On the cup-length of certain classes of flag manifolds (Q2338590)

The \(\mathbb{Z}_2\)-cup-length of a topological space \(X\) is the largest number of elements in the cohomology ring of \(X\), with coefficients in \(\mathbb{Z}_2\), whose product is non-trivial (this gives a lower bound for the Ljusternik-Schnirelmann category of \(X\)). The elements of the flag manifold \(F(n_1, \ldots, n_q)\), for positive integers \(n_i\) whose sum is \(n\), are the sets of all flags of type \((n_1, \ldots, n_q)\) in \(\mathbb{R}^n\), i.e. of all sets of \(q\) mutually orthogonal subspaces of \(\mathbb{R}^n\) of dimensions \((n_1, \ldots, n_q)\) (which can be identified with the homogeneous space \(O(n)/O(n_1) \times \cdots \times O(n_q)\)); a special case are the Grassmann manifolds \(F(k,n)\). \par In the present paper, the cup-length of the flag manifolds \(F(2,2,n_3)\) and \(F(1,3,2^{s+1}-3)\) is computed; similar computations for various flag manifolds are contained in papers by \textit{J. Korbaš} and \textit{J. Lörinc} [Fundam. Math. 178, No. 2, 143--158 (2003; Zbl 1052.55006)] and by \textit{Z. Z. Petrović} et al. [Acta Math. Hung. 149, No. 2, 448--461 (2016; Zbl 1389.57006)]. In the second part of the present paper the authors compute the height (the largest nontrivial power) of the third Stiefel-Whitney characteristic class of the canonical vector bundle over the Grassmann manifold \(F(4,n)\); analogous results for the first and second Stiefel-Whitney classes are contained in papers by \textit{R. E. Stong} [Topology Appl. 13, 103--113 (1982; Zbl 0469.55005)] and by \textit{S. Dutta} and \textit{S. S. Khare} [J. Indian Math. Soc., New Ser. 69, No. 1--4, 237--251 (2002; Zbl 1104.57301)].