[@GK]. Bott manifolds are known to have algebraic torus actions, hence they constitute an important family of toric manifolds. A *cohomology Bott manifold* is a quasitoric manifold whose cohomology ring is isomorphic to that of a Bott manifold.

The question we are interested in here is whether the class of (cohomology) Bott manifolds is cohomologically rigid. So far, there is no counter example to the question, but some positive results. Masuda and Panov considered the problem and showed that any $n$-stage Bott manifold is diffeomorphic to the trivial Bott manifold $(\mathbb CP^1)^n$ if its cohomology ring is isomorphic to that of $(\mathbb CP^1)^n$.

The notion of Bott tower is generalized to a [*generalized Bott tower*]{} in [@ch-ma-su08] which is an iterated complex projective space bundles obtained from projectivization of sum of line bundles over a complex projective space, and the result in [@MP] is extended to generalized Bott manifolds in [@CMS]. Furthermore any three-stage Bott manifolds and $2$-stage generalized Bott manifolds are shown to be cohomologically rigid there.

Davis and Januszkiewicz also introduced a real analogue of a quasitoric manifold called a [*small cover*]{} in [@DJ]. But for small covers the corresponding cohomologies are with $\mathbb Z_2$-coefficients. Moreover we can 