Note on the characteristic rank of vector bundles

Abstract

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.

On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes

“…Then the Z 2 -cup-length cup( G n,3 ) is bounded above by 3(n−3) In another direction we have studied characteristic rank of G n,k (k ≥ 5) and upper characteristic rank of G n, 3 . If X is a connected finite CW complex and ξ is a real (finite rank) vector bundle over X, recall from [10] that the characteristic rank of ξ over X, denoted by charrank X (ξ), is by definition the largest integer k ≤ dim(X) such that every cohomology class x ∈ H j (X; Z 2 ), 0 ≤ j ≤ k, is a polynomial in the Stiefel-Whitney classes w i (ξ). The upper characteristic rank of X, denoted by ucharrank(X), is the maximum of charrank X (ξ) as ξ varies over vector bundles over X.…”

Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

“…The characteristic rank of a manifold was introduced by Korbaš [4] and later generalized by Naolekar and Thakur [9] to the characteristic rank of a vector bundle. Definition 1.1.…”

Characteristic rank of vector bundles over Stiefel manifolds

“…Let X be a connected finite CW -complex and ξ a real vector bundle over X. Recall [5] that the characteristic rank of ξ over X, denoted by charrank X (ξ), is by definition the largest integer k, 0 ≤ k ≤ dim(X), such that every cohomology class x ∈ H j (X; Z 2 ), 0 ≤ j ≤ k, is a polynomial in the Stiefel-Whitney classes w i (ξ). The upper characteristic rank of X, denoted by ucharrank(X), is the maximum of charrank X (ξ) as ξ varies over all vector bundles over X.…”