel-Whitney class w n−k (ξ) = 0 if n − k = 1, 2, 4, 8 and w n−k+1 (ξ) = 0 if n − k = 2, 4, 8. We extend this observation to get the following theorem where we show that there are at most two integers up to 2(n − k), which can occur as the degrees of nonzero Stiefel-Whitney classes of any vector bundle over V k (R n ).…”

“…Recall ( [10]) that ucharrank(X) of X is the maximal degree up to which every cohomology class of X is a polynomial in the Stiefel-Whitney classes of a vector bundle over X. The ucharrank of V k (R n ) was computed in [6], except for the cases n − k = 4, 8, in which cases it was shown that ucharrank(V k (R n )) is bounded above by n − k. In Example 2.3, we construct a vector bundle ξ over V k (R n ), when n − k = 4, 8, such that w n−k (ξ) = 0 and hence improve the result in [6] to obtain ucharrank(V k (R n )) = n − k.…”

A note on the characteristic rank and related numbers

“…Of course, if T M denotes the tangent bundle of M , then we have charrank(T M ) = charrank(M ). Results on the characteristic rank of vector bundles over the Stiefel manifolds can be found in [4]. In addition to being an interesting question in its own right, there are other reasons for investigating the characteristic rank; one of them is its close relation to the cup-length of a given space ( [3], [6]).…”