On maximality of the cup-length of flag manifolds

On real flag manifolds with cup-length equal to its dimension

“…. + a k has digit 1 in position l. By [6], Lemma 2.3, this implies that ai+ai+1+...+a k ai is odd (for all i ∈ [k − 1]), which completes our proof. Lemma 3.4.…”

“…is odd. Now, the result follows from [6], Lemma 2.3. (⇐) Note that the given condition implies that for every i ∈ [k] and every l ∈ N 0 at most one of the numbers a i and a i+1 + .…”

“…In [5] a family of flag manifolds of the form F (1 ...j , 2 ...d , n) with maximal cup-length was found. This family was extended in [6], where, additionally, a necessary and sufficient condition for cup(F (1 ...j , 2 ...d , n)) = dim(F (1 ...j , 2 ...d , n)) in cases d = 1 and d = 2 were obtained. Up to now, no infinite family of flag manifolds with maximal cuplength and at least two steps greater than 2 was known.…”