the cohomological complexity of $H^\ast(M)$. In particular the twist number of a Bott manifold is well-defined, namely, it does not depend on the choice of Bott manifold structures of a Bott manifold.

A $BQ$-algebra of rank $n$ is defined in [@MP]. In particular, the cohomology ring of any $n$-stage Bott manifold is a $BQ$-algebra of rank $n$ over $\mathbb Z$. The converse of this is proved in Theorem \[theorem:BQ-algebra and Bott tower\].

It is proved in [@CMS] that the class of three-stage Bott manifolds are cohomologically rigid. An immediate consequence of this result together with Theorem \[theorem:BQ-algebra and Bott tower\] is Theorem \[theorem:3 stage Bott manifold\], which says that the class of $6$-dimensional quasitoric manifolds whose cohomolgies are BQ-algebras over $\mathbb Z$ is cohomologically rigid.

So far, all the cohomological results are over ${\mathbb{Z}}$ coefficients. But by careful observation of the proofs we can see that the same conclusion can be derived with the $2$-localized ${\mathbb{Z}}_{(2)}$-coefficients. This is treated in Section \[sectoin:BQ-algebra over Z\_2\].

A sum of two line bundles over Bott manifolds
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Let $\{B_{j}=P( \underline{{\mathbb{C}}} \oplus \xi_{j-1} ) \mid 0 \leq j \leq n \}$ be a complex Bott tower of height $n$. By the standard results on 