### On the $\mathbb Z_2$-cohomology cup-length of some real flag manifolds

#### Abstract

In this paper we discuss two different techniques for $\mathbb Z_2$-cohomology cup-length calculation -- one based on the fiberings and a result of Horanska and Korba\v s, and the other based on Gr\"{o}bner bases. We use these techniques to obtain $\mathbb Z_2$-cohomology cup-length and some bounds for the $\mathbb Z_2$-cohomology cup-length of some real flag manifolds of type $F(1,\ldots,1,2\ldots,2,n)$.

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