define a [*real Bott tower*]{} to be an iterated $\mathbb RP^1$ bundles over $\mathbb RP^1$, and a [*generalized real Bott tower*]{} is defined similarly. So one might ask a similar cohomological rigidity question asking whether two real Bott manifolds are homeomorphic if their mod $2$ cohomology rings are isomorphic. This is shown to be true recently by Kamishima and Masuda [@KM], [@Ma1]. However the same question for generalized real Bott manifolds is not true, see [@Ma2].

However not much is known for the cohomological rigidity of Bott manifolds whose cohomology rings are not isomorphic to that of product of $\mathbb CP^1$. In this article we consider one-twist Bott manifolds, i.e., only one stage has nontrivial fibration in its Bott tower structure. We prove in Theorem \[theorem:rigidity of one-twist Bott tower\] that one-twist Bott towers are cohomologically rigid. Moreover this result is extended to quasitoric manifolds whose $\mathbb Z$-cohomology rings are isomorphic to those of one-twist Bott towers in Theorem \[theorem:final theorem\]. Theorem \[theorem:final theorem\] is an immediate consequence of Theorem \[theorem:rigidity of one-twist Bott tower\] together with two properties related with Bott towers. They are Theorem \[theorem:twist number = complexity\] and Theorem \[theorem:BQ-algebra and Bott tower\].

A Bott manifold $M$ 