A note on the characteristic rank of oriented Grassmann manifolds

We derive new lower bound for the characteristic rank of the canonical 3-plane bundle γ˜n,3 over the oriented Grassmann manifold G˜n,3≅SO(n)/(SO(3)×SO(n−3)) for all n in range 2t+2t−1≤n≤2t+1−4, t≥3. Using this knowledge we determine the Z2-cup-length of G˜2t+2t−1+1,3 and G˜2t+2t−1+2,3, verifying the corresponding claim of T. Fukaya's conjecture.