For n=2m+1-4, m ≥ 2, we determine the cup-length of H*(Gn,3; Z/2) by finding a Gröbner basis associated with a certain subring, where Gn,3 is the oriented Grassmann manifold SO(n+3)/SO(n) × SO(3). As an application, we provide not only a lower but also an upper bound for the LS-category of Gn,3. We also study the immersion problem of Gn,3.
Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.195-209.